Paper Title: Things we can learn by considering random locally symmetric manifolds.



Bimodal Scaling Law and Size Effect In Superelastic Nanopillars
    Xian Chen
    August 5, 2024
===============================================================



§ ABSTRACT 
In recent years various results about locally symmetric manifolds were proven using probabilistic approaches. One of the approaches is to consider random manifolds by associating a probability measure to the space of discrete subgroups of the isometry Lie group. 
The main goals are to prove results about deterministic groups and manifolds by considering appropriate measures. 
In this overview paper we describe several such results, observing the evolution process of the measures involved. Starting with a result whose proof considered finitely supported measures (more precisely, measures supported on finitely many conjugacy classes) and proceeding with results which were outcome of the successful and popular theory of IRS (invariant random subgroups). In the last couple of years the theory has expanded to SRS (stationary random subgroups) allowing to deal with a lot more problems and establish stronger results. In the last section we shall review a very recent (yet unpublished) result whose proof make use of random subgroups which are not even stationary. 






  
  



§ INTRODUCTION


The study of discrete subgroups of Lie groups has played a central role in many parts of mathematics since the middle of the previous century, and has been continuously extremely active. In recent years the theory has been boosted by the development of several probabilistic approaches to study these deterministic highly structured objects. In this overview paper I will focus on one of these approaches, namely the consideration of random discrete subgroups. Particularly, I will try to describe my personal journey in the development of the theory.

By a random discrete subgroup we mean a Borel regular probability measure m on the (metrizable, locally compact) space _d(G) of discrete subgroups of G. Considering an adjoint semisimple Lie group without compact factors G and letting X=G/K be the associated Riemannian symmetric space, any discrete subgroup ≤ G corresponds to a locally symmetric orbifold M=\ X (which is a manifold if Λ is torsion free). Thus a random discrete subgroup m corresponds to a random X-orbifold.

Below we shall describe various theorems about deterministic discrete subgroups and locally symmetric manifolds which are proven by considerations of random subgroups. We wish to demonstrate the development of the theory by showing that the measures m with which we have been working evolved over the years and become more general and sophisticated.  This evolution was required by the type of the results the problems that were considered. 

We start with the result from <cit.>. In that paper A. Levit and the author showed that, counted appropriately, most hyperbolic manifolds are non-arithmetic. More precisely, fixing any dimension d (suppose d≥ 4) the number of non-arithmetic complete hyperbolic d-manifolds of volume at most v considered up to commensurability is super-exponential as a function of v, while the number of arithmetic ones is sub-exponential (see <ref> for more precise statements). The proof relies on a model for constructing random hyperbolic manifolds. However, the probability spaces considered in this work are finite, namely we analyze a model for constructing random graphs of manifolds which results on finitely many possibilities corresponding to graphs of bounded size. Therefore, one can argue that this is not a probabilistic proof but a combinatorial one. 
The corresponding measures m on _d(G) are not finitely supported but rather supported on finitely many conjugacy classes. This is a very primitive form of the general notion of invariant random subgroups (IRS).

An IRS on G is a probability measure on (G) which is invariant under the G-action by conjugation. I believe it is fare to say that the theory of IRS has proven to be much more powerful than the expectation. In particular IRS in semisimple Lie groups turned out to be an extremely significant tool to the study of lattices and their asymptotic invariants. In <ref> we shall review some of the basics of IRS in semisimple Lie groups. We shall then demonstrate the power of IRS by presenting two theorems about lattices. 
The first is the celebrated theorem of Kazhdan and Margulis establishing the lower bound on the co-volume of lattices. In <ref> we will present a surprisingly short proof of this classical theorem using a direct analysis of the space IRS(G) of invariant random subgroups of G. The novelty here is not in the result (although the IRS version is not covered by the original proof of Kazhdan and Margulis) but rather in the simplicity and the straightforwardness of the argument.  

One of the highlights of the theory of IRS is the Stuck Zimmer rigidity theorem:


Let G be a simple Lie group of rank at least 2. Let μ be an IRS in G without atoms. Then a μ random subgroup is almost surely a lattice in G.


The theorem also has a variant for semisimple groups but then the requirement is that G or at least one of its factors has Kazhdan's property (T). The Stuck–Zimmer theorem is a generalization of the normal subgroup theorem of Margulis and as such it is expected that it should hold also for higher rank semisimple groups without property (T), e.g. for G=_2()×_2(). This is the famous Stuck–Zimmer conjecture. In <ref> we briefly discuss the Stuck–Zimmer theorem and the conjecture. In <ref> we shall recall the failure of the IRS rigidity in rank one groups by sketching a simple example from <cit.>.



In <ref> we review the Bengamini–Schramm space ℬ𝒮(X) associated to a symmetric space X=G/K and its relation with IRS in G. The interplay between the geometric point of view naturally associated with ℬ𝒮(X) and the analytic point of view associated with IRS(G) is crucial in the proof of the result of the 7 samurai <cit.>. In <ref> we describe the result of <cit.> concerning the asymptotic geometry of higher rank manifold of finite volume when the volume tends to infinity.
The result is that the injectivity radius cannot be bounded when the volume tends to infinity, and in fact that it grows to infinity at `almost all points'. More precisely, if rank(G)≥ 2 then ∀ R,>0 there is V such that if M=\ X has volume ≥ V then (M_≥ R)/(M)≥ 1-, where M_≥ R is the R-thick part of M where the injectivity radius is ≥ R. This geometric result has plenty of applications in the theory of analytic invariants. The most famous application is the convergence of normalized Betti numbers (see <cit.>), but there are plenty of more refined applications (see <cit.>). In <ref> 
we will give the proof of the result concerning the Betti numbers in a special case.

While the notion of invariant random subgroups is quite powerful, the same properties that make it so remarkable are also the ones that demonstrate it limitations. For instance, in view of the Stuck–Zimmer rigidity theorem every non-atomic IRS in G=_3() is supported on lattices. Therefore IRS are useless in the study of thin subgroups or general discrete subgroups of infinite covolume in _3(). More generally, as semisimple Lie groups are very far from being amenable, focusing on invariant measures is a huge compromise and restricts by much the scope of problems that can be investigated. For this reason the theory of random subgroups has been expended in the recent years to non-invariant measures.

Let now μ be a probability measure on G. A probability measure ν on (G) is called μ-stationary random subgroup (or μ-SRS) if ν=μ*ν, where as always G acts on (G) by conjugation. 
The advantage of SRS on IRS is that while it is a much more general notion, a great deal of the theory can be extended to this context. 
Moreover, unlike IRS which are limited to special subgroups, any closed subgroup Λ is related to some SRS. Indeed, one can consider the μ random walk in (G) starting at Λ and hope that some stationary limits retain interesting properties of Λ. More precisely, consider 
the Cesaro averages

    ν_n=1/n∑_i=1^nμ^(i)*_Λ,

and let ν_∞ be a weak-* limit. Hopefully by analyzing ν_∞ one can learn about the initial point Λ. In order to process this philosophy, one should chose the probability measure μ carefully.
In <cit.> A. Levit, G.A. Margulis and myself have worked out a specific measure μ=μ_G which posses some wonderful properties. In particular, as shown in <cit.> for every discrete group Λ, an SRS ν_∞ defined as a weak-* limit of Cesaro averages as above is supported on discrete groups. 

In <cit.>, M. Fraczyk and I proved a stiffness result for SRS supported on discrete subgroups. This allowed us to prove the following conjecture of Margulis. Recall that a subgroup ≤ G is called confined if there is a compact set C⊂ G such that 

    C∩Λ^g∖{1}∅
 
for every g∈ G.


Let G be a simple Lie group of rank at least 2. A discrete subgroup Λ≤ G is confined iff it is a lattice.


Note that this result is new even for subgroups of lattices (e.g. subgroups of (3,)) in which case it establishes a strong version of Margulis' normal subgroup theorem.
 
 Recall that X=G/K denotes the associated symmetric space. 
 Theorem <ref> is equivalent to the following: an X-orbifold M=\ X of infinite volume
admits injected contractible balls of any radius. In other words, bounded injectivity radius implies finite (and in view of <cit.> bounded) volume. 

Section <ref> is dedicated to an overview of SRS in semisimple Lie groups, and  in <ref> we give the proof of Theorem <ref>.

In the final section <ref>  I will report on an ongoing joint project with Arie Levit and Uri Bader where we proved in particular the following theorem:

Let G be a semisimple Lie group without compact factors and suppose rank(G)≥ 2. Let Γ≤ G be an irreducible lattice and Λ≤Γ a confined subgroup. Then Λ is of finite index in Γ.


The novelty of this theorem is that it holds regardless of property (T). This is a generalization of the normal subgroup theorem that holds in all classical cases. In order to prove this theorem we had to establish a new spectral gap for products. The proof relies on the study of certain random subgroups which are not even stationary. 






§ MOST HYPERBOLIC MANIFOLDS ARE NON-ARITHMETIC


In this section revisits my joint work with Arie Levit <cit.>.


A celebrated result of Margulis <cit.> asserts that finite volume locally symmetric manifolds of rank >1 are arithmetic.
Corlette <cit.> and Gromov–Schoen <cit.> extended this result to rank one spaces, with the two exceptions of real and complex hyperbolic.

A remarkable paper of Gromov and Piatetski–Shapiro <cit.> establishes the existence of a non-arithmetic (real) hyperbolic manifold of finite volume, in any given dimension.

We prove that in fact almost all hyperbolic manifolds are non-arithmetic, with respect to a certain way of counting. Recall that two manifolds are commensurable if they share a common finite cover. 
Fixing the dimension d>3, and counting up to commensurability, we show that the number of non-arithmetic hyperbolic d-manifolds of volume bounded by V is super-exponential in V, 
while the number of arithmetic ones tends to be polynomial.




















 §.§ Arithmetic Groups


Let us start with some examples:




  * SL(n,) is a non-uniform lattice in SL(n,).

  * Let Q(x,y,z,w)=x^2+y^2+z^2-√(2)w^2, let 𝔾=𝕊𝕆(Q) and let Γ=𝔾([√(2)]). Then Γ=ℍ() for some algebraic group ℍ satisfying 

    ℍ()≅SO(3,1)× SO(4).
 
(The factors correspond to the Galois conjugates of Q.)
Γ is a lattice in ℍ().
Since SO(4) is compact, Γ projects to a lattice in SO(3,1).
 



A subgroup Γ≤ G is called arithmetic if there is a -algebraic group ℍ and a surjective map f:ℍ()→ G with compact kernel, such that f(ℍ()) 
is commensurable with Γ.


One cornerstone of the theory of lattices is the following:

Suppose that G is semisimple. Then every arithmetic group is a lattice.


A locally symmetric manifold M=Γ\ G/K associated to an arithmetic group Γ≤ G is called arithmetic. In particular, arithmetic manifolds have finite Riemannian volume.

Selberg conjectured that under certain conditions the converse of Borel–Harish-Chandra theorem is also true, i.e. that finiteness of volume implies arithmeticity. The formulation of the conjecture was corrected by Piatetski–Shapiro (who properly defined arithmetic groups) and was later proved by Margulis:


Irreducible[Recall that a locally symmetric manifold is irreducible if it is not commensurable to a direct product of manifolds of smaller dimensions. 
If G is simple all manifolds are locally symmetric.] locally symmetric manifolds associated to a semisimple Lie group of real rank >1 are arithmetic.


This was extended for certain rank one spaces:


Finite volume locally symmetric manifolds associated to Sp(n,1) and F_4(-20) are arithmetic.


The remaining cases are the real and the complex hyperbolic spaces.[In the complex hyperbolic case there are few examples of non-arithmetic finite volume manifolds in complex dimensions 2 and 3, but the question is wide open for higher dimensions.]





 §.§ The Gromov–Piatetski-Shapiro construction


In <cit.> Gromov and Piatetski-Shapiro constructed a non-arithmetic hyperbolic manifold in every dimension. (Prior to that only finitely many examples associated to certain reflection groups were known.) Their idea was to take two arithmetic non-commensurable manifolds M_1 and M_2 which can be cut along isometric totally geodesic co-dimension one manifolds, cut them and then glue by identifying the boundaries of the two. 

Recall that two manifolds M_1 and M_2 are commensurable if they admit a common finite cover. Two subgroups 
Γ_i, i=1,2≤ G are commensurable if their intersection Γ_1∩Γ_2 has finite index in both.



Here is the recipe of the proof of their theorem:




  * Let H≤ G be algebraic groups, and Γ≤ G an arithmetic subgroup. If Γ∩ H is Zariski dense in H then it is arithmetic there. (This is a consequence of the Borel–Harish-Chandra theorem.)


  * Let Γ_1,Γ_2≤ SO(n,1) be two arithmetic subgroups. If Γ_1∩Γ_2 is Zariski dense in SO(n,1) then Γ_1 and Γ_2 are commensurable. (Consider G=SO(n,1)× SO(n,1) and H as the diagonal, take Γ=Γ_1×Γ_2 and apply (1).)


  * Let M be a finite volume complete hyperbolic n-manifold and U⊂ M a submanifold with a totally geodesic boundary and a non-empty interior. Then π_1(U) is Zariski dense in SO(n,1).


  * Let K be a totally real number field. Let Q_1,Q_2 be two quadratic forms over K which have signature (n,1) over ℝ with n≥ 2, while all their non-trivial Galois conjugates are positive definite. Let Γ_i be the arithmetic group in SO(n,1) corresponding to Q_i and M_i the corresponding hyperbolic orbifold. Then M_1 and M_2 are commensurable if and only if Q_1 is isometric to λ Q_2 for some scalar λ∈ K.




Using Items (2),(3), show that if M_1 and M_2 are two non-commensurable arithmetic hyperbolic n-manifolds, that admit isometric totally geodesic co-dimension one submanifolds N_1 and N_2. Then the manifold M obtained by cutting the M_i's along the N_i's and then gluing the two pieces along the boundaries is non-arithmetic.


Item (4) implies that in even dimension, the discriminant of the form is invariant under commensurability. This is what stands behind the following:


1) Suppose that n+1 is even, consider the forms:


    Q_1= x_1^2+x_2^2+…+x_n^2-x_n+1^2, and Q_2= 2x_1^2+x_2^2+…+x_n^2-x_n+1^2

and let M_1,M_2 be the corresponding manifolds (take finite covers to get rid of torsion). Then M_1 and M_2 are non-commensurable and admit 
isometric co-dimension one totally geodesic cuts (along x_1≡ 0). Note that M_1 and M_2 are non-compact.



2) To get a compact example, take 

    R_1= x_1^2+x_2^2+…+x_n^2-√(2)x_n+1^2, and R_2= 3x_1^2+x_2^2+…+x_n^2-√(2)x_n+1^2.

 


These examples thus provide compact and non-compact finite volume non-arithmetic hyperbolic manifolds in any odd dimension. To obtain the even dimensional examples, Gromov and Piatetski-Shapiro took, inside the above example, a co-dimension 1 sub-manifold which intersect the N_i's transversally-orthogonally.






 §.§ Counting hyperbolic manifolds


Denote by ρ_n(v) the number of complete hyperbolic n-manifolds of volume ≤ v, considered up to isometry. 
By Kazhdan–Margulis theorem ρ_n(v)=0 for sufficiently small v. Teichmuller theory implies that ρ_2(4π)=2^ℵ_0. Thurston showed that ρ_3(v)=ℵ_0, for v≥ v_0, where v_0 is the smallest volume of a complete non-compact hyperbolic 3 manifold. For n≥ 4 however we have


For n≥ 4, ρ_n(v)<∞ for all v>0.


Wang's proof is non-effective and gives no estimate on ρ_n(v). Indeed, Wang's beautiful argument basically shows that with respect to a certain topology, the set of conjugacy classes of lattices in SO(n,1) is discrete and the co-volume function is proper. I.e. the set of conjugacy classes of lattices of bounded volume is discrete and compact hence finite.

The first effective bound was given by Gromov <cit.> who showed that, for n≥ 4

    ρ_n(v)≤ vexp(exp(exp(n+v))).


The correct growth type was determined in my joint work with M. Burger, A. Lubotzky and S. Mozes:


For n≥ 4,

    logρ_n(v)≈ vlog v.



By '≈' I mean that the ratio between the two sides is bounded below and above by constants.


Although in <cit.> we proved the lower bound only for non-arithmetic manifolds, 
it can be shown that the same growth type applies if we restrict to each of the following four classes: compact-arithmetic, noncompact-arithmetic, compact-nonarithmetic or noncompact-nonarithmetic (see <cit.>).



The first motivation for these effective estimates came from theoretical phisics <cit.>. 






 §.§ More refined counting

Let us say that a complete hyperbolic manifold is minimal if it does not properly cover any other manifold. It is natural to consider the growth of the number of minimal hyperbolic manifolds, i.e. of

    M_n(v):= #{minimal hyperbolic n-manifolds of volume≤ v},

where again we consider manifolds up to isometry.

An even more refined quantity is the number C_n(v) of hyperbolic n-manifolds of volume ≤ v, considered up to commensurability. Obviously:

    C_n(v)≤ M_n(v)≤ρ_n(v).


Furthermore, denote by C_n^c(v) the number of compact manifolds, and by C_n^nc(v) the number of non-compact ones (of volume ≤ v, up to commensurability). Then 

    C_n(v)=C_n^c(v)+C_n^nc(v).


A first lower bound was given by J. Raimbault[Raimbault's proof is also probabilistic. The reason that he obtained only exponential lower bound while we obtained super-exponential is that he constructed manifolds moduled over circles while we used general regular graphs. Both proofs were inspired by the construction from <cit.> of IRS in (n,1) which is not contained in any lattice.] <cit.> who proved  log C_n^c(v)≥Const· v. Jointly with A. Levit <cit.> we determined the precise growth:


For n≥ 4 we have

    log C_n^c(v)≈log C_n^nc(v)≈ vlog v.
 


Another number that one may wish to consider is that of torsion free lattices of covolume ≤ v considered up to quasi-isometries, denoted QI_n(v). However, while all the cocompact lattices in SO(n,1) are quasi-isometric to each other, a beautiful theorem of R. Schwartz <cit.> says that two non-co-compact lattices in SO(n,1) are quasi isometric iff they are, up to conjugacy, commensurable. I.e.

    QI_n(v)=C_n^nc(v)+1,

whenever v is at least the minimal volume of a compact hyperbolic n-manifold.





 §.§ Graphs of spaces


Let me now explain how one constructs plenty of non-commensurable manifolds. The idea is to construct `random' manifolds modeled over regular graphs. 
Starting with two copies of non-commensurable manifolds with isometric boundaries consisting of 4 components each, 



    < g r a p h i c s >


 

for every finite 4-regular graph, e.g.




    < g r a p h i c s >



 
we replace each vertex by a random copy of either A or B above




    < g r a p h i c s >


 
 and glue boundary components following the edges pattern.

By a simple counting argument the logarithm of the number of 4-regular graphs with v vertices is ≈ vlog v. However, they are all commensurable. By Leighton's theorem <cit.>, two colored finite graphs are commensurable iff they share the same universal covering colored 
 tree. 
 
Using elementary asymptotic group theory, we are able to construct sufficiently many non-commensurable colored graphs. However, it is not clear that manifolds constructed over non-commensurable graphs cannot be commensurable as manifolds. Let me mention a geometric lemma that helps us dealing with this issue:



Let X_1 and X_2 be two submanifolds with totally geodesic boundary and finite volume inside two non-commensurable arithmetic hyperbolic n-manifolds. Assume either that ∂ X_1 and ∂ X_2 are compact and n ≥ 3, or that ∂ X_1 and ∂ X_2 have finite volume and n ≥ 4.

If W is any complete hyperbolic manifold with two embedded submanifolds U_1, U_2 ↪ W that admit finite isometric covers p_i : U_i → X_i for i=1,2, then the intersection U_1 ∩ U_2 has an empty interior.


The proof of this lemma relies on:



Let M be an n-dimensional complete finite volume hyperbolic manifold without boundary and let N be a properly embedded totally geodesic k-dimensional sub-manifold with 1<k≤ n. Then N has finite volume.

 



 §.§ The building blocks


Some technical issues lead us to use 6 building blocks, instead of 2. Moreover, we produce one admissible parcel of compact building blocks, and another one consisting of non-compact building blocks, establishing the desired lower bounds on C_n^c(v) as well as on C_n^nc(v).

The harder case is the even dimension as the discriminant is useless. Instead we address the Legendre and the Hilbert symbols. In the non-compact case we may then use

    Q_p=px_1^2+x_2^2+…+x_n^2-2x_n+1^2, with p=5,13,29,37,53,61.

In the compact case we use

    R_p=px_1^2+x_2^2+…+x_n^2-√(2)x_n+1^2, with p=17,41,97,137,193,241.

The proof that the associated manifolds are non-commensurable relies of the following:


For prime p≡ 1(mod 4) the equation x^4=2 has an integer solution modulo p iff p=x^2+64y^2 for some x,y∈.



















§ INVARIANT RANDOM SUBGROUPS


The results described in the previous section were established by considering the uniform measure on the finite set of 4-regular graphs of a given size. This corresponds to a very primitive example of an IRS in (n,1). We will now consider general IRS, which we will define for every locally compact second countable group. For a more comprehensive exposition of IRS we refer to the lecture note <cit.> (or <cit.>).

Consider a locally compact second countable group G. We denote by (G) the Chabauty space of closed subgroups of G. The Chabauty topology on (G) is metrizable and can be defined as follows. Let d be a proper metric on G which is compatible with the topology, and define the distance ρ(H_1,H_2) between two closed subgroups H_1,H_2 by:

    ρ(H_1,H_2):=∫_0^∞Hd(H_1∩ B(1,r),H_2∩ B(1,r))e^-rdr,

where B(1,r) is the closed r-ball around the identity in G and Hd is the Hausdorff distance defined on compact sets 

    Hd(A_1,A_2)=max{d(x,A_i): i=1,2 and x∈ A_3-i}.
 
Thus H_1 and H_2 are close to each other if their intersections with a large ball in G are close in the Hausdorff distance. 

The space (G) is always compact but in general it is rather complicated. However it is a canonical G space as G acts on it by conjugation. 


An IRS is a G-invariant Borel regular probability measure on (G). 


Basic examples of IRS comes from normal subgroups and lattices:


  * A Dirac IRS, δ_N correspond to a closed normal subgroup N G. 

  * Let Γ≤ G be a lattice and let μ be the G-invariant probability measure on G/Γ. Consider the map 
G/Γ→(G), gΓ↦ gΓ g^-1 and set μ_Γ as the push-forward of μ. Then μ_Γ is an IRS associated with the conjugacy class of Γ and a μ_Γ-random subgroup is a conjugate of .



For instance let Σ be a closed hyperbolic surface and normalize its Riemannian measure. Every unit tangent vector yields an embedding of π_1(Σ) in PSL_2(). Thus the probability measure on the unit tangent bundle corresponds to an IRS of Type (2) above.  




    < g r a p h i c s >


 



Remarkably, the study of IRS plays an important role in the development of the modern theory of lattices and their asymptotic invariants. We shall give several illustrations for this phenomena in subsequent sections. 

IRS are strongly related to p.m.p actions. Indeed, given a probability space (X,m) and a measure preserving G-action G↷ X, it follows by Varadarajan compact model that the stabilizer of almost every point in X is closed in G. Thus the push-forward of m under the stabilizer map x↦ G_x is an IRS in G. By the following theorem every IRS arises in this way:


<cit.>
Let G be a locally compact group and μ an IRS in G. Then there is a probability space (X,m) and a measure preserving action G↷ X such that μ is the push-forward of the stabilizer map X→Sub(G).



This was proved in <cit.> for discrete groups and in <cit.> for general G. The first thing that comes to mind is to take the given G action on (Sub(G),μ), but then the stabilizer of a point H∈Sub(G) is the normalizer N_G(H) of H rather than H. To correct this one considers the larger space Cos_G of all cosets of all closed subgroups, as a measurable G-bundle over Sub(G). Defining an appropriate invariant measure on Cos_G×ℝ and replacing each fiber by a Poisson process on it, gives the desired probability space.





 §.§ IRS in semisimple groups


Let us now restrict to the case where G is a semisimple Lie group. Treating IRS's as a generalization of lattices, a natural thing to do is to establish IRS analogs for theorems about lattices. An important example is the Borel density theorem.  






Let G be a connected non-compact simple (center-free) Lie group. Let μ be an IRS on G without atoms. Then a random subgroup is μ-a.s. discrete and Zariski dense.  
 



By the Von-Neumann–Cartan theorem, every closed subgroup of G is a Lie group. Consider the map ψ:(G)→Gr(Lie(G)), H↦Lie(H). It is upper semi-continuous, hence measurable. Let ν=ψ_*(μ). Then ν is Ad(G)-invariant and hence, by the Furstenberg lemma <cit.>, is supported on {Lie(G),{0}}. Finally observe that ψ^-1(Lie(G))=G and ψ^-1({0}) is the set of discrete subgroups of G. Assuming that G is not an atom of μ the result about discreteness follows. 

Arguing similarely with respect to the map H↦Lie(H^Z) one obtains the result about Zariski density. For this part one has to show that an IRS in G cannot be supported on finite groups (see <cit.>).


We refer to <cit.> for more details and more general versions including the case of semisimple groups over general local fields.  

Let us denote by _d(G) the space of discrete subgroups of G. This is an open subset of the compact space (G). We shall say that an IRS μ is discrete if μ(_d(G))=1, and let IRS_d(G) denote the space of discrete IRS of G.
Since G is isolated in (G) (see <cit.>, see <cit.>) we obtain the following useful fact:


Let G be a connected non-compact simple Lie group. Then IRS_d(G) is compact. 




For the simplicity of the exposition we supposed above that G is simple rather than semisimple. However, both results can be extended to semisimple Lie groups without compact factors. 
We refer to <cit.> for the Borel density theorem (it is proved there for a wider class of semisimple analytic groups over local fields) and to <cit.> for the analog of Corollary <ref> in that context.
 









§ KAZHDAN MARGULIS THEOREM FOR IRS


Recall the celebrated theorem of Kazhdan and Margulis:

<cit.>
Let G be a connected semisimple Lie group without compact factors. There is an identity neighborhood U⊂ G such that for every discrete subgroup Λ≤ G there is some g∈ G for which 

    gΛ g^-1∩ U={1}.



The equivalent geometric formulation of this theorem is:


Given a symmetric space of non-compact type X, there is ϵ>0 such that for every complete X-orbifold M=Λ\ X there is a point where the injectivity radius is at least ϵ. In other words, the ϵ-thick part of M is non-empty.
 

The immediate remarkable consequence is that there is a positive lower bound on the volume of X-orbifolds, or equivalently, on the co-volume of lattices in G. 






There are by now three different proofs of Theorem <ref>. The original proof <cit.>, a proof by Gromov <cit.> which is more geometric and the proof from <cit.> which relies on IRS.  Each of these proof has some advantages on the others. For example the original proof applies to all discrete subgroups (not just lattices) while the proof of Gromov applies to general Hadamard spaces (rather than symmetric spaces). The proof from <cit.> does not apply for general discrete groups but it does apply for general IRS for which it gives a stronger statement than the original theorem, even when restricting to lattices. However, the main advantage of that proof is in its simplicity. 
It happens occasionally in mathematics that a consideration of more general notions allows softer and less technical treatments. This depends however on formulating appropriate definitions. 

Recall that a family of subgroups ℱ⊂(G) is called uniformly discrete if there is an identity neighborhood U⊂ G such that ∩ U={1} for all ∈ℱ. 



A family ℱ⊂_d(G) of invariant random subgroups is said to be weakly uniformly discrete if for every >0 there is an identity neighbourhood U_⊂ G such that

    μ ({∈_G:∩ U_{1}})<

for every μ∈ℱ.





Let G be a connected non-compact semisimple Lie group. Then _d(G) is weakly uniformly discrete.

By restricting to the family μ_ of IRS associated to lattices ≤ G and picking for instance =1/2 one recovers the original Kazhdan–Margulis theorem. Indeed, for every lattice  there is g∈ G such that g g^-1∩ U_1/2={1}. Moreover, “for at least 1/2" of the choices of g this will hold, i.e.  

    m{g∈ G/: g g^-1={1}}≥ 1/2
 
where m is the G-invariant probability measure on G/.


Proof of Theorem <ref>.
Let U_n, n∈ be a descending sequence of compact sets in G which form a base of identity neighbourhoods, and set

    K_n={∈_G:∩ U_n={𝕀 G }}.


Since G has NSS (no small subgroups), i.e. there is an identity neighbourhood which contains no non-trivial subgroups, we have:


The sets K_n are open in (G).



Fix n and
let V⊂ U_n be an open identity neighbourhood which contains no non-trivial subgroups, such that V^2⊂ U_n. It follows that a subgroup  intersects U_n non-trivially iff it intersects U_n∖ V. Since U_n∖ V is compact, the lemma is proved.  


In addition, observe that the ascending union ⋃_n K_n exhausts _d(G), the set of all discrete subgroups of G. Therefore we have:


For every μ∈_d(G) and >0 we have μ(K_n)>1- for some n.

Let

    𝒦_n,:= {μ∈_d(G):μ(K_n)>1-}.

Since (G) is metrizable, it follows from Lemma <ref> that 𝒦_n, is open. 
By Claim <ref>, for any given >0, the sets 𝒦_n,, n∈ form an ascending cover of _d(G). Since the latter is compact, we have _d(G)⊂𝒦_m, for some m=m(). 
It follows that 

    μ({∈(G): intersects U_m trivially})>1-,

for every μ∈_d(G). This completes the proof of Theorem <ref>.




































§ THE STUCK–ZIMMER THEOREM


One of the most striking results about IRS is the Stuck–Zimmer theorem <cit.> which was proved almost two decades before the study of IRS has become popular. 
Recall that a p.m.p action of a semisimple Lie group G is irreducible if every factor of G acts ergodically.


Let G be a semisimple Lie group of rank at least two and with Kazhdan's property (T). Then every irreducible p.m.p action of G is essentially free or transitive. 


As follows from the Borel density theorem every transitive irreducible p.m.p action of a semisimple Lie group G is of the form G G/Γ where Γ≤ G is an irreducible lattice. Thus, in view of Theorem <ref>, Theorem <ref> can be reformulated as follows:


For G as in Theorem <ref>, every non-trivial irreducible IRS is of the form μ_Γ where Γ≤ G is an irreducible lattice.


The Stuck–Zimmer theorem is an ergodic counterpart of the celebrated normal subgroup theorem of Margulis. In order to see why it imply the NST one can argue as follows. Let G be as in Theorem <ref>, let ≤ G be an irreducible lattice and let N be a non-trivial normal subgroup of . Consider the map 

    G/→(G), g↦ gNg^1,

and let μ be the push-forward of the G-invariant probability measure on G/. Then μ is a non-trivial irreducible IRS. Hence by Theorem <ref> it is supported on lattices in G. It follows that N is a lattice in G and hence of finite index in .

Like the NST of Margulis, the Stuck–Zimmer theorem is a consequence of the tension between amenability and property (T). However, while Margulis was able to prove the NST regardless of property (T) <cit.>, the ergodic counterpart (the Stuck–Zimmer Theorem) is still unknown under that generality. 


The analog of Theorem <ref> holds regardless of property (T). That is every irreducible p.m.p action of a semisimple Lie group of rank at least 2 is essentially free or transitive. 


We remark that if at least one of the factors of G has property (T) then the result holds, as was confirmed by Hartman and Tamuz <cit.>. 
But without this assumption, the conjecture is wide open. The first case to consider is G=_2()×_2(). What Harman and Tamuz realized is that the argument of Stuck and Zimmer implies in general that in higher rank every non-trivial irreducible IRS is supported on co-amenable subgroups. 


Let G be a locally compact group. A discrete subgroup Λ≤ G is co-amenable if the Hilbert space L_2(G/Λ) admits G asymptotically invariant vectors. 




Let G be a semisimple Lie group of real rank at least 2. Then every non-trivial irreducible IRS in G is supported on co-ameanable subgroups.


Thus, the following general conjecture implies the Stuck–Zimmer conjecture <ref>:


Let G be a semisimple Lie group without compact factors and rank at least two. Let Λ≤ G be a discrete co-amenable subgroup which projects densely to every proper factor of G. Then Λ is a lattice in G.


In <ref> below I will report about a recent progress in this direction.



Let us recall that the Stuck–Zimmer theorem relies heavily on the Nevo-Zimmer intermediate factor theorem which was actually established later <cit.>. Later on Arie Levit extended the Nevo–Zimmer and the Stuck–Zimmer theorem for semisimple analytic groups over non-archimedean local fields <cit.>.



















§ AN EXOTIC IRS IN RANK ONE


In the lack of Margulis' normal subgroup theorem there are IRS supported on non-lattices. Indeed, if G has a lattice with an infinite index normal subgroup N, arguing as in the previous section, one obtains an ergodic p.m.p. space for which almost any stabilizer  is a conjugate of N. 

We shall now give a more interesting example, in the lack of rigidity: 




[An exotic IRS in _2(), <cit.>]











































Let A,B be two copies of a surface with 2 open discs removed equipped with distinguishable hyperbolic metrics, such that all the 4 boundary components are geodesic circles of the same length.






    < g r a p h i c s >


 




 


Now consider the space {A,B}^ with the Bernoulli measure (1/2,1 2)^. Any element  in this space is a two sided infinite sequence of A's and B's and we can glue copies of A,B `along a bi-infinite line' following this sequence. This produces a random surface M^. Choosing a probability measure on the unit tangent bundle of A (resp. of B) we define an IRS in _2() as follows. First choose M^ randomly, next choose a point and a unit tangent vector in the copy of A or B which lies at the place M^_0 (above 0 in the sequence ), then take the fundamental group of M^ according to the chosen point and direction. 
 



    < g r a p h i c s >


 

It can be shown that almost surely the corresponding group is not contained in a lattice in _2().
Analog constructions can be made in (n,1) for all n's (see <cit.>).
We refer to <cit.> for more examples of exotic IRS in rank one.





















§ THE BENJAMINI–SCHRAMM, THE WEAK-* AND THE IRS TOPOLOGIES










 §.§ The Gromov–Hausdorff topology


Given a compact metric space X, the Hausdorff distance Hd_X(A,B) between two closed subsets is defined as

    Hd_X(A,B):=inf{: N_(A)⊃ B and N_(B)⊃ A},

where N_(A) is the -neighborhood of A. The space 2^X of closed subsets of X equipped with the Hausdorff metric, is compact.

Given two compact metric spaces X,Y, the Gromov distance Gd(X,Y) is defined as

    Gd(X,Y):=inf_Z{Hd_Z(i(X),j(Y))},
 
where the infimum is taken
over all compact metric spaces Z admitting isometric copies i(X),j(Y) of X,Y respectively. 
If (X,p),(Y,q) are pointed compact metric spaces, i.e. ones with a chosen point, we define the Gromov distance 

    Gd((X,p),(Y,q)):=inf_Z{Hd_Z(i(X),j(Y))+d_Z(i(p),j(q))}.
 

The Gromov–Hausdorff distance between two pointed proper (not necessarily bounded) metric spaces (X,p),(Y,q) can be defined as

    GHd((X,p),(Y,q)):=∑_n∈1 2^nGd((B_X(n),p),(B_Y(n),q)),
 
where B_X(n) (resp. B_Y(n)) is the ball of radius n around p in X (resp. around q in Y).



 §.§ The Benjamini–Schramm topology


Let ℳ be the be the space of all (isometry classes of) pointed proper metric spaces equipped with the Gromov–Hausdorff topology. This is a huge space and for many applications it is enough to consider compact subspaces of it obtained by bounding the geometry. That is, let f(,r) be an integer valued function defined on (0,1)×^>0, and let ℳ_f consist of those spaces for which ∀,r, the -entropy of the r-ball 
B_X(r,p) around the special point is bounded by f(,r), i.e. no f(,r)+1 points in B_X(r,p) form an -discrete set. Then ℳ_f is a compact subspace of ℳ.   

In many situations one prefers to consider some variants of ℳ which carry more information about the spaces. 
For instance when considering graphs, it may be useful to add colors and orientations to the edges. The Gromov–Hausdorff distance defined on these objects should take into account the coloring and orientation.

Another example is smooth Riemannian manifolds, in which case it is better to consider framed manifolds, i.e. manifold with a chosen point and a chosen frame at the tangent space at that point. In that case, one replace the Gromov–Hausdorff topology by the ones determined by (,r) relations (see <cit.> for details), which remembers also the directions from the special point.

We define the Benjamini–Schramm space ℬ𝒮=Prob(ℳ) to be the space of all Borel probability measures on ℳ equipped with the weak-* topology. Given f as above, we set ℬ𝒮_f:=Prob(ℳ_f). Note that ℬ𝒮_f is compact.

The name of the space is chosen to hint that this is the same topology induced by `local convergence', introduced by Benjamini and Schramm in <cit.>, when restricting to measures on rooted graphs. Recall that a sequence of random rooted bounded degree graphs converges to a limiting distribution iff for every n the statistics of the n ball around the root (i.e. the probability vector corresponding to the finitely many possibilities for n-balls) converges to the limit. 

The case of general proper metric spaces can be described similarly. A sequence μ_n∈ℬ𝒮_f converges to a limit μ iff for any compact pointed `test-space' M∈ℳ, any r and arbitrarily small[This doesn't mean that it happens for all .] >0, the μ_n probability that the r ball around the special point is `-close' to M tends to the μ-probability of the same event.


An example of a point in ℬ𝒮 is a measured metric space, i.e. a metric space with a Borel probability measure. 
A particular case is a finite volume Riemannian manifold — in which case we scale the Riemannian measure to be one, and then randomly choose a point (and a frame).


Thus a finite volume locally symmetric space M=\ G/K produces both a point in the Benjamini–Schramm space and an IRS in G. This is a special case of a general analogy that we will now describe. Given a symmetric space X, let us denote by ℳ(X) the space of all pointed (or framed) complete Riemannian orbifolds whose universal cover is X, and by
ℬ𝒮(X)=Prob(ℳ(X)) the corresponding subspace of the Benjamini–Schramm space. 

Let G be a semisimple Lie group without compact factors, with maximal compact subgroup K≤ G and an associated Riemannian symmetric space X=G/K. There is a natural map 

    {discrete subgroups of  G}→ℳ(X), ↦\ X.
  
It can be shown that this map is continuous, hence inducing a continuous map

    _d(G)→ℬ𝒮(X).

It can be shown that the later map is one to one, and since _d(G) is compact, it is
a homeomorphism to its image (see <cit.>). One can also characterize the image of this map which is the space of probability measures on ℳ(X) which are invariant under the geodesic flow.




























§ A RESULT OF THE 7 SAMURAI





The following result from <cit.> can be interpreted as `large higher rank manifolds are almost everywhere fat':


Let X be an irreducible symmetric space of rank at least 2.
Let M_n=_n\ X be a sequence of finite volume X-orbifolds with (M_n)→∞. Then M_n→ X in the Benjamini–Schramm topology.
 

This means that for any r and  there is V(r,) such that if M is an X-manifold of volume v≥ V(r,) then (M_≥ r)/v≥ 1-, where M_≥ r denotes the r-thick part of M, i.e. the set of points in M where the injectivity radius is at least r.




Using the dictionary from the previous section we may reformulate Theorem <ref> in the language of IRS:


Let G be a simple Lie group of rank at least 2.
Let _n≤ G be a sequence of lattices with (G/_n)→∞ and denote by μ_n the corresponding IRS. Then μ_n→_{1}.
 

The proof makes use of the equivalence between the two formulations. 


Recall that by Kazhdan's theorem, G has property (T). This implies that a limit of ergodic measures is ergodic:


Let G be a group with property (T) acting by homeomorphisms on a compact Hausdorff space X. Then the set of ergodic G-invariant probability Borel measures on X is w^*-closed.
   

The idea behind Theorem <ref> is that if μ_n are probability measures converging to a limit μ and μ is not ergodic, then there is an L_2(μ) function on X which is G-invariant, orthogonal to the constants and with norm 1. Approximating it one gets 
continuous function f on X which, as a function in L_2(μ) is almost G-invariant, orthogonal to the constants and with norm 1. Thus for large n we have that f is almost invariant in L_2(μ_n), almost orthogonal to the constants and with norm almost 1. Since G has property (T) it follows that there is an invariant L_2(μ_n) function close to f, so μ_n cannot be ergodic.  

Sketch of proof of Theorem <ref>.
Let M_n=_n\ X be X-orbifolds of finite volume with (M_n)→∞ as in <ref>, let μ_n be the corresponding IRS and let μ be a weak-^* limit of μ_n. Our aim is to show that μ=_{1}. Up to replacing μ_n by a subnet, we may suppose that μ_n→μ. By Theorem <ref> we know that μ is ergodic. However, in view of Theorem <ref>,
the only ergodic IRS on G are _G,_{1} and μ_ for ≤ G a lattice. 

Thus, in order to prove Theorem <ref> (equivalently <ref>) we have to exclude the cases μ=_G and μ=μ_. The case μ=_G is impossible since G is an isolated point in Sub_G (see <cit.> or <cit.>). Let us now suppose that μ=μ_ for some lattice ≤ G and aim towards a contradiction. For this, we will adopt the formulation of Theorem <ref> again. Thus we suppose that M_n→ M=\ X in the Benjamini–Schramm topology.

Recall that Property (T) of G implies that there is a lower bound C>0 for the Cheeger constant of all finite volume X-orbifolds. For our purposes, the Cheeger constant of an orbifold M can be defined as the infimum

    C=inf_S  (N_1(S))/min{(M'),(M”)},

where S is any subset which disconnects the manifold to two sides M',M” and N_1(S) is its 1-neighbourhood.[The sides M',M” are not assumed to be connected but any path from M' to M” must cross S.]

Since M=\ X has finite volume, for an arbitrary point p∈ M if we pick R large enough then

    (B_M(p,R-1))>(1-C)(M),

where B_M(p,R-1), that is the (R-1)-ball around p in M. 

In particular, taking 
S={x∈ M:d(x,p)=R} and M'=B_M(p,R) we have that:

    (N_1(S))/(M')≤(N_1(S))/(B_M(p,R-1))≤(B_M(p,R+1)∖ B_M(p,R-1))/(B_M(p,R-1))<C.



This on itself does not contradict property (T) since the complement M”=M∖ M' could (and actually has to) be very small (or even empty if M is compact). 

Now since M_n converges to M in the BS-topology, it follows that for n sufficiently large, there must be a point p_n∈ M_n, whose R+1 neighborhood B_M_n(p_n,R+1) is arbitrarily close (in the Gromov–Housdorff sense) to B_M(p,R+1). 
Setting 

    S_n={x∈ M_n:d(x,p_n)=R}, M_n':={x∈ M_n:d(x,p_n)<R}
 
we get that

    (N_1(S_n))/(M_n')<C.
 
Bearing in mind that (M_n)→∞, we get that for large n, the complement M_n”:=M_n∖ M_n' has arbitrarily large volume, and in particular (M_n')≤(M_n”).
Now, this contradicts the assumption that C is the Cheeger constant of X.













An alternative proof.
Instead of relying on Kazhdan's property (T) one could make use of local rigidity. If μ_n→μ where μ is the IRS corresponding to some lattice  then there must be points in the support of μ_n which converge to points in the support of μ. However, μ_n is supported on the conjugacy class of _n while μ is supported on the conjugacy class of . It follows that there must be conjugates of _n which converges to  in the Chabauty topology. However, by Margulis' super-rigidity theorem
 is locally rigid and hence also Chabauty locally rigid. It follows that for large n, _n must contain a conjugate of . This however implies that (G/_n)≤(G/) in contrast to the assumption that (G/_n)→∞. This approach was taken in <cit.> where the work <cit.> has been generalized to semisimple groups over non-archimedean local fields. We refer to <cit.> for more details.




As explained in the `alternative proof' above, the argument using Cheeger constant could be replaced by local rigidity. However, property (T) is crucial in two other parts of the proof. Indeed, the proof  relies on Theorem <ref> as well as the Stuck–Zimmer theorem <ref> (which in tern relies on property (T)).


In <cit.> Theorem <ref> was proved in the greater generality of irreducible lattices in higher rank semisimple groups with property (T). In fact it is enough to suppose that G is higher rank and at least one of the factors has property (T). In <cit.> A. Levit proved the analog theorem for higher rank groups regardless of property (T) but assuming that the lattices _n are congruence (instead of Kazhdan's property (T) he made use of Clozel's property τ). If Serre's conjecture is true, then every higher rank irreducible lattice is congruence. 




Obviously the analog of Theorem <ref> is false for G=(n,1). For example, one can start with a compact hyperbolic manifold M=\ℋ^n and take a descending sequence of finite index normal subgroups N_i such that the infinite intersection ∩ N_i is non-trivial and consider the covering manifolds M_i=N_i\ℋ^n. Also the examples constructed in <ref> have uniformly bounded injectivity radius. However, maybe for some `more rigid'  rank one groups (for instance, this is interesting for SP(n,1)) the analog of Theorem <ref> holds for some mysterious reason.   

















§ CONVERGENCE OF NORMALIZED BETTI NUMBERS


The most famous application of Theorem <ref> is the convergence of normalized Betti numbers. This was first proved in <cit.> for sequences of compact manifolds with injectivity radius bounded below (i.e. uniformly discrete sequences of uniform lattices) and in <cit.> for general sequences of manifolds. For the simplicity we shall restrict to the special case considered in <cit.>.

Suppose that G is a simple Lie group with ℝ-rank at least 2 and associated symmetric space X = G/K.
We denote β _k (X) the k^th L^2-Betti number of X (see also the analytic definition below).
Then

    β _k (X)= {[                0,           k≠1/2 X; χ (X^∗)/vol(X^∗),          k=1/2 X. ].

Note also that χ (X^∗)=0 unless rank_ℂ(G)=
rank_ℂ(K).
For an X-manifold M  we denote by b_k(M) the k^th Betti number of M.



 Let (M_n) be a sequence of closed X-manifolds with
injectivity radius uniformly bounded away from 0, and vol(M_n)→∞. Then:

    lim_n→∞b_k(M_n)/vol(M_n)=β
    _k(X)

for 0≤ k≤ (X).




















 §.§ The Laplacian and heat kernel on differential forms





Denote by e^-t Δ_k^(2) (x,y) (x,y∈ X) the heat kernel on L^2 k-forms. The corresponding bounded integral operator in End(Ω^k_(2)(X)) defined by 

    (e^-tΔ_k^(2) f) (x) = ∫_X e^-tΔ_k^(2) (x,y) f(y)   dy,   ∀ f ∈Ω^k_(2)(X)

is the fundamental solution of the heat equation (cf. <cit.>).

A standard result from local index theory (see e.g. <cit.>) implies:


Let m >0. There exists a positive constant c_1=c_1(G,m) such that

    || e^-t Δ_k^(2) (x,y) || ≤ c_1 t^-d/2 e^- d (x, y)^2 / 5t  ,      0 < t ≤ m.



Now let M=Γ\ X be a compact  X-manifold. Let Δ_k be the Laplacian on differentiable k-forms on
M. It is a symmetric, positive definite, elliptic operator with pure point spectrum.
Write e^-tΔ_k (x,y) (x,y ∈ M) for the heat kernel on k-forms on M, then for each positive t we have:

    e^-tΔ_k (x,y) = ∑_γ∈Γ (γ_ỹ)^* e^-tΔ_k^(2) (x , γy),

where x, y are lifts of x,y to X and by (γ_y)^*,
we mean pullback by the map (x,y) ↦ (x̃, γỹ).
The sum converges absolutely and uniformly for x̃, ỹ in compacta; this follows from Lemma <ref>, together with the following estimate:



    |{γ∈Γ  :   d(x̃,γỹ) ≤ r }| ≤ c_2 e ^ c_2 r InjRad_M(x)^-d .

Here, c_2=c_2(G) is some positive constant and d=(X).












 §.§ (L^2-)Betti numbers





The trace of the heat kernel e^-t Δ_k^(2)(x,x) on the diagonal
is independent of x ∈ X, being G-invariant. We denote it by 
    Tr e^-t Δ_k^(2):=tr  e^-t Δ_k^(2)(x,x).

The L_2-Betti numbers can be defined as:

    β_k (X) :=  lim_t →∞Tr e^-t Δ_k^(2) .

Recall also that the usual Betti numbers of M satisfies

    b_k (M) = lim_t →∞∫_M tr e^-t Δ_k(x,x)dx.







The following is a consequence of Lemma <ref> and Equations (<ref>) and (<ref>).

 
Let m >0 be a real number. There exists a constant c = c(m,G) such that for any x ∈ M and t∈ (0,m],

    | tr e^-tΔ_k  (x,x) - Tr e^-t Δ_k^(2)| ≤ c ·InjRad_M(x)^-d.



Proof of Theorem <ref>.
Let now M_n=Γ_n\ X be as in the statement of <ref>. 
By the result of the previous section (Theorem <ref>), for every r>0 we have

    lim_n→∞ ((M_n)_<r)/(M_n)=0,

where M_<r:={x∈ M  |  InjRad_M(x)<r} is the r-thin part of M.
Since the injectivity radius is uniformly bounded away from 0
it follows that

    1/vol (M_n)∫_(M_n)_< rInjRad_M_n(x)^- d dx → 0
 
for every r>0.
Hence by Corrolary <ref>:


    1/(M_n)∫_M_ntr e^-tΔ_k^M_n (x,x)   dx  ⟶ Tr e^-t Δ_k^(2)

uniformly for t on compact subintervals of (0, ∞). Since each term in the limit above is decreasing as a function of t, we deduce that


    lim sup_n →∞b_k (M_n)/vol (M_n)≤β_k (X).

Finally, using that the usual Euler characteristic is equal to its L^2 analogue and that Δ_k^(2) has
zero kernel if k ≠1/2 X we derive Theorem <ref>.



The assumption that G is simple can be replaced by semisimple with property (T) again. 
In <cit.> the analog of Theorem <ref> was proved in a much greater generality, removing the assumption that the injectivity radius is bounded below and also allowing the manifolds to be non-compact. This theorem was also extended to the non-Archimedean case in <cit.>. In <cit.> (as well as <cit.>) results concerning the asymptotic of much more general analytic invariants have been established relying on the geometric Theorem <ref>.













§ STATIONARY RANDOM SUBGROUPS



The notion of invariant random subgroups has proven very effective to the study of certain type of problems. However, the restriction to `invariant' measures limits the scope of problems that one can investigate, in particular as the groups we are studying are non-amenable. For example, the rigidity theorem of Stuck and Zimmer says that for higher rank semisimple Lie groups (with property (T)) all IRS comes from lattices. In particular one cannot hope to study thin subgroups or general discrete subgroups which are non-lattices using the theory of IRS. Also for simple groups of rank one, it is known that there are discrete subgroups which cannot be in the support of an IRS. On the other hand every discrete subgroup is associated with some stationary random subgroups (hereafter SRS) as we shall explain below.  



 §.§ First properties


Let G be a connected semisimple Lie groups. Let μ be a Borel regular probability measure on G whose support generates a Zariski dense subgroup. Let X be a compact G-space. A probability measure ν∈Prob(X) is called μ-stationary if 
μ*ν=ν where 

    μ*ν:=∫ g_*ν dμ(g).

We let (B,ν) denote the Poisson boundary of G associated to μ. Then ν is μ-stationary and has the following universal property. 
Recall that on a compact convex space every probability measure has a unique barycenter. In particular this applies to probability measure on the  
compact convex space Prob(X).   


Let X be a G-space. A probability measure ν_0 on X is μ-stationary iff it is the ν-barycenter of the measure f_*ν for some measurable map f:B→Prob(X). 
Moreover, the map f is unique.


One useful observation that follows from this theorem is that if μ_1,μ_2 are two probability measures on G which share the same Poisson boundary (B,ν) then a measure ν_0 on a G-space X is μ_1 stationary iff it is μ_2 stationary. 

Let K be a fixed maximal compact subgroup of G. Below we will only consider measures μ which are bi-K-invariant. For such measures the Poisson boundary is (G/P, ν) where P is a minimal parabolic subgroup and ν is the unique K-invariant measure on G/P (the uniqueness is due to the fact that K acts transitively on G/P which is a consequence of the Iwasawa decomposition) . By the observation above we may vary the measure μ within this class (acording to our needs) without effecting the property of being stationary. 




 §.§ A special measure


Let G be a connected semisimple Lie group without compact factors.
In <cit.> we have constructed a specific measure μ_G which is quite effective in various situations. 
The measure M_G has the form

    η_K*δ_s*η_K

where η_K is the Haar measure of the compact group K and s is a certain semisimple element (see <cit.> for details).  





 §.§ The discreteness radius









Fix a norm · on the Lie algebra Lie(G) such that exp:Lie(G)→ G restricted to the unit ball B(1)={X∈Lie(G): X<1} is a well defined diffeomorphism. For r≤ 1 denote B(r)={X∈Lie(G): X<r}.
For a discrete group ⊂ G set

    ℐ()=sup{r≤ 1: expB(r)∩Γ={1}}.
 
We call ℐ(Γ) the discreteness radius of .









 §.§ The Margulis function on the space of discrete subgroups of G







A fundamental result established in <cit.> is that there is a positive constant =(G) such that 

    u():=ℐ()^-

satisfies Inequality (<ref>) below, that is, it is a Margulis function on the space _d(G) of discrete subgroups of G with respect to μ_G. 


There exist 0<c<1, b> 0 such that, for every discrete subgroup ≤ G,

    ∫_G u(^g) dμ_G(g)≤ c u() +b.
 


The constants , c and b are computed explicitly in <cit.> in order to prove certain effective results and in particular a quantitative version of the Kazhdan–Margulis theorem. We will not make use of the explicit values here. 




 §.§ Effective weak uniform discreteness of SRS


Motivated by <cit.> and the original paper of Kazhdan and Margulis <cit.> we have establish in <cit.> the following result which is a strong effective version of the Kazhdan–Margulis theorem. It is obviously stronger than the result from <cit.> as it is effective and applies to all SRS rather than IRS. Moreover, since it applies to all SRS it implies the Kazhdan–Margulis theorem for all discrete subgroups.

 


There are   constants β, ρ, δ   > 0   such that for every  

discrete stationary random subgroups ν on G

    ν({  :  ∩B(r) ≠{id_G}}) ≤β r^δ  ∀ r < ρ.



As a straight forward consequence we obtain:


The space of discrete stationary random subgroups SRS_d(G) is compact.








 §.§ Random walks on the space of discrete subgroups





The main result of this section is that any stationary limit of a measure supported on discrete subgroups of G is almost surely discrete. 
This is a key ingredient in the proofs of the results of <ref> (in particular Theorem <ref> and Theorem <ref>). 






Let ν be a probability measure on _d(G). Let 

    ν_n=1/n∑_i=0^n-1μ_G^(n)*ν
 
and let ν_∞ be a weak-* limit of ν_n. Then ν_∞ is supported on the set of discrete subgroups of G, that is, ν_∞ is a discrete stationary random subgroup. 



The proof is reminiscent of <cit.>. 


 
Recall the discreteness function ℐ and the Margulis function u defined in the previous page.
By restricting to compact subsets of _d(G), we may allow ourself to suppose that ν is compactly supported and that A:=∫ u(Λ)dν(Λ) is finite. 

To prove Theorem <ref> we need to show that 




    lim_→ 0ν_∞({Λ:ℐ(Λ)<})= 0.



Inequality (<ref>) implies that 

    ∫ u(Λ)d(μ_G*ν)(Λ)=∫ u(Λ^g)dμ_G(g)dν(Λ)≤ cA+b.

Furthermore, iterating Condition (<ref>) and summing the resulting geometric series we get

    ∫ u(Λ)d(μ_G^(n)*ν)(Λ)< c^n A+C,
 
with C:=b/(1-c), uniformly for all n. Set M=A+C.
Then, for every >0 and n≥ 1 we have




    ν_n({Λ:u(Λ)≥ M^-1})< .


Setting =(/M)^1/ we get

    ν_n({Λ:ℐ(Λ)≤})<.

Taking n→∞ gives 
ν_∞({Λ:ℐ(Λ)≤})<,
and letting → 0 we get (<ref>).




















§ CONFINED SUBGROUPS HAVE FINITE CO-VOLUME


Let G be a Lie group. A discrete subgroup ≤ G is called confined if there is a compact set C⊂ G such that ∀ g∈ G, ^g∩ C∖{1}∅.



 §.§ Margulis' conjecture

In this section we will explain the proof of the following result from <cit.> which confirmed a conjecture of Margulis. 


Let G be connected simple Lie group of rank at least 2. Let ≤ G be a discrete subgroup. Then  is confined iff it is a lattice in G.


Let X=G/K be the associated symmetric space.
Geometrically, the theorem says that if  M=\ X is an X-orbifold with bounded injectivity radius then it must have finite volume.
In different words, the theorem says that if ≤ G is a discrete subgroup of infinite co-volume, then it admits a sequence of conjugates ^g_n which converges to the trivial group. The following statement (from <cit.>) is much stronger:


Let μ=μ_G be the probability measure on G constructed in <cit.>. 
Let ≤ G be a discrete subgroup of infinite co-volume.
Then

    1/n∑_i=1^n μ^(i)*_→_⟨ 1⟩.



Geometrically, the theorem says that every the random walk on a locally X-manifold of infinite volume, spans most of the time in the r-thick part for every r>0. 

Let G be as above and Λ≤ G a discrete subgroup of infinite co-volume. Set 

    ν_n:=1/n∑_i=1^n μ^(i)*_.
 
The aim is to prove that ν_n→_⟨ 1⟩.
Since Prob((G)) (with the week-* topology) is compact, it is enough to prove that _⟨ 1⟩ is the only limit of ν_n. Thus let ν_∞ be a week-* limit of ν_n, and we shall argue to show that ν_∞=_⟨ 1⟩. In view of Theorem <ref> ν_∞ is a discrete SRS, i.e. a ν_∞-random subgroup is almost surely discrete. 









 §.§ Discrete stationary random subgroups are invariant.

We shall start by proving the following stiffness result which is of independent interest:


A discrete SRS on (G) is an IRS.


Fix a minimal parabolic subgroup P≤ G and recall that the Poisson boundary of (G,μ) is (G/P,ν_P) where ν_P is the unique K-invariant probability measure on G/P. Let ν_0 be an SRS. In view of Furstenberg's theorem (see <ref>) ν_0 is the baricenter of a measurable map ϕ:G/P→Prob((G)), that is:

    ν_0=∫_G/Pϕ (gP)dν_P.

Moreover, if ν_0 is a discrete SRS, i.e. if ν_0(_d(G))=1 then for ν_P-almost every gP∈ G/P, ϕ(gP) is a discrete random subgroup as well. 

Theorem <ref> is a consequence of the following celebrated theorem of Nevo and Zimmer <cit.>:


Let G be a higher rank simple Lie group. Let μ be a smooth probability measure on G and let (X,ν) be a probability μ-stationary action of G. Then either
ν is G-invariant, or there exists a proper parabolic subgroup Q⊂ G and a measure preserving G-equivariant map[ By measure preserving we only mean that ν(π^-1(A))=μ_Q(A) for every Borel set A⊂ G/Q. In particular we do not require that it is a relatively measure preserving extension.] π (X,ν)→ (G/Q,μ_Q), where μ_Q is the unique μ-stationary measure on G/Q.


Our specific measure μ=μ_G is not smooth. However, in view of the remark after Theorem <ref>, since there exists some smooth bi-K-invariant probability measure on G, Theorem <ref> applies also for μ_G.

Arguing by way of contradiction, suppose that ν_0 is a non-invariant discrete SRS. It follows from Theorem <ref> that there is a parabolic subgroup Q containing P such that for ν_P almost all gP∈ G/P the measure ϕ(gP) is a gPg^-1-invariant discrete random subgroup of gQg^-1. The contradiction, thus follow from the following:


Let P≤ Q be parabolic subgroups of G. The only P-invariant discrete random subgroup of Q is _⟨ 1⟩.





 Let Q be a parabolic subgroup of G. Let Q=LN be a Levi decomposition of Q and let A_L be the centre of L. Then, any discrete A_L-invariant random subgroup of Q is contained in L almost surely. 




Let a∈ A_L be an element such that the norm of the restriction of Ad(a^-1) to the Lie algebra of N is less than 1. It follows that for any two identity neighbourhoods U_1,U_2⊂ N in N such that U_2 is bounded there is n_0∈ such that Ad(a^-n_0)(U_2)⊂ U_1.

Let λ be a discrete A_L-invariant random subgroup of Q. Suppose by way of contradiction that λ ({⊂ L})<1. Then there is a bounded identity neighbourhood V_2 in Q such that 
λ()>0 where :={:∩ V_2∖ L∅}.
Furthermore, we may take V_2 to be of the form V_2=W· U_2 where W⊂ L and U_2⊂ N and suppose that they are preserved by a^-1 (e.g. we can suppose that log U_2 is a norm ball in the Lie algebra of N). Since λ is discrete we can chose a small identity neighborhood U_1⊂ N so that setting V_1=W· U_1 we have 

    :=λ({∈:∩ (V_1∖ L)=∅})>0.

Choosing n_0 as above we get that λ (^a^n_0)≤λ()- in contrast with the assumption that λ is A_L-invariant.



In view of Lemma <ref>, Proposition <ref> follows from the following Lie theoretic result (see <cit.>):



The intersection of all Levi subgroups of Q is trivial.


The contradiction is now achieved, i.e. assuming that ν_0 is non-invariant we deduce that ϕ(gP)=_⟨ 1⟩ and hence ν_0=∫_G/P_⟨ 1⟩dν_P=_⟨ 1⟩ which is absurd. 
This completes the proof of Theorem <ref>. 


We remark that the analog of stiffness Theorem <ref> does not hold for rank one groups. We refer to <cit.> for an example of a discrete SRS in _2() which is not an IRS, and note that such examples can be constructed in any rank one simple group.




















 §.§ The proof of Theorem <ref>.


Recall that we fixed a discrete subgroup of infinite co-volume  ≤ G and denoted ν_n:=1/n∑_i=1^n μ^(i)*_ and ν_∞ an arbitrary accumulation point of ν_n. We will prove Theorem <ref> by showing that ν_∞ must be _⟨ 1⟩. Since ν_∞ is stationary, it follows from Theorem <ref> that it is invariant, i.e. an IRS. Moreover, in view of Theorem <ref>, ν_∞ must be discrete. By the Stuck–Zimmer theorem (Theorem <ref>) any non-trivial subgroup in the support of ν_∞ must be a lattice. Therefore in order to prove that ν_∞=_⟨ 1⟩ it is enough to show that there are no lattices in its support. Recall that the ν_n are supported on the conjugacy class of . Therefore ν_∞ is supported on the closure of ^G. It is therefore enough to show that the closure of the conjugacy class ^G cannot contain lattices. In order to complete the proof we shall refer to the classical local rigidity theorem.



  
Local rigidity. Let  be a finitely generated group. Consider the deformation space Hom(,G) of homeomorphisms from  to G with the point-wise convergence topology. A map ψ∈Hom(,G) is called locally rigid if there is a neighborhood of ψ which consists of conjugates. That is if ϕ∈Hom(,G) is sufficiently close to ψ then there is g∈ G such that ϕ=ψ^g. A subgroup  of G is called locally rigid if the inclusion i:↪ G is locally rigid.
The classical local rigidity theorem due to Selbeg, Weil and Margulis, says the following:


Let G be a non-compact simple Lie group not locally isomorphic to _2() or _2(). The every lattice in G is locally rigid. 


We can now proceed with the proof of Theorem <ref>.
Suppose by way of contradiction that there is a sequence of conjugates ^g_n which converges to a lattice ≤ G. Since  is discrete there is a compact identity neighborhood U⊂ G with U∩={1}. Recall that , being a lattice is finitely presented (see <cit.>). Let ⟨:R⟩ be a finite presentation of . For n large enough we have both that


  * ^g_n∩ U={1}, and

  * if _n∈^g_n is a nearest element to  for any ∈ then for every w∈ R, the evaluation of w on _n, ∈ falls in U.

It follows that all the relations w∈ R are satisfied on the set {_n:∈} and hence the assignment ↦_n an homomorphism of  which is close to the inclusion. By local rigidity we deduce that ^g_n contains a conjugate of . It follows that  is a lattice, in contrast to the assumption. This completes the proof of  Theorem <ref>. 



  
Summery of the proof: Theorem <ref> is proven in three steps:


  * Discreteness — Theorem <ref>.

  * Stiffness — Theorem <ref>.

  * Rigidity — the use of Stuck–Zimmer rigidity theorem and local rigidity, as described above.










 §.§ The case of semisimple groups


In <cit.> several analogs of Theorem <ref> were proved also for general semisimple groups. This relied on the following decomposition theorems for discrete IRS and SRS (see <cit.>):



Let G=G_1×⋯× G_n be a connected centre-free semisimple Lie group without compact factors and with simple factors G_i, i=1,…,n.
Let ν be an ergodic discrete invariant random subgroup in G. Then G decomposes to a product of semisimple factors G=H_1×…× H_k with 1≤ k≤ n, such that almost surely the projection of a random subgroup to each H_i is discrete while the projection of the intersection with H_i to each proper factor of H_i is dense. 




Let G be a connected centre-free semisimple Lie group without compact factors and ν a discrete μ-stationary random subgroup of G. Then G decomposes to a product of three semisimple factors G=G_ℐ× G_ℋ× G_𝒯 such that


  * ν projects to an IRS in G_ℐ for which all the irreducible factors are of rank at least 2.

  * G_ℋ is a product of rank one factors and ν projects discretely to every factor of G_ℋ.

  * ν projects trivially to G_𝒯.

Furthermore, the intersection of a random subgroup with every simple factor of G_ℋ as well as with every irreducible factor of G_ℐ is  almost surely Zariski dense in that factor.


By the irreducible factors of G_ℐ we mean the irreducible factors associated with the decomposition of IRS as in Theorem <ref>. The subscripts ℐ, ℋ,𝒯 stands for invariant, hyperbolic and trivial (respectively). 

Theorem <ref> allows the extension of Theorem <ref> to semisimple groups under certain conditions. The simplest case is when all the factors of G are of rank at least 2: 



Let G=G_1×⋯× G_n be a connected centre-free semisimple Lie group such that each G_i is simple of rank at least 2.
Let Λ be a discrete subgroup. Then Λ is confined if and only if there is a nontrivial semisimple factor H G such that Λ∩ H is a lattice in H. If Λ is not confined then 
 1/n∑_i=0^n-1μ_G^(i)*_→_{1}.


We refer to <cit.> for additional generalizations of Theorem <ref> which allows also rank one factors. 
We remark that Theorem <ref> plays also an important role in the results described in the next section.










§ SPECTRAL GAP FOR PRODUCTS AND A STRONG VERSION OF THE NST




 §.§ Margulis' normal subgroup theorem


Recall the celebrated normal subgroup theorem of Margulis (see <cit.> and <cit.>):


Let G be a center-free semisimple Lie group without compact factors with rank(G)≥ 2, and let ≤ G be an irreducible lattice. Then every non-trivial normal subgroup N is of finite index. 


The impact of this theorem on the field is enormous. The proof is quite remarkable, both in the general idea and in the details. Margulis deduce's the finiteness of /N from the tension between two analytic properties that cannot get along with each other for infinite groups, namely amenability and property (T).  

Recall that a group H is amenable if the regular representation on L_2(H) admits asymptotically invariant vectors, while property (T) is a spectral gap with respect to all unitary representations and in particular the regular representation. Therefore if H is both amenable and has (T) then L_2(H) must admit the constant non-zero functions which implies that H is compact, or finite if it is also discrete.

In case the semisimple group G has property (T) one gets for free that also  and /N have (T). Hence in that case Margulis' proof boils down to showing that /N is amenable. The case where G does not have property (T), e.g. for G=_2()×_2() requires additional (and again quite remarkable) work and is carried out in <cit.>.

There are several works that generalizes and improves the normal subgroup theorem. For instance Thereom <ref> can be regarded as a measure theoretic counterpart of the NST, while Theorem <ref> can be regarded as a geometric variant of the NST and both are much stronger than the NST when they hold. However these works, as well as all other published generalizations of the NST for classical semisimple groups relied on Kazhdan's property (T) and do not cover the general higher rank case.






 §.§ Higher rank irreducibly confined subgroups are lattices


I will now review a 
joint work with Arie Levit and Uri Bader <cit.> that will be posted on arXiv in a couple of weeks.
We have managed to prove the following:


Let G and  be as in Theorem <ref>. Then every confined subgroup of  is of finite index.


In other words, infinite index subgroups of higher rank irreducible lattices are unconfined. That is, if |:|=∞ then there is a sequence _n∈ such that ^_n→⟨ 1⟩. This property is much stronger than being non-normal (assuming  is nontrivial). Indeed, normal subgroups are the fixed points for the action ().

When G (or at least one of its simple factors) has property (T), Theorem <ref> follows from the much stronger results proved in <cit.> (see <ref>).
The novelty in our new work is that it applies regardless of property (T). 

As in <cit.>, we also obtained a general result for discrete groups which are not assumed a priory to be contained in a lattice. For this result we need however to impose some irreducibility condition which passes to limits of conjugates:


Let G be a semisimple Lie group. A discrete subgroup of G is called strongly confined if it has no discrete conjugate limit which is contained in a proper factor of G.
A discrete subgroup is called irreducibly confined if it is strongly confined and its intersection with any proper factor of G is not confined.



Let G be a connected center-free semisimple Lie group without compact factors and with rank(G)≥ 2. A discrete subgroup ≤ G is irreducibly confined in G iff it is an irreducible lattice in G.


The paper <cit.> is self contained, hence can also serve as an alternative to <cit.> for a proof of the NST in this most complicated case.[Note that the proof of the NST in the lack of property (T) is not fully contained in <cit.>.]



 §.§ Hints about the proof


Recall the following lemma <cit.>:

 
Let G=G_1× G_2 where G_1,G_2 are locally compact second countable groups. Let X be a compact, second countable G-space with G_1-invariant probability measure ν such that Stab_G (x) is discrete and has a dense projection onto G_2 for ν-almost every x∈ X. Then ν is G-invariant. 



In view of the ergodic decomposition of probability measure preserving actions, the lemma clearly reduces to the ergodic case, so let us assume that ν is G_1-ergodic. Using the large stabilizers of the action and Kakutani's ergodic theorem for random walks we will prove that such measure must be also G_2-invariant.

Choose a smooth symmetric probability measure η on G_1 whose support generates G_1. Write η^g_1 for the measure η^g_1(A)=η(g_1^-1A g_1). Since the support of η^g_1 generates G_1, the measure ν is ν^g_1 stationary and ergodic with respect to the random walk on X induced by η^g_1, for every g_1∈ G_1. We say that a point x∈ X is η^g_1-generic for ν if

    lim_n→∞1/n∑_i=1^n-1∫ f(g x)d(η^g_1)^∗ i(g)=lim_n→∞1/n∑_i=1^n-1∫ f(g_1^-1g g_1 x)dη^∗ i(g)=∫ f dν(x),
 for all continuous functions f. By Kakutani's pointwise ergodic theorem for random walks <cit.>, the set of η^g_1-generic points has full measure with respect to ν. It follows that the set 
    { (g_1,x)| x  is not η^g_1-generic}⊂ G_1× X
 has zero measure. By Fubini's theorem we deduce that there is a subset X'⊂ X with ν(X')=1, such that every x∈ X' satisfies the following property. For almost every g_1∈ G_1  

    lim_n→∞1/n∑_i=1^n-1  (η^g_1)^∗ iδ_x=ν.
 in the weak-* topology. 

Take a point x∈ X' such that Γ:= Stab_G(x) is discrete and has a dense projection onto G_2. Since Γ is countable, we may fix  g_1∈ G such that (<ref>) holds for g_1γ_1, for every γ=(γ_1,γ_2)∈Γ. Note that η^γ g=η^γ_1g_1. Comparing (<ref>) for g_1 and γ_1 g_1 and using the fact that γ^-1x=x we find that 
    ν=   lim_n→∞1/n∑_i=1^n-1  (η^γ_1g_1)^∗ iδ_x=lim_n→∞1/n∑_i=1^n-1  (η^γ g_1)^∗ iδ_x
    =   lim_n→∞1/n∑_i=1^n-1γ_* (η^g_1)^∗ iδ_γ x=γ_*lim_n→∞1/n∑_i=1^n-1  (η^g_1)^∗ iδ_x=γ_*ν.
 This means that Γ G_1⊂ Stab_G ν. The action of G on the set of probability measures on X is continuous, so Stab_G ν⊃Γ G_1=G.


In a sense, Theorem <ref> is established by proving a carful suitable approximated version of Lemma <ref>. That is, we establish that certain measures which are sufficiently G_1 invariant must also be G_2 almost invariant. However, the precise formulation and the proof are considerably more complicated and we refer the reader to <cit.>. Below we shall outline how data this type implies the desired statements.

Since simple groups with rank at least 2 have property (T), Theorem <ref> (as well as Theorem <ref>) follows from <cit.> in that case. Therefore we may suppose that G has more than one factors. Let ≤ G be an irreducibly confined subgroup. Throughout most of the proof we suppose that  is also co-amenable. That is, there is a an asymptotically invariant sequence of unit vectors f_n∈ L_2(G/).  This means that if Ω is a compact set in G then 

    sup_g∈ L_g(f_n)-f_n→ 0,

Where L_g(f)(x):=f(g^-1x).

Since we suppose that G is not simple we can write it as G_1× G_2. We then proceed in a similar fashion to the argument in <cit.> and show that we can modify the sequence f_n to another sequence h_n which is asymptotically invariant with respect to G_1 but uniformly non-invariant with respect to G_2. That means that for some fixed compact generating sets _i for G_i, i=1,2 we have that 

    sup_g∈_1 L_g(h_n)-h_n→ 0 while sup_g∈_2 L_g(h_n)-h_n≥

for some fixed >0. This part of our argument is similar to the proof from <cit.> but we do it in a much more general setup, and in a more direct way. 

We then consider h_n^2 as functions in L_1(G/) and as such we view them as distributions of probability measures. Then we consider the map

    G/→ (G), g↦ g g^-1,

and push these measures to probability measures on (G) obtaining a sequence of discrete random subgroups which are asymptotically G_1-invariant but uniformly non-invariant under G_2. We then derive a contradiction by a certain approximated version of Lemma <ref>. The proof however is considerably more delicate. 


The above sketch is an over-simplification of the actual proof avoiding various nuances. For instance, it is crucial for us the the sequence h_n is uniform in the sense that 

    lim_g→ 1sup_n { L_g(h_n)-h_n}=0.







 §.§ From general subgroups to co-amenable ones


We sketched in the previous subsection some ideas of the proof of Theorems <ref> and <ref> under the additional assumption that the irreducibly confined subgroup ≤ G is also co-amenable in G. Let us now explain how to remove the co-amenability assumption.

Supposing we already know that irreducibly confined co-amenable subgroups are lattices, we wish to show that every irreducibly confined subgroup is a lattice.
Let ≤ G be an irreducibly confined subgroup. Consider the ergodic averages of the μ=μ_G random walk on _d(G) starting at _:

    1/n∑_i=1^nμ^(i)*_
 
and let ν be an arbitrary weak-* limit. Then ν is μ-stationary and by Theorem <ref> it is a discrete SRS. Applying the decomposition Theorem
<ref>, keeping in mind that ν is supported on conjugate limits of  and hence on irreducibly confined groups, we get that G=G_ℐ. That is ν is an irreducible IRS. In view of Theorem <ref>, a ν-random subgroup is almost surely 
co-amenable. It follows that  admits a co-amenable conjugate limit . By what we already agreed on  must be an irreducible lattice in G. In particular  is locally rigid. This implies that  admits a subgroup conjugated to , hence  is also a lattice.





 §.§ Spectral gap for products


The presentation of the previous two subsection may make the impression that the `irreducibly confined' assumption is only required in the reduction to the co-amenable case, and that for co-amenable groups a weaker irreducibility assumption might be sufficient. However, this is not the case. 
Even in the co-amenable case the `irreducibly confined' assumption is crucial in our argument. The reason is that we apply a process of changing the probability measures and we have limited control on the way this is done. Therefore we need to assume some irreducibility also for Chabauty limits of stabalizers. The spectral gap result that we obtain in this case is quite general however:




Let G_1 and G_2 be a pair of locally compact second countable compactly generated groups such that G_2 has compact abelianization. Set G = G_1 × G_2. 
Let X be a locally compact topological G-space endowed with a G-invariant (finite or infinite) measure m. Assume that

    
  * L^2_0(X,m)^G_2=0, and
    
  * There is a G-invariant closed subset of (G) containing Stab_G(x) for m-almost every  point x∈ X such that every subgroup H in  this subset satisfies G_1 H = G.

Then the Koopman G-representation L^2_0(X,m) has a spectral gap.


For semismmple groups we obtain the following. 




Let G be an adjoint semisimple group without compact factors. Write G = G_1 × G_2 where G_1 is semisimple and G_2 is simple. Let X be a locally compact topological G-space endowed with a G-invariant  measure m, either finite or infinite. Assume that


  * L^2_0(X,m)^G_2 = 0,

  * m-almost every point x has Stab(x) ∩ G_1 = Stab(x) ∩ G_2 = {e} and

  * if f_n ∈ L^2(X,m) is an asymptotically G_1-invariant sequence of unit vectors then every  accumulation point  μ∈Prob( G) of the sequence of probability measures Stab_*(|f_n|^2 · m) is such that μ-almost every subgroup is  discrete,  not contained in the factor G_2  and admits a Zariski-dense projection to G_2.

Then the Koopman G-representation L^2_0(X,m) has a spectral gap.




9999999999










[7's11]7-note M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, I. Samet, On the growth of Betti numbers of locally symmetric spaces. C. R. Math. Acad. Sci. Paris, 349(15-16):831–835, 2011.


[7's12]7S
M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, I.  Samet,  On the growth of L^ 2-invariants for sequences of lattices in Lie groups. arXiv preprint arXiv:1210.2961, 2012.


[7's17]7A
M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, I.  Samet,
On the growth of L2-invariants for sequences of lattices in Lie groups.  
Ann. of Math. (2) 185 (2017), no. 3, 711–790. 

[7's20]7B
M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, I.  Samet,
On the growth of L2-invariants of locally symmetric spaces, II: exotic invariant random subgroups in rank one, Int. Math. Res. Not. IMRN 2020, no. 9, 2588–2625.


[ABBG23]ABBG M. Abert, N. Bergeron, I. Biringer, T. Gelander, Convergence of normalized Betti numbers in nonpositive curvature, Duke Math. J., 172, 2023, 4, 633–700,


[AGV14]AGV M. Abert, Y. Glasner, B. Virag. Kesten's theorem for invariant random subgroups. Duke Math. J., 163(3):465–488, 2014.



[AG81]AG
D. Angluin, A. Gardner, Finite common coverings of pairs of regular graphs, J. Combin. Theory Ser. B 30 (1981), 184–187.











[BGL24]BGL U. Bader, T. Gelander, A. Levit, Confined subgroups of irreducible lattices, TBPS.

[BS06]BaSha U. Bader, Y. Shalom, Factor and normal subgroup theorems for lattices in products of groups. Invent.
Math. 163 (2006), no. 2, 415–454.



[BGS85]BGS W. Ballmann, M. Gromov, V. Schroeder, Manifolds of nonpositive curvature,
   Progress in Mathematics, 61, Birkhäuser Boston, Inc., Boston, MA, 1985, vi+263.



[Bar83]BarbaschMoscovici
D. Barbasch, H. Moscovici,
L2-index and the Selberg trace formula.
J. Funct. Anal., 53(2):151–201, 1983.


[BGLS10]BGLS M. Belolipetsky, T. Gelander, A. Lubotzky, A. Shalev, Counting Arithmetic Lattices and Surfaces,  Ann. of Math. (2) 172 (2010), no. 3, 2197–2221.


[BS01]BS I. Benjamini and O. Schramm. Recurrence of distributional limits of finite planar
graphs. Electron. J. Probab., 6:no. 23, 13 pp. (electronic), 2001.

[BV13]BV N. Bergeron and A. Venkatesh. The asymptotic growth of torsion homology for arithmetic
groups. J. Inst. Math. Jussieu, 12(2): 391–447, 2013.


[B81]Bor A. Borel, Commensurability classes and volumes of hyperbolic 
3-manifolds, Ann. Scuola Norm. Sup. Pisa, Ser. IV, 8 (1981) 1-33.

[BHC62]BoHC A. Borel, Harish-Chandra, Arithmetic subgroups of algebraic groups. Ann. Math. (2) 75 (1962) 485-535.

[BGLM02]BGLM
M. Burger, T. Gelander, A. Lubotzky, and S. Mozes, Counting hyperbolic
  manifolds, Geometric and Functional Analysis 12 (2002), no. 6,
  1161–1173.
  
  
  
[Ca97]Ca1 S. Carlip, Spacetime foam and the cosmological constant, UCD-97-19, gr-qc/9708026, Phys.Rev.Lett, 79,
1997, 4071–4074

[Ca98]Ca2 S. Carlip, Dominant topologies in Euclidean quantum gravity, UCD-97-21, gr-qc/9710114, Class.Quant.Grav, 15, 1998, 2629–2638.
 
[Co92]cor K. Corlette, Archimedean superrigidity and hyperbolic geometry. 
Ann. of Math. 135 (1992), no. 1, 165–182. 
 
 
[EM02]EM A. Eskin, G. Margulis. "Recurrence properties of random walks on finite volume homogeneous manifolds." Random walks and geometry (2002), 431–444.


[FG23]FG M. Fraczyk, T. Gelander,  
 Infinite volume and infinite injectivity radius
Ann. of Math. (2) 197 (2023), no. 1, 389–421.

[F76]Fu H. Furstenberg, A note on Borel's density theorem, Proc. Amer. Math. Soc. 55, (1976),
209–212.

[Ge04]HV T. Gelander, Homotopy type and volume of locally symmetric manifolds.  Duke Math. J.  124  (2004),  no. 3, 459–515.

[Ge11]crells T. Gelander, Volume versus rank of lattices. J. Reine Angew. Math. 661 (2011), 237–248. 

[Ge18a]G T. Gelander, Lecture notes on Invariant Random Subgroups and Lattices in rank one and higher rank, London Math. Soc. Lecture Note Ser., 447
Cambridge University Press, Cambridge, 2018, 186–204.


[Ge18b]WUD T. Gelander, Kazhdan–Margulis theorem for invariant random subgroups, Adv. Math., (2018) 327, 47–51.


[Ge18c]ICMT. Gelander, A view on invariant random subgroups and lattices. In Proceedings of
the International Congress of Mathematicians: Rio de Janeiro 2018, pages 1321–1344.
World Scientific, 2018.

[GL14]ccc T. Gelander, A. Levit, Counting commensurability classes of hyperbolic manifolds, Geom. Funct. Anal. 24 (2014), no. 5, 1431–1447.


[GL16]GL T. Gelander, A. Levit, nvariant random subgroups over non-archimedean
local fields. Mathematische Annalen, 372: 1503–1544, 2018.


[GLM22]GLM T. Gelander, A. Levit, G.A. Margulis, Effective discreteness radius of stabilisers for stationary actions,
Michigan Math. J. 72 (2022), 389–438.

[Gek,Le24]GekLe I.Gekhtman, A. Levit, Stationary random subgroups in negative curvature, https://arxiv.org/abs/2303.04237.

[GW97]GW E. Glasner, B. Weiss, 
Kazhdan's property T and the geometry of the collection of invariant measures. (English summary) 
Geom. Funct. Anal. 7 (1997), no. 5, 917–935. 

[GS92]GS M. Gromov, R. Schoen, Harmonic maps into singular spaces and p-adic superrigidity 
for lattices in groups of rank one. Inst. Hautes Ètudes Sci. Publ. Math. 76 (1992), 165–246.
 


[GPS87]GPS
M. Gromov and I. Piatetski-Shapiro, Non-arithmetic groups in
  lobachevsky spaces, Publications Mathématiques de l'IHÉS 66
  (1987), no. 1, 93–103.

[Gr81]Gromov M. Gromov, Hyperbolic manifolds according to Thurston and Jorgensen, Semin. 
Bourbaki, 32e annee, vol. 1979/80, exp. 546, Lect. Notes Math. 842, (1981), 40-53.


[HT16]HT Y. Hartman, O. Tamuz, Stabilizer rigidity in irreducible group actions. Israel J. Math. 216 (2016), no. 2, 679–705.
 
[Kak51]Kak51 S. Kakutani, Random Ergodic Theorems and Markoff Processes with a Stable Distribution, Second Berkeley Symposium on Mathematical Statistics and Probability, 247–261, 1951.

[KM68]KM D. Kazhdan, G.Margulis, A proof of Selberg's hypothesis, Mat. Sb. (N.S.) 75(117) (1968)
163–168.

[Lei82]Le F. T. Leighton, Finite Common Coverings of Graphs, J. Comb. Theory, Series B
33 (1982), 231–138.


[Lev14]Levit A. Levit, The Nevo-Zimmer intermediate factor theorem over local fields, Geom. Dedicata 186 (2017), 149–171.

[Lev20]Levit1A. Levit, On Benjamini-Schramm limits of congruence subgroups. Israel Journal of
Mathematics, 239: 59–73, 2020.

[M91]Ma G.A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer- Verlag, 1991.

[M79]Ma1 G.A. Margulis. Finiteness of quotient groups of discrete subgroups.
Functional Analysis and Its Applications, 13(3):178–187, 1979.


[M84]Ma2 G.A. Margulis, Arithmeticity of the irreducible lattices in the semi-simple groups
of rank greater then 1, (Russian), Invent. Math. 76 (1984) 93-120.
















[NZ99]NZ1 A. Nevo, R.J. Zimmer, Homogenous projective factors for actions of semi-simple Lie groups. Invent. Math. 138(2), 229–252 (1999).

[NZ00]NZ2 A. Nevo, R.J. Zimmer,  Rigidity of Furstenberg entropy for semisimple Lie group actions. Ann. Sci. Ec. Norm. Super. 33(3), 321–343 (2000).

[NZ02]NZ3 A. Nevo, R.J. Zimmer,   A structure theorem for actions of semisimple Lie groups. Ann. Math. 156(2), 565–594 (2002).

[NZ02]NZ4 A. Nevo, R.J. Zimmer,  A generalization of the intermediate factors theorem. J. Anal. Math. 86, 93–104(2002).


[R12]Raimbault
J. Raimbault, A note on maximal subgroup growth in SO(1, n), IMRN,
  doi 10 (2012).

[Sch96]Schwartz
R.E. Schwartz, The quasi-isometry classification of rank one lattices,
Inst. Hautes tudes Sci. Publ. Math. No. 82 (1995), 133–168 (1996).

[Se93]serre1993course
J.P. Serre, A course in arithmetic, 1993.

[Sh00]ShalomShalom, Rigidity of commensurators and irreducible lattices. Invent. Math. 141, 1–54 (2000).



[Wa72]Wang H.C. Wang, Topics on totally discontinuous groups,
Symmetric Spaces, edited by W. Boothby and G. Weiss (1972), 
M. Dekker, 460-487. 



[SZ94]SZ
G. Stuck and R.J. Zimmer.
Stabilizers for ergodic actions of higher rank semisimple groups.
Annals of Mathematics, pages 723–747, 1994.




[Z37]Za H. Zassenhaus, Beweis eines satzesuber diskrete gruppen. (German) Abh. Math. Sem. Univ. Hamburg 12 (1937), no. 1, 289–312.











 Paper Title: A general dynamical theory of Schwarz reflections, B-involutions, and algebraic correspondences.



Slices of Schwarz reflections, and B-involutions]A general dynamical theory
 of Schwarz reflections, B-involutions,
 and algebraic correspondences




[Y. Luo]Yusheng Luo
Department of Mathematics, Cornell University, 212 Garden Ave, Ithaca, NY 14853, USA
yusheng.s.luo@gmail.com

Y.L. was partially supported by NSF Grant DMS-2349929.

[M. Lyubich]Mikhail Lyubich
Institute for Mathematical Sciences, Stony Brook University, 100 Nicolls Rd, Stony Brook, NY 11794-3660, USA
mlyubich@math.stonybrook.edu

M.L. was partially supported by NSF grants DMS-2247613 and 1901357, the Clay Fellowship, and the Centre for Nonlinear Analysis and Modeling (CNAM)

[S. Mukherjee]Sabyasachi Mukherjee
School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005, India
sabya@math.tifr.res.in
S.M. was supported by the Department of Atomic Energy, Government of India, under project no.12-R&D-TFR-5.01-0500, an endowment of the Infosys Foundation, and SERB research project grants MTR/2022/000248.


§ ABSTRACT 
In this paper, we study matings of (anti-)polynomials and Fuchsian/reflection groups as Schwarz reflections/B-involutions or as (anti-)holomorphic correspondences, as well as their parameter spaces.
We prove the existence of matings of generic (anti-)polynomials, such as periodically repelling, or geometrically finite (anti-)polynomials, with circle maps arising from the corresponding groups.
These matings emerge naturally as degenerate (anti-)polynomial-like
maps, and we show that the corresponding parameter space slices for such matings bear strong resemblance with parameter spaces of polynomial maps.
Furthermore, we provide algebraic descriptions for these matings, and construct algebraic correspondences that combine generic (anti-)polynomials and genus zero orbifolds in a common dynamical plane, providing a new concrete evidence to Fatou’s vision of a unified theory of groups and maps.


[
    Amy Navarathna,1,2 Benjamin J. Carey,1,2 James S. Bennett,3 Soroush Khademi,1,2 and Warwick P. Bowen1,2,4*
    August 5, 2024
==============================================================================================================














§ INTRODUCTION







A dynamical theory of Schwarz reflection maps has emerged in the past decade
that provided a general framework for matings between anti-polynomial  maps and Fuchsian reflection groups as (anti-)algebraic correspondences, leading to relations between the corresponding parameter spaces (which also gave a new insight into the holomorphic case). Many special families have been thoroughly explored <cit.>. In this paper we will unify these various pieces into
a general theory.

A prototype of the theory we develop is the classical theory of polynomial-like maps.
Polynomial-like maps were first defined by Douady and Hubbard as a tool to study similarities between parameter spaces of complex polynomials and those of more general families of holomorphic maps <cit.>. This led to great deal of work on the understanding of parameter spaces of complex dynamical systems. In the process, intimate connections between polynomial-like maps and renormalization theory emerged (see <cit.>). 

Inou and Kiwi developed a more general theory of polynomial-like maps and the associated straightening maps (that relate parameter spaces of holomorphic maps to appropriate standard models) which address the additional complexity furnished by higher degree renormalizations and critical orbit interactions <cit.>. 

However, the above theory has some limitations: over the years, various situations related to parabolic external dynamics started to emerge which necessitated a more general set-up of pinched polynomial-like maps <cit.>. 

The presence of parabolic elements is also an important trait of all the groups that appear in our mating theory, which brings us to the general theory of pinched polynomial-like maps (that generalizes a theory that has been recently developed in <cit.>). 


The principal results of this paper can be summarized as follows.


  *  We prove the existence of conformal matings between `generic' polynomials (with connected Julia set) and large classes of Fuchsian/reflection groups. Such matings are realized both as maps in one complex variable as well as algebraic correspondences (Theorems <ref>, <ref>, <ref>, <ref>).


  * We introduce the class of degenerate polynomial-like maps that unifies the above mating constructions. This framework allows us

    
  * to interpret the mating structure of algebraic correspondences in terms of the classical description of polynomial-like maps as matings of `hybrid classes' and `external maps' (Sections <ref>, <ref>, <ref>, <ref>), and

    
  * to prove general bijection/homeomorphism statements between connectedness loci of polynomials and various parameter spaces of matings (Theorems <ref>, <ref>, <ref>, <ref>, <ref>).


In item (<ref>) above, the term `generic' refers to the collection of at most finitely renormalizable, periodically repelling polynomials and geometrically finite polynomials.

Before elaborating on the main theorems, let us briefly mention some of the technical novelties involved in carrying out the tasks mentioned above. 



  * We use the structure of degenerate polynomial-like maps to adapt classical `puzzle piece' machinery for our setting. This plays a pivotal role in establishing various combinatorial continuity and rigidity results, which lie at the heart of our conformal mateability theorems (Sections <ref> and <ref>).


  * The construction of algebraic correspondences (global objects) from one variable matings between polynomials and groups (i.e., certain degenerate polynomial-like maps that are semi-global objects) requires algebraic descriptions of such matings. This is straightforward for Schwarz reflection maps, which are known to be `rationally semi-conjugate' to reflection in the unit circle. To deal with the holomorphic situation, we introduce a class of meromorphic maps called B-involutions, and give an explicit algebraic description to facilitate the task of globalization (see Section <ref>, cf. <cit.>).


  * In order to manufacture algebraic correspondences that are matings between generic polynomials and groups with multiple parabolics, we need to define such correspondences on nodal spheres or cacti of spheres. This extends previously
known mating frameworks for correspondences; and in particular, gives first examples of correspondences associated with piecewise Schwarz reflection maps (Sections <ref>, <ref>). Even for the simplest examples of piecewise Schwarz reflection maps (namely, the Circle-and-Cardioid Schwarz reflections studied in <cit.>), such a correspondence description is new.


In Appendix <ref>, we investigate canonical product structures in certain families of degenerate polynomial-like maps. This product structure is based on the classical idea of decomposing a polynomial-like map into a hybrid class and an external map <cit.>. Loosely speaking, these families bind connectedness loci of polynomials (parameter space of hybrid classes) and Teichmüller spaces of certain surfaces (parameter space of external maps) together in a dynamically natural fashion. A more detailed analysis of the boundary of such families of degenerate polynomial-like maps and the associated degeneration phenomena will be presented in a subsequent work.




 §.§ Antiholomorphic matings: Schwarz reflections

A domain Ω in the Riemann sphere is called a quadrature domain if it admits an anti-meromorphic map S:Ω→ that continuously extends to the identity map on the boundary ∂Ω. Such a map  S:Ω→ is called the Schwarz reflection map of Ω. A simply connected domain Ω⊊ is a quadrature domain if and only if its Riemann uniformization ϕ:→Ω extends as a rational map of the Riemann sphere. 

A quadrature multi-domain  is the union of finitely many disjoint simply connected domains {Ω_j}_j=1^k admitting a continuous map S:→ which is anti-meromorphic on  and which fixes ∂ pointwise. Each component of a quadrature multi-domain is a quadrature domain. The map S, which we call a Schwarz reflection of a quadrature multi-domain, acts as the Schwarz reflection of Ω_j on each Ω_j, for j∈{1,⋯,k}.


The compact set T(S):=∖ is called a droplet. Its boundary consists of real-analytic curves possibly with finitely many cusp and double point singularities. Removing these finitely many singular points from the droplet yields the desingularized droplet or the fundamental tile of S. We denote the fundamental tile of S by T^0(S).

The dynamical plane of S admits an invariant partition into the escaping/tiling set T^∞(S) and the non-escaping set K(S). The former set  comprises all points that eventually escape to the fundamental tile T^0(S), and the latter set is its complement (see Section <ref> for precise definitions).


The exploration of Schwarz reflection dynamical systems was initiated in <cit.>, and it was observed that in many cases, the Schwarz reflection map S of a quadrature multi-domain is a combination/mating of an anti-holomorphic polynomial (anti-polynomial for short) and an ideal polygon reflection group. Roughly, a Schwarz reflection S is called a mating of an anti-polynomial and  an ideal polygon reflection group if the dynamics of S on its non-escaping set is topologically conjugate (conformally conjugate on the interior) to that of an anti-polynomial on its filled Julia set, while the action of S on the tiling set is conjugate to a piecewise anti-Möbius map, called the Nielsen map, that is orbit equivalent to the ideal polygon reflection group (see Definition <ref> for the precise notion of conformal mating and Figure <ref> for an illustration of the Nielsen map of the ideal triangle reflection group). The Nielsen map of an ideal (d+1)-gon reflection group is also compatible with degree d anti-polynomials as the action of the Nielsen map on the unit circle is topologically conjugate to the map z^d.
In <cit.>, the so-called Circle-and-Cardioid (or C&C) family of Schwarz reflections was studied in details, and it was shown that many maps in the C&C family arise as matings of quadratic anti-polynomials and the ideal triangle reflection group.

In the following, we will discuss the general theory of two natural classes of Schwarz reflections:


  * polygonal Schwarz reflections; and

  * anti-Farey Schwarz reflections.




  §.§.§ Polygonal Schwarz reflections


We say that  is a tree-like quadrature multi-domain if  is connected and simply connected. In this case, each Ω_j is a Jordan domain, and their touching structure can be recorded by a plane tree (see Figure <ref>).
A Schwarz reflection S:→ associated with a tree-like quadrature multi-domain is polygonal if the fundamental tile T^0(S) is conformally equivalent to an ideal (d+1)-gon in the hyperbolic plane (where d is the degree of the Schwarz reflection), and S has no critical value in T^0(S). The terminology `polygonal' is motivated by the fact that a polygonal Schwarz reflection is conformally conjugate to the Nielsen map of an ideal polygon reflection group near its droplet (see Section <ref> for more details).
We remark that Schwarz reflections in the C&C family are prototypical examples of polygonal Schwarz reflections. 


As in the degree two case, a degree d polygonal Schwarz reflection map S has a connected non-escaping set if and only if no critical point of S escapes to the fundamental tile. This property is also equivalent to the fact that the tiling set dynamics of S is conformally conjugate to the Nielsen map of an ideal (d+1)-gon reflection group. 



We denote the connectedness locus of (normalized) degree d polygonal Schwarz reflections by 𝒮__d. Similarly, we denote by 𝒞^-_d the connectedness locus of monic and centered anti-polynomials of degree d. We say that two maps f, g∈𝒞^-_d (respectively, f, g∈𝒮__d) are combinatorially equivalent, denoted by f∼ g, if they have the same rational laminations (in analogy with polynomial dynamics, the rational lamination of a polygonal Schwarz reflection captures the co-landing patterns of its pre-periodic dynamical rays, see Definition <ref>).

Let 𝒞^-_d, r⊂𝒞^-_d be the subset of periodically repelling anti-polynomials. For polygonal Schwarz reflections, the set 𝔖 of singular (fixed) points on the droplet boundary is special as these points always have parabolic dynamics. Hence, we define the analogous subspace 𝒮__d, r for polygonal Schwarz reflections as the collection of those maps S in 𝒮__d for which no point of 𝔖 has an attracting direction in K(S) and each periodic point of S not in 𝔖 is repelling. Finally, let 𝒞^-_d, r = 𝒞^-_d, r/∼ (respectively, 𝒮__d, r = 𝒮__d, r/∼) be the space of corresponding combinatorial classes.

The following theorems generalize and sharpen the main results of <cit.>. We remark that in <cit.>, only the space of geometrically finite degree two polygonal Schwarz reflections were studied.


There is a dynamically natural homeomorphism 

    Φ: 𝒞^-_d, r⟶𝒮__d, r.

Here `dynamically natural' means that the circle homeomorphism conjugating the Nielsen map to the power map z^d carries the rational lamination of a Schwarz reflection to the rational lamination of an anti-polynomial.


(See Section <ref> for the proof of Theorem <ref>.)

Let 𝒮__d, fr⊂𝒮__d, r be the subset of at most finitely renormalizable periodically repelling Schwarz reflections, and let 𝒞^-_d, fr⊂𝒞^-_d,r be its counterpart for anti-polynomials.
Adapting similar methods as in the polynomial setting, we show that each map in 𝒮__d, fr (or 𝒞^-_d, fr) is combinatorially rigid; i.e., its combinatorial class consists of a single map.
As an immediate corollary of this rigidity, we have the following result, which is proved in Section <ref>.

There is a dynamically natural homeomorphism 

    Φ: 𝒞^-_d, fr⟶𝒮__d, fr.

Here dynamically natural means that for any f∈𝒞^-_d, fr, the Schwarz reflection Φ(f) is the unique conformal mating of f and the Nielsen map of the ideal (d+1)-gon reflection group.









It is natural to ask if the above natural homeomorphism can be extended to a homeomorphism between 𝒞^-_d and 𝒮__d.
The next result shows that it is not the case.

There is a dynamically natural bijection 

    Φ: 𝒞^-_d, gf⟶𝒮__d, gf
 
between geometrically finite anti-polynomials and polygonal Schwarz reflections with connected Julia/limit set, such that for any f∈𝒞^-_d, gf, the Schwarz reflection Φ(f) is the unique conformal mating of f and the Nielsen map of the ideal (d+1)-gon reflection group.
However, both Φ and Φ^-1 are discontinuous.


The existence of the bijection Φ appearing in Theorem <ref> is established in Section <ref>, and the discontinuity of Φ, Φ^-1 is proved in Section <ref>.


The above theorem is analogous to a famous theorem of Kerckhoff-Thurston which states that the natural identification between two different Bers embeddings fails to extend to a continuous map on their boundaries.

It is worth emphasizing that the map Φ that relates the connectedness loci 𝒞^-_d and 𝒮__d (appearing in the previous two theorems) replaces the external dynamics of an anti-polynomial f (i.e., the dynamics of f on its basin of infinity) with the action of the Nielsen map of an ideal (d+1)-gon reflection group, or equivalently, mates the filled Julia set dynamics of f with the dynamics of the Nielsen map. The following implication of Theorems <ref>, <ref> and <ref> is important in its own right (see Definition <ref> for the definition of combinatorial mating).


Let f be a degree d anti-polynomial with connected Julia set.
Suppose that f is either 


  * geometrically finite; or 

  * periodically repelling, finitely renormalizable.

Then, f can be conformally mated with the Nielsen map of an ideal (d+1)-gon reflection group, and the resulting Schwarz reflection map is unique up to Möbius conjugacy.

Further, each periodically repelling map in 𝒞_d^- can be combinatorially mated with the Nielsen map of an ideal (d+1)-gon reflection group.





We remark that notions of mating without assuming local connectivity of Julia sets also appeared <cit.>.



  §.§.§ Anti-Farey Schwarz reflections

In the antiholomorphic world, there is another important expansive piecewise real-analytic circle endomorphism, called the anti-Farey map, that is closely related to reflection groups. The anti-Farey map, which was first formally introduced in <cit.> (and had its precursors in <cit.>), is a factor of the Nielsen map of a regular ideal polygon reflection group (see Section <ref>). Schwarz reflection maps that admit the anti-Farey map as the conformal model of their escaping set dynamics will be referred to as anti-Farey Schwarz reflections.


The proofs of Theorems <ref>–<ref> can be adapted for the family of anti-Farey Schwarz reflections, demonstrating close parameter space relations between the space of anti-Farey Schwarz reflections and the connectedness loci of anti-polynomials. We refer the reader to Theorem <ref> for the precise statements. As a consequence of these results, we have the following generalization of <cit.>.

Let f be a degree d anti-polynomial with connected Julia set.
Suppose that f is either 


  * geometrically finite; or 

  * periodically repelling, finitely renormalizable.

Then, f can be conformally mated with the degree d anti-Farey map, and the resulting Schwarz reflection map is unique up to Möbius conjugacy.

Further, each periodically repelling map in 𝒞_d^- can be combinatorially mated with the degree d anti-Farey map.




 §.§ Holomorphic matings: B-involutions

We now pass to the holomorphic analog of Schwarz reflection maps that we will study in this paper. 
In order to develop a mating framework for complex polynomials and Fuchsian groups, we define a suitable class of holomorphic maps which we call B-involutions. 
Special cases of B-involutions were introduced in <cit.> in the context of mating parabolic rational maps with Hecke groups (cf. Section <ref>).

An inversive multi-domain  is the union of finitely many disjoint simply connected domains {Ω_j}_j=1^k admitting a continuous map S:→ which is meromorphic on  and which restricts to an orientation-reversing order two self-homeomorphism of the boundary ∂. The map S is called a B-involution of an inversive multi-domain (see Section <ref> for a precise definition). 
We use a conformal welding argument to give an algebraic characterization of inversive multi-domains and B-involutions in terms of uniformizing rational maps. This parallels the characterization of quadrature domains as univalent images of disks under global rational maps (see Theorem <ref>).






To study the dynamics of a B-involution of an inversive multi-domain, we partition its dynamical plane into escaping and non-escaping sets (see Section <ref>). The action of a B-involution of an inversive multi-domain on its escaping set will be referred to as its external dynamics.

Natural examples of such B-involutions are provided by matings of complex polynomials with connected Julia set and certain orientation-preserving external maps, called factor Bowen-Series maps (see Theorem <ref> and Proposition <ref>). Factor Bowen-Series maps, which first appeared in <cit.>, can be thought of as holomorphic cousins of Nielsen or anti-Farey maps. These maps are semi-conjugates of classical Bowen-Series maps associated with certain genus zero orbifolds (more precisely, with the Fuchsian models of such orbifolds); in particular, they are piecewise analytic, expansive, circle coverings of degree greater than one. A formal description of factor Bowen-Series maps is given in Section <ref>.

Our analysis shows that spaces of B-involutions having factor Bowen-Series maps as the conformal model of their external dynamics are closely related to the connectedness loci of complex polynomials (see Theorem <ref>). As a consequence of  Theorem <ref>, we have an analog of Theorem <ref> that confirms the existence of conformal matings of generic complex polynomials (with connected Julia set) and appropriate factor Bowen-Series maps.




Let f be a degree d polynomial with connected Julia set.
Suppose that f is either 


  * geometrically finite; or 

  * periodically repelling, finitely renormalizable.

Then, f can be conformally mated with any degree d factor Bowen-Series map, and the resulting B-involution is unique up to Möbius conjugacy. 

Further, each periodically repelling degree d polynomial with connected Julia set can be combinatorially mated with any degree d factor Bowen-Series map.





 §.§ Matings as degenerate (anti-)polynomial-like maps


To provide a common language for all the mating results stated above, we now put forward an appropriate generalization of polynomial-like maps, called degenerate (anti-)polynomial-like maps. Pinched versions of polynomial-like maps were introduced in <cit.>, and more general notions (still not sufficient for the purpose of the current paper) appeared in the papers mentioned above.

A polygon is a closed Jordan disk in  with a piecewise smooth boundary. The points where two smooth local boundary arcs meet are called the corners of the polygon (note that cusps are also considered to be corners).

A pinched polygon is a set in  which is homeomorphic to a closed disk quotiented by a finite geodesic lamination, and which has a piecewise smooth boundary. The cut-points of a pinched polygon will be called its pinched points, and the non-cut points where two smooth local boundary arcs meet are called the corners of the pinched polygon. 


Let P_1, P_2⊂ be pinched polygons such that 


  * P_1⊂ P_2,

  * the pinched points (respectively, corners) of P_2 are also pinched points (respectively, corners) of P_1, and

  * ∂ P_1∩∂ P_2 consists of the corners and pinched points of P_2.

Suppose that there is a degree d continuous map g:P_1 → P_2 which is a proper holomorphic/anti-holomorphic map from each component of P_1 onto some component of P_2, and such that the singular points of ∂ P_1 are the g-preimages of the singular points of ∂ P_2.



We then call the triple (g, P_1, P_2) a degenerate (anti-)polynomial-like map of degree d.



As with polynomial-like maps, we define the filled Julia set or non-escaping set of a degenerate (anti-)polynomial-like map to be 

    K(g)=⋂_n≥ 0 g^-n( P_1).
 
Analogous to classical polynomial-like maps, the filled Julia set K(g) of a degenerate (anti-)polynomial-like map is connected if and only if it contains all of the critical values of g.

For a degree d degenerate (anti-)polynomial-like map (g, P_1, P_2) with connected filled Julia set, one can define the notion of an external map following the classical construction of external maps of polynomial-like maps (cf. <cit.>). Note that the external map of such a g is a piecewise real-analytic, degree d, covering map of 𝕊^1. Since the fundamental domain P_2∖P_1 of g is a pinched annulus, the external map of g is in general not expanding on 𝕊^1. This causes significant difficulty in the study of degenerate polynomial-like maps.
In fact, the external maps of all degenerate polynomial-like maps that we will encounter in this paper have parabolic cycles and hence are not expanding. However, they enjoy the weaker property of being expansive on 𝕊^1.


Polygonal or anti-Farey Schwarz reflections and B-involutions are some natural examples of degenerate (anti-)polynomial-like maps.
For concreteness, consider S:→ as a polygonal Schwarz reflection map.
The restriction S:S^-1()→ is a finite degree proper anti-holomorphic map from a pinched polygon to a larger pinched polygon (see Figure <ref>). We note that S^-1() is not compactly contained in . Thus, the map S:S^-1()→ is a degenerate anti-polynomial-like maps as in Definition <ref>.

The description of S as a mating of an anti-polynomial and an ideal polygon reflection group can be recast in the classical language of polynomial-like maps: the internal class of S:S^-1()→ is that of a degree d anti-polynomial with connected Julia set and its external class is represented by the Nielsen map of an ideal (d+1)-gon reflection group.
The parameter space homeomorphisms appearing in the theorems stated in the previous subsections can be regarded as straightening theorems between slices of degenerate (anti-)polynomial-like maps and connectedness loci of (anti-)polynomials. 






 §.§ Matings as (anti-)holomorphic correspondences


The first obstruction to mating a polynomial map with a reflection/Fuchsian group is the fundamental mismatch between invertible and non-invertible dynamics. 
The passage from a group to its Nielsen map, anti-Farey map or factor Bowen-Series map (i.e., reduction from group dynamics to semi-group dynamics) allowed us to address this discord and combine the two dynamical systems into a single plane. However, as the conformal mating is a non-invertible map, its dynamics does not showcase the whole structure of the underlying group.

It was originally proposed by Fatou in the 1920s that the category of dynamical systems which truly encompasses rational dynamics as well as actions of Kleinian group is given by algebraic correspondences <cit.>. The first precise formulation of this proposal was given by Bullett and Penrose <cit.>, who studied certain algebraic correspondences of bi-degree 2:2 on the Riemann sphere that combine the actions of quadratic polynomials with the modular group in a precise sense. 


Let f be a degree d (anti-)polynomial with connected Julia set and let G be a discrete subgroup of (anti-)conformal automorphism group of the unit disk .
Let ℭ be an (anti-)holomorphic correspondence on a compact, simply connected (possibly noded) Riemann surface 𝔚.

We say ℭ a mating of f and G if there is a ℭ-invariant partition 𝔚=⊔ such that the following hold.

	
  * On , the dynamics of ℭ is equivalent to the action of a group of (anti-)conformal automorphisms acting properly discontinuously. Further, /ℭ is biholomorphic to /G.
	
	
  *  can be written as the union of two copies _1, _2 of (f) (where (f) is the filled Julia set of f), such that _1 and _2 intersect in finitely many points. Furthermore, ℭ has a forward (respectively, backward) branch carrying _1 (respectively, _2) onto itself with degree d, and this branch is conformally (respectively, anti-conformally) conjugate to f|_𝒦(f). 



It turns out that in all three cases we considered in this paper, we can define (anti-)holomorphic correspondences that combine actions of (anti-)polynomials with the corresponding groups.
Specifically, such a correspondence is manufactured by pulling back the dynamics of the mating via the rational maps that uniformize the associated simply connected quadrature/inversive domains.
The first instance of this construction appeared in <cit.>, where matings of z^d with regular ideal polygon reflection groups were realized as antiholomorphic correspondences. 
Our next theorem provides a systematic way of producing such examples.


Let G be an ideal (d+1)-gon reflection group or the anti-Hecke group.

Let f be a degree d anti-polynomial with connected Julia set which is either


  * geometrically finite; or 

  * periodically repelling, finitely renormalizable.


Then there exists an antiholomorphic correspondence ℭ which is a mating of f and G.

Similarly, let Σ be a hyperbolic orbifold of genus zero with arbitrarily many (at least one) punctures, at most one order two orbifold point, and at most one order ν≥ 3 orbifold point, with the corresponding Fuchsian group G.
Let 

    d(Σ)=
    1-2ν·χ_orb(Σ)  if Σ has an order ν≥ 3 orbifold point,
    
    1-2χ_orb(Σ)  otherwise.
 
Let f be a degree d(Σ) polynomial with connected Julia set which is either


  * geometrically finite; or 

  * periodically repelling, finitely renormalizable.

Then there exists a holomorphic correspondence ℭ which is a mating of f and G.


The antiholomorphic parts of Theorem <ref> are proved in Sections <ref>, <ref>, and the holomorphic part of it is proved in Section <ref>.

We remark that the correspondences produced by Theorem <ref> are reversible; i.e., their forward branches are (anti-)conformally conjugate to their backward branches. This property facilitates the dynamical study of such correspondences.



 §.§ Connections with known combination theorems for correspondences




  §.§.§ Special cases of Theorem <ref>

We now list several known results in the literature that are generalized by Theorem <ref>.


  * The algebraic correspondences that arise as matings of quadratic polynomials and the modular group are precisely the bi-degree 2:2 correspondences introduced by Bullett and Penrose <cit.> (cf. <cit.>). Bullett and Haïssinsky showed in <cit.> that this space of correspondences contains matings of Collet-Eckmann quadratic polynomials (with connected Julia set) and the modular group.



  * The antiholomorphic correspondences that arise from Theorem <ref> as matings of quadratic anti-polynomials and the anti-Hecke group isomorphic to /2∗/3 were investigated in <cit.>.


  * Some examples of correspondences realizing matings of degree d complex polynomials with the Hecke group (isomorphic to /2∗/(d+1)) were furnished in <cit.>. In <cit.>, it was conjectured that all complex polynomials with connected Julia set can be mated with the corresponding Hecke group. Theorem <ref> proves this conjecture for all  geometrically finite polynomials and finitely renormalizable, periodically repelling polynomials.


  * In <cit.>, correspondences were constructed as matings of degree d semi-hyperbolic anti-polynomials (i.e., anti-polynomials with no parabolic cycles and no recurrent Julia critical points) with connected Julia set and the anti-Hecke group (isomorphic to /2∗/(d+1)).


  * In <cit.>, correspondences were constructed as matings of semi-hyperbolic polynomials with connected Julia set and the corresponding Fuchsian group under an additionally imposed real-symmetry condition. These examples also become special cases of Theorem <ref>.





  §.§.§ (Anti-)polynomials vs parabolic (anti-)rational maps


We now compare Theorem <ref> with related (but technically different) constructions of algebraic correspondences as matings of parabolic (anti-)rational maps and (anti-)Hecke groups.

The external maps arising from reflection/Fuchsian groups (namely, Nielsen, anti-Farey, and factor Bowen-Series maps) have at least one parabolic fixed point on the circle, and hence are not quasisymmetrically conjugate to the power map z^d or z^d. This discrepancy prohibits one from applying standard quasiconformal surgery tools to mate Nielsen/anti-Farey/factor Bowen-Series maps with (anti-)polynomials. In other words, the conformal matings described in Theorems <ref>, <ref>, and <ref> cannot be constructed by quasiconformal surgery.

However, it turns out that the anti-Farey maps and the factor Bowen-Series maps associated with Hecke groups are somewhat special as they have a unique parabolic fixed point on the circle. Equivalently, the (anti-)Hecke groups have a unique conjugacy class of parabolic elements. This implies that the above maps are quasisymmetrically conjugate to parabolic (anti-)Blaschke products since the latter class of maps also have a unique parabolic fixed point on the circle. This quasisymmetric compatibility property was utilized


  * to prove that every quadratic parabolic rational map (with connected Julia set) can be mated with the modular group as a holomorphic correspondence <cit.>,


  * to construct anti-holomorphic correspondences as matings of arbitrary degree d parabolic anti-rational maps (with connected Julia set) with anti-Hecke groups <cit.>, and 


  * to construct holomorphic correspondences as matings of arbitrary degree d parabolic rational maps (with connected Julia set) with Hecke groups <cit.>.


To conclude, let us note that for semi-hyperbolic (anti-)polynomials with connected Julia set, the conformal matings furnished by Theorems <ref>, <ref>, and <ref> have David regularity. Specifically, for a semi-hyperbolic (anti-)polynomial f with connected Julia set, the topological conjugacy (conformal on the interior) from the filled Julia set dynamics of f to the non-escaping set dynamics of the conformal mating between f and the Nielsen/anti-Farey/factor Bowen-Series map is the restriction of a global David homeomorphism (cf. <cit.>, <cit.>). In general, we ask the following question:


What is the regularity of the mating between a general periodically repelling, finitely renormalizable (anti-)polynomial  and the Nielsen/anti-Farey/factor Bowen-Series map?




 §.§ Proof techniques

Here we present an overview of the techniques employed in this paper:

    
  * Utilizing the David surgery method, we construct matings of post-critically finite (anti-)polynomials / geometrically finite (anti-)polynomials with appropriate ideal polygon reflection groups / anti-Hecke groups / Fuchsian genus zero orbifold groups.
    It is important to note that while David surgery is a powerful tool, there are analytic and geometric constraints, as the current techniques only apply to John domains.
    
  * To extend our constructions to more general setting, we take limits of post-critically finite matings.
    These matings emerge naturally as degenerate (anti-)polynomial-like maps.
    As a preliminary step, we prove a boundedness result for the connectivity locus of such degenerate (anti-)polynomial-like maps.
    
  * The structure of degenerate (anti-)polynomial-like maps enables us to adapt puzzle techniques, which allow us to establish combinatorial continuity of the association between periodically repelling (anti-)polynomials and their corresponding matings.
    
  * By dividing the plane into pieces using puzzles, we apply generalized renormalization methods to establish combinatorial rigidity for periodically repelling, finitely renormalizable matings.

These results permit us to conclude the existence of conformal matings and combinatorial matings, as well as continuity and combinatorial continuity of `straightening maps' between the parameter spaces.
Notably, the fixed ray puzzles give a natural stratification of the space of Schwarz reflections. 
This stratification leads to discontinuity of the straightening map between geometrically finite parameters.
Furthermore, we derive algebraic descriptions for these matings, which enable us to transfer the previous mating results to the framework of correspondences.






Acknowledgements. Part of the work on this project was carried out during the authors' visit to The Fields Institute, Urgench State University, and Simons Laufer Mathematical Sciences Institute (MSRI).



Notation: 


  * For X⊂, the sets ∂ X, X^c stand for the boundary and the complement of X in .


  * η^+(z)=1/z.


  * η^-(z)=1/z.


  * ^*=∖.


  * B(a,r):={| z-a|<r}, where a∈ and r>0.


  * B(a,r):={| z-a|≤ r}, where a∈ and r>0.





  * Aut() = the group of all Möbius automorphisms of ,
 
Aut^±() = the group of all Möbius and anti-Möbius automorphisms of .






PART: 
Antiholomorphic world





§ REFLECTION GROUPS AND ASSOCIATED EXTERNAL MAPS


We denote the regular (d+1)-gon reflection group by G_d. It is generated by reflections ρ_1,⋯,ρ_d+1 in the circles C_1,⋯,C_d+1 such that C_i∩ is a hyperbolic geodesic of  connecting ω^i-1 and ω^i, where ω:=e^2π i/d+1. Let Π be the ideal polygon in  with vertices at the (d+1)-st roots of unity. Then Π is a fundamental domain for the G_d-action on . 

The group of conformal and anti-conformal automorphisms of  generated by G_d and ⟨ M_ω⟩, where M_ω(z)=ω z and ω:=e^2π i/d+1, is called the anti-Hecke group. We denote the anti-Hecke group by G_d. It is an index d+1 extension of G_d.





 §.§ Nielsen map
 The Nielsen map _d associated with the regular ideal polygon group G_d is defined as

    _d:∖Π⟶,   z↦ρ_i(z)  if   z∈H_i,

where H_i is the hyperbolic half-plane cut out by the bi-infinite geodesic C_i∩ and the counter-clockwise circular arc ω^i-1,ω^i, for i∈{1,⋯,d+1} (see Figure <ref>). The Nielsen map _d fixes ∂Π pointwise, and restricts to an orientation-reversing, C^1, expansive, degree d covering map of 𝕊^1. Hence, it is topologically conjugate to the map z^d: 𝕊^1 ⟶𝕊^1. There is a unique such conjugacy that sends 1 to 1. We denote this map by ℰ_d:𝕊^1→𝕊^1, and call it the d-th Minkowski circle homeomorphism.



 §.§ Anti-Farey map


Let us now associate a circle endomorphism with the anti-Hecke group G_d.
Roughly speaking, we first pass to the index d subgroup G_d⩽G_d, look at its Nielsen map _d, and then pass to a semi-conjugate (or factor) of the Nielsen map using the torsion element M_ω.

More precisely, observe that due to the rotational symmetry of the fundamental domain Π, the Nielsen map _d of commutes with M_ω. This allows one to construct a factor dynamical system

    _d:(∖Π)⟨ M_ω⟩⟶⟨ M_ω⟩

that is semi-conjugate to _d:∖Π→ via the quotient map → /⟨ M_ω⟩.


Finally, as z↦ z^d+1 defines a conformal isomorphism ξ:/⟨ M_ω⟩⟶, we have the so-called degree d anti-Farey map

    _d:= ξ∘_d∘ξ^-1:∖ℋ→,

where ℋ:=ξ(Π/⟨ M_ω⟩) (see Figure <ref>).

The fact that _d fixes ∂Π pointwise, implies that _d fixes ∂ℋ pointwise. Moreover, the map _d restricts to an expansive, orientation-reversing, degree d, piecewise anti-analytic covering of 𝕊^1. Hence, _d|_𝕊^1 is topologically conjugate to z^d_𝕊^1.

The anti-Farey map _d has a couple of key differences from the Nielsen map _d. These dissimilarities come from passing to a factor dynamical system. Note that the piecewise Möbius map _d has no critical point, but the anti-Farey map has a unique critical point (of multiplicity d) at ξ(ρ_1(0)) with associated critical value at 0. Further, while the Nielsen map has d+1 parabolic fixed points on the circle, the anti-Farey map has a unique parabolic fixed point on 𝕊^1 (at the point 1).


We refer the reader to <cit.> for a detailed discussion of the anti-Farey map and the anti-Hecke group.


The idea of constructing an external map as a factor of another will also be used in Section <ref> to define factor Bowen-Series maps.




§ SCHWARZ REFLECTION MAPS




 §.§ Simply connected quadrature domains



A domain Ω⊊ with ∞∉∂Ω and int(Ω)=Ω is called a quadrature domain if there exists a continuous function S:Ω→ satisfying the following two properties:


  * S=id on ∂Ω, and


  * S is anti-meromorphic on Ω.


The map S is called the Schwarz reflection map of Ω.


The boundary of a quadrature domain is real-analytic with conformal cusps and double points as only possible singularities <cit.>. Thus, the map S can be regarded as a semi-global anti-meromorphic extension of the local Schwarz reflection maps with respect to the non-singular points of ∂Ω.




<cit.>,<cit.>
A simply connected domain Ω⊊ (with ∞∉∂Ω and int(Ω)=Ω) is a quadrature domain if and only if the Riemann map ϕ:𝔻→Ω is rational.
In this case, the Schwarz reflection map S of Ω is given by ϕ∘η^-∘(ϕ|_)^-1. 

Moreover, if the degree of the rational map ϕ is d, then S:S^-1(Ω)→Ω is a (branched) covering of degree (d-1), and S:S^-1(Ω^c)→Ω^c is a (branched) covering of degree d.


    d[swap]η^-rϕ   ΩdS
    rϕ   ℂ
 

We will call the degree of the rational map ϕ the degree of the associated quadrature domain Ω.



 §.§ Schwarz reflections in quadrature multi-domains



Let {Ω_1,⋯,Ω_k} be a finite collection of disjoint simply connected domains with Ω_j⊊, ∞∉∂Ω_j and int(Ω_j)=Ω_j, for j=1,⋯,k. We call =_j=1^k Ω_j a quadrature multi-domain if there exists a continuous function S:→ satisfying the following two properties:


  * S=id on ∂, and


  * S is anti-meromorphic on .


The map S is called the Schwarz reflection map of the quadrature multi-domain .


Note that for a quadrature multi-domain :=_j=1^kΩ_j, each component Ω_j is a quadrature domain and S|_Ω_j agrees with the Schwarz reflection map S_j of Ω_j. 

Let ϕ_j:→Ω_j be the Riemann uniformizations of the simply connected quadrature domains Ω_j such that each ϕ_j extends as a rational map of  of degree d_j. It follows that S_j:S_j^-1(Ω_j)→Ω_j is a branched covering of degree (d_j-1), and S_j:S_j^-1(Ω_j^c)→Ω_j^c is a branched covering of degree d_j. Therefore, S:S^-1()→ is a branched covering of degree 

    d:=∑_j=1^k d_j-1.

On the other hand, S:S^-1(^c)→^c is a branched covering of degree d+1.
We refer to S as a degree d Schwarz reflection map.

Set 

    T(S):=∖,  and   T^0(S):=T∖𝔖,

where 𝔖 is the set of singular points (cusps and double points) on ∂ T(S). We call the components of T^0(S) the fundamental tile and the components of S^-n(T^0(S)) the tiles of rank n.
The escaping/tiling set T^∞(S) of S is defined as the union of all tiles; i.e., 

    T^∞(S):=⋃_n=0^∞ S^-n(T^0(S)).
 
The non-escaping set of S is defined as K(S):=∖ T^∞(S). A connected component of K(S) is called a Fatou component of S. The common boundary of K(S) and T^∞(S) is called the limit set of S, and is denoted by Λ(S).



 §.§ Conformal mating and Schwarz reflections



Let f be a monic, centered complex polynomial of degree d with a connected and locally connected Julia set. We denote the Böttcher coordinate of f by

    ψ_f:∖→ℬ_∞(f):=∖𝒦(f).
 
This map, which conjugates z^d to f, is normalized to be tangent to the identity map near infinity.
As 𝒥(f) is locally connected, ψ_f extends continuously to 𝕊^1 to yield a semi-conjugacy between z^d|_𝕊^1 and f|_𝒥(f). 
By Section <ref>, there exists a homeomorphism 𝔥:𝕊^1→𝕊^1 that conjugates z^d to _d (respectively, _d). We normalize 𝔥 so that it sends the fixed point 1 of z^d to the fixed point 1 of _d (respectively, _d).

We define an equivalence relation ∼ on 𝒦(f) generated by 

    ψ_f(ζ)∼𝔥(ζ), for all z∈𝕊^1.
 

We now precise what it means to conformally mate f with the Nielsen map _d (the case of the anti-Farey map _d is exactly the same).


The maps f and _d are said to be conformally mateable if there exist a continuous map S: Dom(S)⊊→ (called a conformal mating of _d and f) that is complex-analytic in the interior of Dom(S) and continuous maps 
	
    𝔛_f:𝒦(f)→ and 𝔛__d: →,

	conformal on 𝒦(f) and  (respectively), satisfying
	
		
  *  𝔛_f(𝒦(f))∪𝔛_Σ() =,
		
		
  *  Dom(S)= 𝔛_f(𝒦(f))∪𝔛__d(∖Π),
		
		
  * 𝔛_f∘ f(z) = S∘𝔛_f(z),  for z∈𝒦(f),
		
		
  * 𝔛__d∘_d (w) = F∘𝔛__d(w),  for w∈∖Π,  and 
		
		
  *  𝔛_f(z)=𝔛__d(w) if and only if z∼ w where ∼ is the equivalence relation on 𝒦(f)⊔ defined by Relation (<ref>).
	


Using David surgery techniques, it was proved in <cit.> that many anti-polynomials with connected Julia set can be conformally mated with the Nielsen/anti-Farey map. Crucially, the property of _d:∖Π⟶ (respectively, of _d:∖ℋ⟶) that it fixes ∂Π (respectively, ∂ℋ) pointwise, translates to the fact that the resulting conformal matings are Schwarz reflection maps.

<cit.><cit.>
Let f be a degree d hyperbolic anti-polynomial with connected Julia set.
Then, there exists a conformal mating S of f:𝒦(f)→𝒦(f) and _d:∖Π⟶ (respectively, _d:∖ℋ⟶). Moreover, S is a Schwarz reflection map in a quadrature multi-domain.







§ POLYGONAL SCHWARZ REFLECTION MAPS




 §.§ Basic definitions and properties


We continue to use the notation of the previous section.
Suppose further that  is connected and simply connected. Since the boundary of a quadrature domain consists of real-analytic curves possibly with cusps and double point singularities, no more than two distinct components of  can touch at a point. Thus, the above condition is equivalent to following ones.


  * Each Ω_j is a Jordan domain, and the contact graph of the quadrature multi-domain (i.e., a graph having a vertex for each Ω_j and an edge connecting two vertices if the corresponding quadrature domains touch) is a tree.

  *  is a pinched polygon, and the associated finite geodesic lamination contains no polygon.

We will refer to such a  as a tree-like quadrature multi-domain, and call the Schwarz reflection of the tree-like quadrature multi-domain  a Schwarz reflection of a pinched polygon.


For a tree-like quadrature multi-domain, the desingularized droplet T^0(S) is homeomorphic to an ideal polygon in the hyperbolic plane.


A degree d Schwarz reflection map S:→ of a pinched polygon 
is said to be polygonal if  T^0(S) is homeomorphic to an ideal (d+1)-gon in  and S has no critical values in T^0(S).
 
A degree d polygonal (piecewise) Schwarz reflection map is called regular polygonal if T^0(S) is conformally equivalent to the regular ideal (d+1)-gon Π.  
We denote by 𝒮_d the space of degree d regular polygonal Schwarz reflections.

We will often omit the word `regular' for brevity. We normalize a polygonal Schwarz reflection S so that the conformal map between Π and T^0(S) sends the origin to ∞ and has asymptotics z↦ 1/z+O(z) as z→ 0.


In the above definition, the condition that T^0(S) is homeomorphic to an ideal (d+1)-gon in  is equivalent to requiring that the sum of the number of cusps and twice the number of double points on ∂ is equal to d+1.


The following result justifies the nomenclature `polygonal Schwarz reflection maps'. Indeed, it shows that such maps are characterized by having the Nielsen map of a polygonal reflection group as the conformal model of their dynamics near the droplet.


Let S:→ be a degree d Schwarz reflection map of a pinched polygon. Then, S is regular polygonal if and only if S:S^-1(T^0(S))→ T^0(S) is conformally conjugate to _d:_d^-1(Π)→Π.


It is clear that if S:S^-1(T^0(S))→ T^0(S) is conformally conjugate to _d:_d^-1(Π)→Π, then S is regular polygonal.

For the converse, let us assume that S is regular polygonal. Then, there exists a conformal map ψ_S:Π→ T^0(S) that preserves the ideal vertices. The fact that T^0(S) is simply connected and S has no critical point in S^-1(T^0(S)) together imply that S^-1(T^0(S)) consists of d+1 components, each of which maps diffeomorphically onto T^0(S) under S. Similarly, _d^-1(Π) consists of d+1 components, each of which maps diffeomorphically onto Π under _d.
Moreover, S and _d act as identity maps on ∂ T^0(S) and ∂Π, respectively. Hence, ψ_S can be lifted to a conformal isomorphism from _d^-1(Π) onto S^-1(T^0(S)). Note that the trivial actions of S and _d on ∂ T^0(S) and ∂Π (respectively) ensure that the lifts match on the boundaries of Π, T^0(S). By construction, the extended map conjugates S:S^-1(T^0(S))→ T^0(S) to _d:_d^-1(Π)→Π.


An important consequence of Lemma <ref> is that for a degree d polygonal (piecewise) Schwarz reflection S:→, the restriction S:S^-1()→ is a degree d degenerate anti-polynomial-like map. Many of the following results can be regarded as analogs of basic facts about classical polynomial-like maps in the current setting.




Let S∈𝒮_d.

  * T^∞(S) and K(S) are completely invariant under S.


  * The escaping set T^∞(S) is an open and connected. The non-escaping set K(S) is a full compact subset of . 


  * Each Fatou component of S is simply connected.



1) This is immediate from the definitions.

2) Let us denote the union of the tiles of rank 0 through k by E^k. Since every tile of rank k≥1 is attached to a tile of rank (k-1) along a boundary curve and E^0=T^0(S) is connected, it follows that E^k is connected. Moreover, E^k⊊E^k+1 for each k≥ 0. 

Note that if z∈ T^∞(S) belongs to the interior of a tile of rank k, then it lies in the interior of E^k. On the other hand, if z∈ T^∞(S) belongs to the boundary of a tile of rank k, then it lies in the interior of E^k+1. Hence, T^∞(S)=⋃_k≥ 0E^k. 
Thus, T^∞(S) is an increasing union of open, connected sets, and hence itself is such. Consequently, its complement K(S) is a full compact subset of . 

3) This follows from the fact that K(S) is a full compact.




Let (S:→)∈𝒮_d, μ be an S-invariant Beltrami coefficient supported on K(S), and Φ be a quasiconformal map solving the Beltrami equation with coefficient μ. Then Φ∘ S∘Φ^-1∈𝒮_d.


The assumption that μ is S-invariant implies that Φ∘ S∘Φ^-1 is anti-meromorphic on Φ() that continuously extends to the identity map on ∂Φ(). It follows that each component of Φ() is a simply connected quadrature domain, and hence Φ∘ S∘Φ^-1 is a degree d Schwarz reflection map with connected non-escaping set. Since μ is trivial on T^∞(S), the quasiconformal map Φ is conformal on T^∞(S). Therefore, T^0(Φ∘ S∘Φ^-1)=Φ(T^0(S)) is conformally equivalent to Π; i.e., Φ∘ S∘Φ^-1∈𝒮_d.





 §.§ Connectedness locus of polygonal Schwarz reflections




Let (S:→)∈𝒮_d. Then the following are equivalent.


  * K(S) is connected.

  * S^∘ n(c)∉ T^∞(S), ∀ c∈crit(S).

  * S:T^∞(S)∖T^0(S)⟶ T^∞(S) is conformally conjugate to the Nielsen map _d:∖Π→.



(1)  (2) If some critical point of S lies in the interior of a tile, then this tile would be mapped forward with degree larger than one and its closure would have at least two complementary components. Each of these complementary components would intersect non-escaping set of S, providing a disconnection of K(S).

Next suppose that some critical point of S lies on the boundary of a tile. Then this critical point is a common boundary point of two tiles of equal rank, and the closure of the union of these two tiles has at least two complementary components. Again, each of these components would intersect K(S), proving that K(S) is disconnected. 


(2)  (3) Let ψ_S:Π⟶ T^0(S) be a homeomorphism that is conformal on the interior. Since T^0(S) is simply connected and none of the critical points of S escapes to T^0(S), it follows that each tile of S maps diffeomorphically onto T^0(S) under some iterate of S. Similarly, the tiles of the tessellation of  arising from the regular (d+1)-gon reflection group G_d map diffeomorphically onto Π under iterates of _d. Furthermore, S and _d act as identity maps on ∂ T^0(S) and ∂Π, respectively. This allows us to lift ψ_S to a conformal isomorphism from  (which is the union of all iterated preimages of Π under _d) onto T^∞(S) (which is the union of all iterated preimages of T^0(S) under S). Note that the trivial actions of S and _d on ∂ T^0(S) and ∂Π (respectively) ensure that the iterated lifts match on the boundaries of the tiles. By construction, the extended map conjugates _d:∖Π→ to S:T^∞(S)∖T^0(S)→ T^∞(S).

(3)  (1) The assumption that T^∞(S) is a topological disk implies that K(S) is connected.



We define the  connectedness locus as

    𝒮__d:= {S ∈𝒮_d: K(S)  is connected.}⊆𝒮_d.
 



Note that by Proposition <ref>, the connectedness locus 𝒮__d consists precisely of maps having the Nielsen map _d as the conformal model of their tiling set dynamics.





 §.§ Fatou components and critical points


Let (S:→)∈𝒮_d.


  * Every Fatou component of S is eventually periodic.


  * Every periodic Fatou component U of S is of one of the following types.


  * U is an immediate basin of attraction of an attracting cycle.

  * U is an immediate basin of attraction of a parabolic cycle.

  * U is a Siegel disk.

  * U is an immediate basin of attraction of some w∈𝔖.




(1) Since the space 𝒮_d is parametrized by finitely many rational maps the sum of whose degrees is d+1, it follows that 𝒮_d is finite-dimensional. Moreover, the family 𝒮_d of maps is closed under quasiconformal conjugation in the sense of Lemma <ref>. Hence, the classical proof of non-existence of wandering Fatou components for rational maps can be adapted for the space 𝒮_d (cf. <cit.>).
 
(2) Let U be a periodic Fatou component of period k of S. By Proposition <ref>, U is simply connected. According to <cit.>, the Fatou component U is either the (immediate) basin of attraction of an attracting cycle, or a Siegel disk, or all orbits in U converge to a unique boundary point w. If w lies in , then the classical Snail Lemma implies that w must lie on a parabolic cycle. Otherwise, w∈∂∩ K(S), and hence w is a singular point of ∂ T(S).



Let (S:→)∈𝒮_d.


  * If S has an attracting or parabolic cycle, then the forward orbit of a critical point of S converges to this cycle.  


  * Every Cremer point (i.e. an irrationally neutral, non-linearizable periodic point) and Siegel disk boundary (of S) is contained in the ω-limit set of a recurrent the critical point of S. 


  * If w∈𝔖 has an attracting direction in K(S), then some critical orbit of S converges to w.



(1) and (2) The proof of <cit.> applies to the current setting (cf. <cit.>).

(3) Let us first assume that w∈𝔖 is a cusp and has an attracting direction in K(S). By <cit.>, the asymptotics of S^∘ 2 near w is of the form 

    ζ↦ζ+c_1(ζ-w)^n/2+O((ζ-w)^(n+1)/2),
 
for some c_1∈^*, and n≥ 5.
Standard arguments used in the theory of parabolic germs allow one to count the number of attracting/repelling directions at w (cf. <cit.>). These directions occur in an alternating fashion. Let U be a Fatou component of S such that the S^∘ 2-orbit of every point in U converges to w asymptotic to some attracting direction. A change of coordinate of the form ζ↦ 1/(ζ-w)^n/2-1 that sends w to ∞ can now be applied to a sector containing this attracting direction to bring the map S^∘ 2 to the form ξ↦ξ+1+O(1/ξ) on a right-half plane. This enables one to argue as in <cit.> to prove the existence of an attracting petal in U on which the action of S^∘ 2 is conformally conjugate to translation by +1 on a right half-plane. Finally, the proof of <cit.>, applied to the current setting, shows that the boundary of the maximal domain of definition of this conjugacy contains a critical point of S^∘ 2. It follows that U contains a critical point of S.

Let us now consider a double point w∈𝔖 that has an attracting direction in K(S). After possibly renumbering the quadrature domains in , we may assume that ∂Ω_1∩∂Ω_2={w}. Then, S^∘ 2 is given by the two tangent-to-identity parabolic germs S_2∘ S_1, S_1∘ S_2 near w. These two germs are inverses of each other, and hence have the same number of attracting/repelling directions. Let U be a Fatou component of S such that the S^∘ 2-orbit of every point in U converges to w asymptotic to some attracting direction. Since S^∘ 2 is given by a parabolic germ, it follows that there exists an attracting petal in U on which the action of S^∘ 2 is conformally conjugate to translation by +1 on a right half-plane, and that the boundary of the maximal domain of definition of this conjugacy contains a critical point of S^∘ 2 (cf. <cit.>. Thus, U contains a critical point of S.




 §.§ Dynamical rays and lamination


We now define G_d-rays in  arising from the Cayley graph of G_d with respect to the generating set ρ_1,⋯, ρ_d+1. 


Let (i_1, i_2, ⋯)∈{1,⋯, d+1}^ with i_j≠ i_j+1 for all j. The corresponding infinite sequence of tiles {Π,ρ_i_1(Π),ρ_i_1∘ρ_i_2(Π),⋯} shrinks to a single point of 𝕊^1≅/, which we denote by θ(i_1, i_2, ⋯). We define a G_d-ray at angle θ(i_1, i_2, ⋯) to be the concatenation of hyperbolic geodesics (in ) connecting the consecutive points of the sequence {0,ρ_i_1(0),ρ_i_1∘ρ_i_2(0),⋯}. 


We note that in general there may be more than one G_d-rays at a given angle, but all of them land at the point θ(i_1, i_2, ⋯)∈𝕊^1.
For instance, both the symbol sequences (1,d+1,1,d+1,⋯) and (d+1,1,d+1,1,⋯) yield G_d-rays at angle 0, and both land at the point 1.
Moreover, the image of the tail of a ray at angle θ under _d is the tail of a ray at angle _d(θ).





For S∈𝒮__d, the image of a G_d-ray at angle θ under the map ψ_S:⟶ T^∞(S) (that conjugates _d to S) is called a θ-dynamical ray of S.

Evidently, the image of the tail of a dynamical θ-ray (of S) under S is the tail of a dynamical ray at angle _d(θ).
We denote the set of all (pre)periodic points of _d:𝕊^1→𝕊^1 by Per(_d).


Let S∈𝒮__d. Then the following statements hold true.


  * Let θ∈Per(_d). Then a dynamical ray of S at angle θ lands on Λ(S). 


  * The dynamical rays of S at fixed angles land at points of 𝔖. Conversely, each point of 𝔖 is the landing point of a fixed ray.


  * The dynamical rays of S at periodic angles of period larger than two land at repelling or parabolic periodic points of Λ(S). 



1) After possibly applying some iterate of S, we may assume that θ is periodic of period k under _d. Then, the angle θ corresponds to a sequence in {1,⋯, d+1}^ that is periodic under the left-shift map; i.e., θ=θ(i_1, i_2, ⋯, i_k) (see Definition <ref>). Consider the sequence of points w_n:=ψ_S((ρ_i_1∘⋯∘ρ_i_k)^∘ n(0)), n≥1, on the θ-dynamical ray of S. 
Note that if γ_n is the sequence of arcs on this θ-dynamical ray bounded by w_n and w_n+1, then S^∘ k(γ_n)=γ_n-1, and ℓ_hyp(γ_n)=c for all n≥1, where c is a fixed positive constant, and ℓ_hyp denotes the hyperbolic length in T^∞(S). We now consider two cases.

Case a. (The accumulation set of the θ-ray is contained in ∪_n≥ 0 S^-n(𝔖).) Since the accumulation set of a ray is connected, and the set of iterated preimages of 𝔖 is totally disconnected, it follows that the θ-ray lands. 

Case b. (The accumulation set of the θ-ray intersects Γ(S)∖∪_n≥ 0 S^-n(𝔖).) Let w∈Γ(S)∖∪_n≥ 0 S^-n(𝔖) be an accumulation point of the θ-ray. By our assumption, the map S^∘ k admits an (anti-)holomorphic extension in a neighborhood of w. Moreover, since the arcs γ_n of fixed hyperbolic length accumulate on the limit set Γ(S), their Euclidean lengths must shrink to 0. One can now apply the arguments of <cit.> to the current setting, and deduce that S^∘ k(w)=w. But the set of fixed points of S^∘ k in Γ(S) is totally disconnected (as the set of fixed points of the holomorphic second iterate S^∘ 2k in Γ(S) is discrete). As before, connectedness of the accumulation set of a ray implies that the θ-ray lands.

2) Let θ be fixed under _d; i.e., θ=j/d+1, for some j∈{0,⋯,d}. Since ∂ T^0(S) is locally connected, the image of the radial line at angle θ under ψ_S lands at some point w∈Λ(S). The fact that _d does not admit an anti-conformal extension to a neighborhood of an ideal boundary point of Π implies that S does not have an anti-conformal extension to a neighborhood of w. Hence, w∈𝔖. We will show that the θ-dynamical ray of S lands at w.

By the asymptotic development of S^∘ 2 (which is given by parabolic germs if w is a double point and by Equation (<ref>) if w is a cusp), the map S^∘ 2 admits an inverse branch g in a relative neighborhood of w with the following properties:

i) g(w)=w, and

ii) for w'∈ S^-2(T^0(S))∩Domain(g) is sufficiently close to w, the sequence g^∘ n(w') converges to w horocyclically (in the simply connected domain T^∞(S)).

It is easy to see that the sequence g^∘ n(w') is within a bounded distance from the dynamical θ-ray of S, and hence the θ-ray lands at w.

For the converse statement, note that each w∈𝔖 is the landing point of the curve {ψ_S(re^2π iθ):r∈[0,1)}, for some angle θ fixed under _d. By the previous paragraph, the dynamical ray of S at angle θ lands at w.
 
3) We claim that the dynamical rays of S of period larger than two do not land on 𝔖. Once this claim is established, the result follow by the classical Snail lemma.

Let us fix w∈𝔖. Once again, by the asymptotic development of S^∘ 2 near w, the map S^∘ 2|_S^-2() extends to an orientation-preserving local homeomorphism in a neighborhood of w. One can apply the arguments of <cit.> to this map to conclude that all rays landing at w have the same period under S^∘ 2 (cf. <cit.>). Since w is the landing point of some fixed ray of S (by the previous part), it follows that all rays landing at w are fixed by S^∘ 2. Therefore, the dynamical rays of S of period larger than two cannot land on 𝔖




Let S∈𝒮__d. Then, every repelling and parabolic periodic point of S is the landing point of finitely many (at least one) dynamical rays. Moreover, all these rays are periodic of the same period under S^∘ 2.


The proof of the corresponding statement for polynomials with connected Julia set (cf. <cit.>) can be easily adapted for the current setting.



Using the same techniques as in <cit.>, we can prove stability of dynamical rays landing at pre-repelling points whose orbit avoids critical points.

Let S∈𝒮__d. Assume that a dynamical ray lands at a repelling pre-periodic point z such that the orbit of z avoids the critical points of S.
Then, the corresponding dynamical ray lands at the real-analytic continuation z' (of the pre-periodic point z) for all S'∈𝒮__d sufficiently close to S.





In light of Proposition <ref>, one can define a landing map L:Per(_d)→Λ(S) that associates to every (pre)periodic angle θ  the landing point of the θ-dynamical ray of S.


For S∈𝒮__d, the rational lamination of S is defined as the equivalence relation on Per(_d)⊂/ such that θ, θ'∈Per(_d) are related if and only if L(θ)=L(θ'). We denote the rational lamination of S by λ(S).


Rational laminations of Schwarz reflections in 𝒮__d are analogous to rational laminations of anti-polynomials with connected Julia set (cf. <cit.>). In particular, λ(S) is a _d-invariant equivalence relation.


The push-forward (ℰ_d)_∗(λ(S)) of the rational lamination of S is defined as the image of λ(S)⊂Per(_d)×Per(_d) under ℰ_d×ℰ_d. Clearly, (ℰ_d)_∗(λ(S)) is an equivalence relation on /. Similarly, the pullback (ℰ_d)^∗(λ(f)) of the rational lamination of an anti-polynomial f with connected Julia set is defined as the preimage of λ(f)⊂/×/ under ℰ_d×ℰ_d.



For Schwarz reflections as well as anti-polynomials, we will denote by λ(S) (respectively, λ(f)) the smallest equivalence relation on / that contains the closed set λ(S)⊂/×/ (respectively, λ(f)⊂/×/).
Note that the push-forward operation defined in Definition <ref> can be applied to λ(S) as well.




 §.§ Combinatorial mating of _d with periodically repelling anti-polynomials


Recall that the filled Julia set of a periodically repelling anti-polynomial f∈𝒞_d^- has empty interior. By Proposition <ref>, the same statement holds for the non-escaping set of a periodically repelling Schwarz reflection S∈𝒮__d.
The proof of <cit.> implies that if the Julia/limit set of such a map is locally connected, then its filled Julia set/non-escaping set dynamics is topologically modeled by the induced map

    m_-d:/λ(f)⟶/λ(f)

or

    _d:/λ(S)⟶/λ(S).

Further, if 

    (ℰ_d)_*(λ(S))=λ(f),

then the above induced maps are topologically conjugate via the homeomorphism

    ℰ_d: /λ(S)⟶/λ(f)

induced by the conjugacy ℰ_d:𝕊^1→𝕊^1 between _d and z^d. In particular, local connectedness of the Julia (respectively, limit) set of f (respectively, S) and the condition (ℰ_d)_*(λ(S))=λ(f) together imply that f|_𝒦(f) and S_|K(S) are topologically conjugate.

The above discussion motivates the following weaker definition of mating.


We say that a periodically repelling Schwarz reflection S∈𝒮__d is a combinatorial mating of _d with a periodically repelling anti-polynomial f∈𝒞_d^- if (ℰ_d)_*(λ(S))=λ(f).







§ GEOMETRICALLY FINITE POLYGONAL SCHWARZ REFLECTIONS



A Schwarz reflection map S∈𝒮_d is said to be geometrically finite if every critical point of S in the
limit set Λ(S) is pre-periodic.


By Propositions <ref> and <ref>, each periodic Fatou component of a geometrically finite Schwarz reflection is an attracting/parabolic basin, or a basin of attraction of some singular point x∈𝔖. Moreover, S has no Cremer cycle.



Let S∈𝒮__d,gf. Then the following hold true.

  * Λ(S) is locally connected.

  * Λ(S) has zero area.



1) The proof is analogous to the proof of local connectivity of connected Julia sets of geometrically finite polynomials (cf. <cit.>, <cit.>).

2) We say that a point w∈Γ(S) is a radial point if there exists δ > 0, an infinite sequence of positive integers { n_k }, and well-defined inverse branches of S^∘ n_k defined on B(S^∘ n_k(w), δ) sending S^∘ n_k(w) to w for all k. The set of all radial points of Γ(S) is called the radial limit set of S. Arguments used in <cit.> show that the radial limit set of S is given by Γ(S)∖⋃_j=0^∞ S^-j(𝔖∪𝒫∪ C_preper), where 𝒫 is the union of the parabolic cycles and C_preper is the set of strictly pre-periodic critical points of S. Finally, the nowhere dense structure of Λ(S) can be transferred from moderate scales to microscopic scales at radial points (using the inverse branches of iterates of S), and this implies that no radial limit point is a point of Lebesgue density for Γ(S) (cf. <cit.> or <cit.>). The above facts, together with the Lebesgue density theorem, imply that Λ(S) has zero area. 


Local connectedness of limit sets of maps in 𝒮__d,gf, combined with the arguments of <cit.>, implies that for S∈𝒮__d,gf, the dynamics of S on its limit set Λ(S) is topologically conjugate to 

    _d: (/)λ(S)⟶(/)λ(S) .

In particular, λ(S) is generated by co-landing of the dynamical rays of S. A straightforward adaptation of <cit.> for Schwarz reflections yields the following properties of λ(S) for S∈𝒮__d,gf.




Let S∈𝒮__d,gf. Then, λ(S) satisfies the following properties.

  * λ(S) is closed in /×/.


  * Each equivalence class A of λ(S) is a finite subset of /.


  * If A is a λ(S)-equivalence class, then _d(A) is also a λ(S)-equivalence class.


  * If A is a λ(S)-equivalence class, then A↦_d(A) is consecutive reversing; i.e., for every connected component (s,t) of /∖ A, we have that (_d(s),_d(t)) is a connected component of / ∖ _d(A).


  * λ(S)-equivalence classes are pairwise unlinked; i.e., if A and B are two distinct equivalence classes of λ(S), then there exist disjoint intervals I_A, I_B⊂/ such that A⊂ I_A and B⊂ I_B.


  * If γ⊂(/)λ(S) is a periodic simple closed curve, then the corresponding return map is not a homeomorphism.


  * If c is a critical point of P on the limit set Λ(S), then τ_S^-1(c)⊂Per(_d), and τ_S^-1(c)↦_d(τ_S^-1(c)) has some degree δ>1. Every other equivalence class of λ(S) maps injectively onto its image equivalence class under _d. Here, τ_S:→ T^∞(S) is the normalized conformal map that conjugates _d to S.





To prove Theorem <ref>, we need the following technical lemma that guarantees the existence of continuous extensions of circle homeomorphisms as quasiconformal/David homeomorphisms of . 




  * The homeomorphism ℰ_d^-1:𝕊^1→𝕊^1, that conjugates z^d to _d, admits a continuous extension to  as a David homeomorphism.



  * Let B:→ be a degree d Blaschke (respectively, anti-Blaschke product) with a parabolic fixed point on 𝕊^1 that has an attracting direction in . Then there exists a homeomorphism H:𝕊^1→𝕊^1 that conjugates z^d (respectively, z^d) to B and admits a continuous extension to  as a David homeomorphism.



  * Let n≥ 2, and 

    B:𝕊^1×/n⟶𝕊^1×/n, B(z,i)=(B_i(z),i+1)

be such that each B_i is a degree d_i anti-Blaschke product. Assume further that each 

    B_i+n-1∘⋯∘ B_i+1∘ B_i:𝕊^1→𝕊^1
 
is a degree d Blaschke or anti-Blaschke product with a parabolic fixed point on 𝕊^1 that has an attracting direction in , depending on whether n is even or odd, where d=d_0 d_1⋯ d_n-1, i∈/n.
Then, there exists a homeomorphism H:𝕊^1×/n⟶𝕊^1×/n conjugating 

    P:𝕊^1×/n⟶𝕊^1×/n, P(z,i)=(z^d_i,i+1)

to B.
Moreover, H continuously extend as a David homeomorphism of ×/n.


  * Let n≥ 2, and 

    B:𝕊^1×/n⟶𝕊^1×/n, B(z,i)=(B_i(z),i+1)

be such that each B_i is a degree d_i anti-Blaschke product. Assume further that each 

    B_i+n-1∘⋯∘ B_i+1∘ B_i:𝕊^1→𝕊^1
 
is a degree d Blaschke or anti-Blaschke product with an attracting fixed point in , depending on whether n is even or odd, where d=d_0 d_1 ⋯ d_n-1, i∈/n.
Then, there exists a homeomorphism H:𝕊^1×/n⟶𝕊^1×/n conjugating 

    P:𝕊^1×/n⟶𝕊^1×/n, P(z,i)=(z^d_i,i+1)

to B.
Moreover, H continuously extends as a quasiconformal homeomorphisms of ×/n.



1) Follows from <cit.>.

2) Note that the parabolic (anti-)Blaschke product B_d^-(z) :=(d+1)z^d+(d-1)/(d+1)+(d-1)z^d (or the parabolic Blaschke product B_d^+(z)=(d+1)z^d+(d-1)/(d+1)+(d-1)z^d) is conjugate to B via a quasisymmetric homeomorphism H_1:𝕊^1→𝕊^1 (cf. <cit.>). We extend H_1 to a quasiconformal homeomorphism of  (also called H_1).

By <cit.>, there exists a circle homeomorphism H_2:𝕊^1→𝕊^1 that conjugates z^d or z^d to B_d^±, such that H_2 admits a continuous extension to  as a David homeomorphism. By <cit.>, the desired map H is given by H_1∘ H_2.



	
3) Let us assume that n is odd. The proof of the even case is completely analogous.

By the previous part, there exists a topological conjugacy H_n-1:𝕊^1→𝕊^1 between z^d and B_n-2∘⋯∘ B_0∘ B_n-1. Since B_j and z^j are circle coverings of the same degree for j∈/n, we can succesively lift H_n to obtain circle homeomorphisms H_n-2,⋯,H_0, H_n-1 such that the following diagram commutes:


    𝕊^1 rz^d_n-1[swap]dH_n-1   𝕊^1 rz^d_0[swap]dH_0   𝕊^1 dH_1rz^d_1   𝕊^1 dH_2rz^d_2   ⋯rz^d_n-3d   𝕊^1 [swap]dH_n-2rz^d_n-2   𝕊^1 dH_n-1
    𝕊^1 [swap]rB_n-1   𝕊^1 [swap]rB_0   𝕊^1 [swap]rB_1   𝕊^1 r[swap]B_2   ⋯[swap]rB_n-3   𝕊^1 [swap]rB_n-2   𝕊^1.

It now follows that

    (B_n-2∘⋯∘ B_0∘ B_n-1) ∘H_n-1 = (B_n-2∘⋯∘ B_0∘ B_n-1) ∘ H_n-1,

and hence H_n-1∘ H_n-1^-1 is a deck transformation for the circle covering B_n-2∘⋯∘ B_0∘ B_n-1. 
Hence, possibly after post-composing H_n-2,⋯,H_0, H_n-1 with deck transformations of appropriate B_js, we can assume that H_n-1=H_n-1. Hence, 

    H:𝕊^1×/n→𝕊^1×/n, (z,i)↦ (H_i(z), i)

is the desired topological conjugacy between P and B.

Finally, as each H_i conjugates z^d to a degree d anti-Blaschke product with a parabolic fixed point on 𝕊^1 that has an attracting direction in , the first part (of this proposition) allows us to conclude that H_i can be continuously extended to David homeomorphisms of . 

4) The proof, which uses the fact that any expanding, analytic, orientation-reversing covering of 𝕊^1 is quasisymmetrically conjugate to z^n (where n is the degree of the map), is similar to the previous part.







The proof will be divided into three parts.


Constructing a map Φ:𝒞^-_d,gf⟶𝒮__d,gf. Let f∈𝒞^-_d,gf be geometrically finite. Our goal is to construct a Schwarz reflection in 𝒮__d,gf that is a conformal mating of f and _d.

By an antiholomorphic version of <cit.> (the proof of the orientation-preserving case applies mutatis mutandis to the orientation-reversing case thanks to the orientation-reversing version of Thurston Realization Theorem, cf. <cit.>, <cit.>), there exists a postcritically finite map f_0∈𝒞^-_d such that f|_𝒥(f) is topologically conjugate to f_0|_𝒥(f_0). We first construct a partially defined continuous map on the sphere from f_0 as follows.


∙ Let us denote the David extension of ℰ_d^-1:𝕊^1→𝕊^1 (that conjugates z^d to _d) to  by ξ. Further, let τ_f_0:→ℬ_∞(f_0) be a conformal map that conjugates z^d to f_0 (where ℬ_∞(f_0) is the basin of infinity of f_0).  We replace the action of f_0 on ℬ_∞(f_0) by τ_f_0∘ξ^-1∘_d∘ξ∘τ_f_0^-1. 

Note that the map τ_f_0∘ξ^-1∘_d∘ξ∘τ_f_0^-1 preserves the pullback of the standard complex structure on  by ξ∘τ_f_0^-1.


∙ Suppose that U_0,⋯,U_n-1 is a cycle of bounded Fatou components of f, and U_0,⋯,U_n-1 is the corresponding cycle of bounded Fatou components of f_0. One can choose conformal maps κ:×/n→_i∈/n U_i and κ:×/n→_i∈/nU_i such that κ^-1∘ f∘κ:×/n→×/n is a fiberwise anti-Blaschke product B satisfying the hypotheses of part (3) or part (4) of Proposition <ref>, and κ^-1∘ f_0∘κ:×/n→×/n is a fiberwise power map P as in part (3) or part (4) of Proposition <ref>. Hence, there exists a homeomorphism H:𝕊^1×/n⟶𝕊^1×/n that conjugates P to B, and admits a David/quasiconformal extension ×/n.
We replace the action of f_0 on _i∈/nU_i by the map κ∘H^-1∘B∘H∘κ^-1.

Again, the map κ∘H^-1∘B∘H∘κ^-1 preserves the pullback of the standard complex structure on ×/n by H∘κ^-1.



∙ Finally, let U be a strictly pre-periodic Fatou component of f with f(U)=V, and U, V be the corresponding components of f_0. The maps f:U→ V and f_0:U→V induce, via Riemann maps of the Fatou components, anti-Blaschke actions of , and these anti-Blaschke actions are quasisymmetrically conjugate on 𝕊^1 (cf. <cit.>). This allows us to replace the action of f_0 on U by the conformal model of the action of f on U as in the previous steps. We note that this step is only necessary for Fatou components U containing a critical point.

The above recipe defines a continuous map S on a subset of the sphere and an invariant Beltrami coefficient on the union of the periodic Fatou components of f_0. We use S to spread this Beltrami coefficient out to all the pre-periodic Fatou components of f_0, and set it equal to zero on the Julia set of f_0. Since f_0 is subhyperbolic, one can now argue as in <cit.> to conclude that this invariant Beltrami coefficient μ satiesfies the David condition and that the conjugate of S by the corresponding integrating map Ψ is antiholomorphic. We denote the conjugated map Ψ∘S∘Ψ^-1 by S.

By construction, S is defined on the complement of the simply connected domain Ψ(τ_f_0(ξ^-1(Π))), and fixes the boundary of its domain of definition pointwise. Thus, S is a Schwarz reflection map whose escaping and non-escaping sets are given by Ψ(ℬ_∞(f_0) and Ψ(K(f_0)), respectively. The tiling set dynamics of S is conformally conjugate to _d via ξ∘τ_f_0^-1∘Ψ^-1; i.e., S∈𝒮__d,gf. It also follows from the construction of S that its non-escaping set dynamics is topologically conjugate (conformally on the interior) to the filled Julia set dynamics of f. It is now easily checked that S is a conformal mating of f and _d. 


We also note that by <cit.> and the fact that ℬ_∞(f_0) is  John domain, we have that the limit set of S is conformally removable.

We claim that such an S∈𝒮__d,gf is unique. Indeed, if S_1 were another conformal mating of f and _d, then the tiling set dynamics and non-escaping set dynamics of S and S_1 are conformally conjugate. Further, these conjugacies match continuously along the limit sets to yield a topological conjugacy between S and S_1 that is conformal away from Λ(S). By conformal removability of Λ(S), this conjugacy is a Möbius map; i.e., S and S_1 are Möbius conjugate. The normalization of the maps in 𝒮__d now implies that S=S_1. 
Alternatively, one can employ classical pullback arguments to deduce uniqueness of the map S (see <cit.> for an implementation of this principle to prove rigidity of geometrically finite Schwarz reflections in the C&C family).

Thus, we have a well-defined map Φ:𝒞^-_d,gf⟶𝒮__d,gf that sends f to the unique conformal mating S of f and _d.


Injectivity of Φ. Suppose that Φ(f)=Φ(f_1)=S. Then, the basin of infinity dynamics and filled Julia set dynamics of f and f_1 are conformally conjugate. Further, these conjugacies match continuously along the Julia sets to yield a topological conjugacy between f and f_1 that is conformal away from 𝒥(f). By conformal removability of 𝒥(f) (see <cit.>), this conjugacy is a Möbius map. (Equivalently, injectivity of Φ follows from well-known results about rigidity of geometrically finite polynomials, cf. <cit.>.) By the normalization of Böttcher coordinates for maps in 𝒞^-_d, this Möbius conjugacy between the monic, centered anti-poynomials f, f_1 is tangent to the identity at ∞. Hence, the Möbius conjugacy must be the identity map; i.e., f=f_1.


Surjectivity of Φ. Let S∈𝒮__d,gf and λ(S) be the rational lamination of S. By Proposition <ref> and the orientation-reversing version of <cit.>, there exists a postcritically finite anti-polynomial f_0∈𝒞^-_d with λ(f_0)= (ℰ_d)_∗(λ(S)). Having f_0 at our disposal, we can now produce a geometrically finite anti-polynomial f∈𝒞^-_d by performing quasiconformal/David surgeries on the bounded Fatou components of f_0 (as in the first step of the proof) so that the filled Julia set dynamics of f is topologically conjugate (conformally on the interior) to the non-escaping set dynamics of S. Evidently, the map f is the desired preimage of S under Φ.



























§ STRATA OF THE CONNECTEDNESS LOCUS 𝒮__D




 §.§ Fixed ray lamination and a coarse partition


In this section, we will explain that the connectedness locus of the space of degree d regular polygonal Schwarz reflections admits a finite partition.
This coarse partition gives a natural stratification of the connectedness locus.



 §.§ Fixed ray lamination



The Nielsen map _d:𝕊^1→𝕊^1 has exactly d+1 fixed points 0,⋯,d/d+1. We denote the set of these fixed points by Fix(_d).






We define a fixed ray lamination as a geodesic lamination ℒ⊂ so that


  * each leaf ℓ = (a,b) of ℒ has end points a, b ∈Fix(_d) ⊂≅∂;

  * two leaves in ℒ have disjoint closures in .


Given a fixed ray lamination ℒ, we call each complementary component G ⊂∖ℒ a gap.
Let G be a gap. Then ∂ G ∩ has length equal to k ·1/d+1 for some multiple k.
We shall call this number k the degree of the gap, and is denoted by (G).
From the definition, we have

    d+1 = ∑_G(G),

where the sum is over all gaps of ℒ.


Let us denote the number of fixed points of _d in (∂ G ∩) by (G), and call it the cusp number of the gap.
We also denote the number of leaves on ∂ G by tan(G), and call it the tangent number of the gap.
An easy computation shows that

    (G) = (G) + tan(G).


Let G_1, G_2 be two gaps.
We say that they are adjacent if ∂ G_1 ∩∂ G_2 ≠∅.
Note that if two gaps are adjacent, then ∂ G_1 ∩∂ G_2 is a leaf of ℒ.
It is easy to see that there are exactly tan(G) number of gaps adjacent to G.



 §.§ Fixed ray laminations associated with quadrature domains



Recall that a polygonal Schwarz reflection map S∈𝒮__d (normalized as in Section <ref>) induces the action of the Nielsen map _d on the ideal boundary I(T^∞(σ))≅/ of the escaping set, and hence has exactly d+1 fixed points on its ideal boundary.






Let ℒ be a fixed ray lamination.
We say it is associated with a marked degree d polygonal Schwarz reflection S if


  * ℓ = (a, b) is a leaf of ℒ if and only a, b ∈ I(T^∞(S)) correspond to the same point on ∂ T^∞(S).



Recall that 𝒮__d is the connectedness locus of degree d regular polygonal Schwarz reflections. As mentioned in Subsection <ref>, the domain of a Schwarz reflection S∈𝒮__d is a pinched polygon, and each cut-point of this pinched polygon is the touching point of precisely two Jordan quadrature domains defining S. 
Thus, we have a coarse partition

    𝒮__d = ⋃_ℒ𝒮__d, ℒ,

where 𝒮__d, ℒ consists of all marked degree d polygonal Schwarz reflections with connected limit set and fixed ray lamination ℒ, and the union is over all possible fixed ray laminations.

The following theorem gives the topological structure of the quadrature domains for S ∈𝒮__d, ℒ.

Let ℒ be a fixed ray lamination. Let S : ⟶∈𝒮__d, ℒ.
Then 


  * the components of  are in one-to-one correspondence with the gaps of ℒ:

    = ⋃_GΩ_G;


  * two quadrature domains Ω_G, Ω_G' share at most one boundary point; moreover, they share a boundary point if and only if G, G' are adjacent;

  * the touching point of two quadrature domains Ω_G, Ω_G' is a non-singular point for both ∂Ω_G and ∂Ω_G' (i.e., it is a double point of ∂);

  * the quadrature domain Ω_G has (G) number of cusps on its boundary; and

  * the uniformizing rational map of the quadrature domain Ω_G has global degree (G).


 
Let ψ:→ T^∞(S) be a conformal conjugacy between _d and S. Also recall that Π denotes the ideal (d+1)-gon in  with ideal vertices at the (d+1)-st roots of unity.
By definition, =∖ψ(Π), where the closure is taken in . By injectivity of ψ on , the only points on Π that may be identified by ψ are the (d+1)-st roots of unity. By design, each such identification is recorded by a leaf of the fixed ray lamination of ℒ. 

1) and 2) It follows from the above discussion that the cut-points of the pinched polygon  correspond bijectively to the leaves of  and hence the components of  correspond bijectively to the gaps of . The touching structure of the components of  also follow from above.

3) As each Ω_G is a Jordan quadrature domain, the only singularities on ∂Ω_G are conformal cusps. Let ϕ_G:→Ω_G be a uniformizing rational map. Then, each conformal cusp on ∂Ω_G is a critical value of ϕ_G with an associated critical point on 𝕊^1. By the injectivity of ϕ_G on , such a cusp is an inward pointing cusp for Ω_G. This shows that the touching point of Ω_G and Ω_G' cannot be a conformal cusp for the boundary of either domain. The result follows.

4) Note that _d does not admit an anti-conformal extension in a neighborhood of any ideal vertex of Π, while it does admit such extensions around all other boundary points of Π. It follows that S does not admit an anti-conformal extension in a neighborhood of any point of ψ(ω^j) (where ω=exp(2π i/d+1), and j∈{0,⋯,d}), but it does extend anti-conformally around other boundary points of ∂. The only obstructions for an anti-conformal extension of S across a boundary point of  is a cusp or a double point. By part 3) of this theorem, the pairs of end-points of the leaves of  are in one-to-one correspondence with the double points on ∂. Hence, the other ideal vertices of Π correspond bijectively to the cusps on ∂. The result is now a consequence of the above analysis and part (1) of this theorem.

5) The map _d carries each arc of 𝕊^1∖Fix(_d) to the complement of its closure. Hence, _d(∂ G∩𝕊^1) covers ∂ G∩𝕊^1 exactly ((G)-1) times. It follows that Λ(S)∩Ω_G covers itself ((G)-1) times under the map S|_Ω_G. Since the limit set is completely invariant, we conclude that S|_Ω_G:(S|_Ω_G)^-1(Ω_G)→Ω_G is a degree ((G)-1) branched covering. This observation and Proposition <ref> together imply that the degree of the uniformizing rational map of Ω_G is equal to (G).









 §.§ Coarse partitions in low degree

There are four fixed ray laminations in degree 2: the trivial lamination _0, and the three symmetric laminations _1:={{1/3,2/3}}, _2:={{0,1/3}}, _3:={{0,2/3}}. The corresponding space 𝒮__2, ℒ_1 (respectively, 𝒮__2, ℒ_0) is given by Schwarz reflections in a fixed cardioid and its circumcircles (respectively, by Schwarz reflections in deltoid-like curves). These families were studied in <cit.>.









 §.§ Compactness of connectedness locus 𝒮__d


Recall that 𝒮__d stands for the space of degree d normalized Schwarz reflection maps whose tiling set dynamics are conformally conjugate to the Nielsen map _d of the regular ideal polygon reflection group G_d. The domain of definition of such a Schwarz reflection is the closure of a disjoint collection of simply connected quadrature domains Ω_1,⋯, Ω_k such that
∪_i=1^k Ω_i is connected and simply connected.


Since there are only finitely many fixed ray laminations, we can assume (after possibly passing to a subsequence) that any sequence in 𝒮__d entirely lies in 𝒮__d,, where  is a particular fixed ray lamination. We enumerate the gaps of  as 𝒢_1,⋯,𝒢_k.
For maps S:→ in 𝒮__d,, we denote the components of  as Ω_1,⋯,Ω_k such that Ω_i corresponds to 𝒢_i. 

We topologize 𝒮__d as follows.

We say that a sequence {S_n:^n=⋃_r=1^kΩ_r^n→}⊂𝒮__d, converges to S:=⋃_r=1^k⋃_j=1^l_rΩ_r,j→ if 


  * {Ω_r,j: j∈{1,⋯,l_r}} is the collection of all Carathéodory limits of the sequence of domains {Ω_r^n}_n, r∈{1,⋯,k}, and 


  * for n large enough, the antiholomorphic maps S_n converge uniformly to S on compact subsets of .




For each Schwarz reflection S_n:^n=⋃_r=1^kΩ_r^n→, there are k Riemann uniformization maps ϕ_n,r: ⟶Ω_r^n, r = 1,..., k which extend as rational maps of  (see <ref>).
For S:=⋃_r=1^k⋃_j=1^l_rΩ_r,j→, there are ∑_r=1^k l_r Riemann uniformization maps ϕ_r, j: ⟶Ω_r, j which extend as rationals maps of .
It follows from the definition of Carathéodory limits that S_n → S if and only if there exist M_n, r, j∈() so that

    ϕ_n,r∘ M_n, r, j→ϕ_r, j

compactly away from finitely many points in  (where each map is viewed as maps on ).




The connectedness locus 𝒮__d is compact.



Let {S_n:^n=∪_r=1^kΩ_r^n→}⊂𝒮__d,. By definition, there exist conformal maps

    ψ_n:→ T^∞(S_n)

that conjugate _d to S (wherever defined). We can normalize ψ_n, possibly after conjugating S_n by Möbius maps, such that ψ_n(0)=∞, and ψ_n(z)=1/z+O(z) near the origin. 

Note that ψ_n(Π)=T^0(S_n). Since Π contains a round disk around the origin, it follows from the Koebe one-quarter theorem that each ψ_n(Π) contains a definite round disk centered at ∞. In other words, _n⊂ B(0,R), for some fixed R>0. 
Also note that the maps ψ_n|_ form a normal family. After possibly passing to a subsequence, we may assume that ψ_n converges normally to a conformal map ψ_∞ on . Let B_s be an open ball compactly contained in ρ_s(Π) centered at ρ_s(0), for s∈{1,…,d+1}. Normal convergence of ψ_n to ψ_∞ implies that for n large, the conformal disks Ω_r^n contain some fixed open set ψ_∞(B_s) (see Figure <ref>). It follows that for each r∈{1,⋯,k}, the sequence of conformal disks {Ω_r^n}_n has at least one non-trivial Carathéodory limit. We will now argue that there are finitely many such Carathéodory limits.


By conformality of ψ_∞, the set ψ_∞(∂Π∩) is a union of d+1 disjoint bi-infinite geodesics in the hyperbolic metric of ψ_∞(). Since each ψ_n(∂Π)=∂_n is a finite union of real-algebraic curves of uniformly bounded degree contained in the fixed compact set B(0,R), one concludes that the ends of the above bi-infinite geodesics have unique limit points on ∂ψ_∞(). Thus, ψ_∞ extends continuously to ∂Π and ψ_∞(Π) has only finitely many identifications on its boundary (i.e., it is a topological polygon possibly with some vertices identified). Specifically, only the ideal vertices of Π may be identified under ψ_∞.

It follows from the above discussion that ∖ψ_∞(Π) has finitely many (at least one) components, and these are precisely the Carathéodory limits of the sequences of domains {Ω_r^n}_n, r∈{1,⋯,k} (with all possible choices of base points). We denote these components as Ω_1,⋯,Ω_l.
As the Riemann uniformizations of Ω_r^n are rational maps, it follows from the Carathéodory kernel convergence theorem that the Riemann uniformization of each Ω_j, j∈{1,⋯,l}, is also a rational map. Thus, each such limit is a simply connected quadrature domain.  


The description of Schwarz reflection maps given in Proposition <ref> now implies that for n large enough, S_n converges to the Schwarz reflection map S associated with the quadrature domains Ω_1,⋯,Ω_l. We also note that 
T^0(S) is equal to ψ_∞(Π). The equivariance property of ψ_n implies that ψ_∞ conjugates _d to S, wherever defined. Hence, the escaping set of S is ψ_∞(), and the dynamics of S on its escaping set is conformally conjugate to _d via ψ_∞^-1. Hence, S:→ lies in 𝒮__d and is a limit point of the sequence {S_n:_n→}.





Since fixed ray laminations are equivalence relations on 𝕊^1, there is a natural partial order ≥ on them. 


If {S_n}⊂𝒮__d, _1 converges to some S∈𝒮__d, _2, then _2≥_1.


If ψ_n(a)=ψ_n(b) for some a,b∈∂Π∩𝕊^1 and for all n∈, then ψ_∞(a)=ψ_∞(b).




By using quasi-PCF degeneration studied in <cit.>, one can show that the converse of Corollary <ref> is also true.
Suppose _2≥_1, then there exists a sequence {S_n}⊂𝒮__d, _1 that converges to some S∈𝒮__d, _2.
Some explicit examples are constructed in section <ref>, which are the key ingredients for proving discontinuity.








 §.§ Discontinuity and change of signature


Recall from Section <ref> that there exists a natural bijective map Φ from geometrically finite polynomials 𝒞^-_d,gf onto geometrically finite Schwarz reflections 𝒮__d,gf.

Let H_0 and H_1 stand for the main hyperbolic component and the period two hyperbolic component contained in the (0,1/d+1)-limb of the Multicorn ℳ_d, which is the connectedness locus of unicritical anti-polynomials f_c(z)=z^d+c. Then, H_1 bifurcates from H_0 along an arc, and barring a unique double parabolic parameter, all points on this arc are simple parabolic parameters (see <cit.>, <cit.>, and <cit.>).

We denote the trivial fixed ray lamination by _0, and the fixed ray lamination {{0,1/d+1}} by _1.


Let c_∞ be a simple parabolic parameter on ∂ H_0∩∂ H_1. Then Φ is discontinuous at c_∞.


Suppose that {c_n}⊂ H_1 converges to c_∞∈∂ H_0∩∂ H_1.
By our assumption, the dynamical rays at angles 0 and 1/d+1 land at a common point of the Julia set of f_c_n. Hence, Φ(f_c_n) lies in 𝒮__d, _1. By Corollary <ref> and unicriticality of the maps, any subsequential limit of {Φ(f_c_n)} also lies in 𝒮__d, _1.

On the other hand, as f_c_∞ has a simple parabolic fixed point and hence a Jordan curve Julia set, it follows that Φ(f_c_∞) lies in 𝒮__d, _0. Hence, the sequence {Φ(f_c_n)} does not converge to Φ(f_c_∞).





Let c_0 be the unique double parabolic parameter on ∂ H_0∩∂ H_1. Then Φ^-1 is discontinuous at Φ(f_c_0).


Let c_∞ be a simple parabolic parameter on ∂ H_0∩∂ H_1, and {c_n} be a sequence in H_1 converging to c_∞. Then, the multiplier of the unique attracting 2-cycle of f_c_n converges to 1. By definition of Φ, the multiplier of the unique attracting 2-cycle of Φ(f_c_n) also converges to 1. This implies that the sequence {Φ(f_c_n)} converges to Φ(f_c_0)∈𝒮__d, while {c_n} converges to c_∞. Hence, Φ^-1 is discontinuous at Φ(f_c_0).


We are now ready to complete the proof of Theorem <ref>.

The discontinuity of Φ and Φ^-1 follow from Lemmas <ref> and <ref>, respectively. 





§ COMBINATORIAL CLASSES FOR POLYNOMIALS AND ANTI-POLYNOMIALS

Let 𝒞^±_d, r be the space of monic and centered polynomials (or anti-polynomials) with connected Julia set and all cycles repelling (also called `periodically repelling').
In this section, we discuss a natural way to organize the data for renormalizations of polynomials, or anti-polynomials in 𝒞^±_d, r.

We will use puzzle pieces to construct a decomposition of the dynamical plane.
To better organize these combinatorial data, we will inductively construct puzzles depending on their level of renormalizations and the depth of the puzzle pieces.



 §.§ Level zero renormalization puzzles

To start our construction, we consider rays landing at the fixed points.
If the polynomial or the anti-polynomial is periodically repelling, the following lemma guarantees that these rays cut through the Julia set.

Let f ∈𝒞^±_d, r.
Then the union of rays landing at all fixed points together with their landing points cut the Julia set of f.


If f is a polynomial, then there are d fixed points in .
Since f has connected Julia set and is periodically repelling, these fixed points are distinct and all lie on the Julia set.
Since there are d-1 fixed rays, at least one fixed point p is the landing point of some periodic rays of period ≥ 2.
Then these rays landing at p together with p cut the Julia set of f.

If f is an anti-polynomial, then there are d fixed points in  by Lefschetz fixed point theorem, as all fixed points are repelling (see <cit.>).
Since there are d+1 fixed rays, at least two fixed rays must land together at a fixed point p.
Then these two rays landing at p together with p cut the Julia set of f.


To define level zero renormalization puzzles, we let ϕ be the Riemann mapping from - to - K(f) (where K(f) is the filled Julia set of f), normalized so ϕ'(∞) = 1.
Consider the disk D bounded by the image of the circle {| z| =2} under ϕ.
This disk encloses K(f) and is bounded by an equipotential of the complement.

By Lemma <ref>, this disk is cut into q>1 pieces by the external rays landing at the fixed points of f that are also cut points.
These pieces form the puzzle pieces of depth zero for level zero renormalization.
Each piece is a closed disk, whose boundary consists of fixed points, segments of external rays landing at those fixed points, and part of ∂ D.

The puzzle pieces at depth k+1 are defined inductively as the closures of the components of f^-1((P)), where P ranges over all puzzle pieces at depth k.

The puzzle pieces at depth k have disjoint interiors and cover the Julia set.
As the depth increases, the puzzles become successively finer.
More precisely, a piece at depth k+1 is contained in a unique piece at depth k.

We will use the notation P_k, f^0(z) to denote the puzzle piece at depth k for level zero renormalization, that contains z in its interior.
Sometimes we simply write it as P_k^0(z) or P_k(z) if the map f and the level of renormalizations are not ambiguous.

We remark that the puzzle piece P_k, f^0(z) is well-defined for z∈ J(f) if f^k(z) is not a fixed point.
If f^k(z) is a fixed point, then z might be on the common boundary of multiple puzzle pieces at depth k.


Let c be a critical point for f.
We say that c is level zero renormalizable if c is not on the boundary of any puzzle piece and there exists an integer p ≥ 1 so that 

    P_k(c) = P_k(f^p(c))  for all  k≥ 0.

The smallest such integer p is called the renormalization period.

We say that f is level zero renormalizable if some critical point of f is level zero renormalizable, and level zero non-renormalizable otherwise.




If c is level zero renormalizable for a map f ∈𝒞^±_d, r, then there exist open sets U∋ c, V∋ c with U compactly contained in V so that 

    f^p: U ⟶ V

is a polynomial-like map with connected small Julia set J(c) := ⋂_k P_k(c).


Since P_k(c) is a nested decreasing sequence of closed disks, 
there exists some integer N so that for any critical point c', either


  * c' ∈ P_k(c) for all k ≥ N; or

  * c' ∉ P_k(c) for all k≥ N.

Let k ≥ N.
Consider the map

    f^p: P_k+p(c) ⟶  P_k(f^p(c)) = P_k(c).

Note that P_k+p(c) ⊆ P_k(c).
Since all cycles of f are repelling, we can find a small neighborhood U, V of P_k+p(c) and P_k(c) so that


  * a critical point c' ∈ U if and only if c' ∈ P_k+p(c);

  * U is compactly contained in V;

  * f^p: U ⟶ V is a branched covering map.

Then any point in J(c) ⊆ U is non-escaping for the map f^p: U ⟶ V.
Let c' ∈ U be a critical point.
Then c' ∈ P_k(c) for all k ≥ N, so c' is non-escaping.
Therefore, f^p: U ⟶ V is a polynomial-like map with connected small Julia set J(c).



If c is level zero renormalizable, then its renormalization period p is at least 2.


Suppose for contradiction that p = 1.
Then we have a polynomial-like map f:U ⟶ V.
By Lemma <ref>, the external rays landing at the fixed points of f: U ⟶ V would cut through the small Julia set J(c) = ⋂_k P_k(c), which is a contradiction.



If c is level zero renormalizable, then f^p: U ⟶ V is a simple renormalization, i.e., the small Julia sets J(c), J(f(c)), ..., J(f^p-1(c)) intersect possibly only at their tip points.


For sufficiently large k, we may assume that the puzzle pieces 

    P_k(c), P_k(f(c)), ..., P_k(f^p-1(c))
 
have disjoint interior.
Therefore, J(f^i(c)) = ⋂_k P_k(f^i(c)) cannot intersect at a cut point.





























 §.§ Level m renormalization puzzles

Let f ∈𝒞^±_d, r be a polynomial or anti-polynomial with connected Julia set and all cycles repelling.
Suppose f is level zero renormalizable.
Let c be a critical point that is level zero renormalizable with renormalization period p.
By Lemma <ref>, we have a polynomial-like map f^p: U ⟶ V with small Julia set J(c). 
The fixed points of f^p in J(c) are periodic points of f with period p.
Note that the polynomial-like map f^p: U ⟶ V has connected Julia set and is periodically repelling.
Thus, the external rays landing at the fixed points of f^p in J(c) cut the small Julia set.

To define the level one renormalization puzzle, we consider 


  * the external rays landing at all fixed points of f; as well as 

  * the external rays landing at the periodic points of f associated with the fixed points of the polynomial-like maps f^p: U ⟶ V for all level zero renormalizable critical points.

Then the closed disk D bounded by the image of the circle {| z| =2} under the Riemann mapping ϕ:  - ⟶ - K(f) is cut by these external rays into finitely many pieces.
These pieces form the puzzle pieces of depth zero for level one renormalization.

Similar as before, the puzzle pieces at depth k+1 are defined inductively by pulling back the puzzle pieces at depth k.
We use the notation P^1_k(z) to denote the puzzle piece at depth k for level one renormalization that contains z in its interior.

Just as in Definition <ref>, we define a critical point c to be level one renormalizable
if c is not on the boundary of any puzzle piece and there exists a number p ≥ 1 so that 

    P^1_k(c) = P^1_k(f^p(c))  for all  k.

If c is level one renormalizable with renormalization period p_1, then the same proof of Lemma <ref> shows that there exists a polynomial-like map

    f^p_1: U^1 ⟶ V^1

with small Julia set J^1(c)= ⋂_k P^1_k(c).

Inductively, if f is level m renormalizable, then we can define the level m+1 renormalization puzzle for f.
We remark that the choice of depth zero puzzle pieces for level n renormalization depends on the polynomial or the anti-polynomial f.

To make our notations uniform in the future, if f is not level m renormalizable, then we define level m+1 renormalization puzzle for f equal to the level m renormalization puzzle.

In this way, for a periodically repelling polynomial or an anti-polynomial f with connected Julia set, we can associate an infinite sequence of puzzles

    𝔓^m(f) = {P^m_k}, m=0, 1, ⋯.

We say that f is finitely renormalizable if this sequence of puzzle 𝔓^m(f)  is eventually constant, and infinitely renormalizable otherwise.

It is clear that we have the following basic nesting property.

If c is level m renormalizable, then c is level j renormalizable for all j ≤ m.
The corresponding renormalization periods satisfy

    p_0 | p_1 | ... | p_m.

Further, the small Julia sets are nested; i.e.,

    J^m(c) ⊆ J^m-1(c) ⊆ ... ⊆ J^0(c).






 §.§ Combinatorial distance


    Let f, g ∈𝒞^±_d, r. 


    We say that g is aligned with f at depth k of level m if for each point p ∈∂ P^m_k,f∩ J(f), there exists a point q ∈ J(g) so that the angles of the external rays landing at p are the same as the angles of those landing at q.

From the definition, it is clear that if g is aligned with f at depth k of level m, then g is aligned with f at depth k' of level m' for all k' ≤ k and m' ≤ m.

Given a puzzle piece P^m_k,f for f, its boundary consists of segments of external rays, their landing points and segments of an equipotential.
If g is aligned with f at depth k of level m, we can construct a puzzle piece for g bounded by the corresponding segments of external rays (with the same angle), their landing points and segments of the equipotential.
We call this the induced puzzle piece for g.
We use the notation P^m_k, g|f(z) to represent the induced puzzle piece for g at depth k of level m that contains z.




We remark that these induced puzzle pieces for g may be different from the level m renormalization puzzle 𝔓^m(g) for g.
Indeed, it is possible for a non-renormalizable polynomial g to align with a renormalizable polynomial f at some depth k and level m with m ≥ 1.
Then there are more induced puzzle pieces at depth zero of level 1 than those in 𝔓^1(g).  As an example, one can take f to be the quadratic Feigenbaum polynomial and pick any non-renormalizable g in the limb of the Mandelbrot set cut out by the parameter rays at angles 2/5 and 3/5.


We define level m renormalization distance from g to f as

    d_m(f, g) = e^-(k_m(f, g)+1)· (m+1),

where 

    k_m(f,g) = inf{j: g  is not aligned with f at depth j of level m}.

We define

    d(f, g) = ∑_m≥ 0 d_m(f, g).

We remark that d is not a distance function as d(f, g) may be different from d(g, f).



































Recall that two periodically repelling polynomials or anti-polynomials are in the same combinatorial class if they have the same rational lamination, and 𝒞^±_d, r consists of combinatorial classes of periodically repelling polynomials or anti-polynomials.

Let f, g ∈𝒞^±_d, r. Then d(f, g) = 0 if and only if f, g have the same rational lamination; i.e., [f] = [g] ∈𝒞^±_d, r.


It is clear that if f, g have the same rational lamination, then the induced puzzle for g agrees with the renormalization puzzle for f at all levels and depths. Thus, d(f, g) = 0.

Conversely, suppose that s, t ∈ form a leaf of the rational lamination for f; i.e., they coland at some pre-periodic point x for f.
If x is on the boundary of some puzzle piece, then s, t also coland for g as d(f, g) = 0.
So s, t form a leaf for the rational lamination for g as well.

Thus, we assume that x is not on the boundary of any puzzle piece.
Assume that x has pre-period l and period p, and y = f^l(x).
Then f^p: (P^m_k+p, f(y)) ⟶(P^m_k, f(y)) is a proper map for all m and k.
We can choose m large enough so that the level m renormalization period for any critical point is larger than p.
(Here we use the convention that the renormalization period is ∞ if the critical point is not renormalizable.)
Thus, we can then choose N large enough so that (P^m_k+p, f(y)) contains no critical points for k ≥ N.
So f^p: (P^m_k+p, f(y)) ⟶(P^m_k, f(y)) has degree 1 for all k ≥ N.
Therefore, ⋂_k P^m_k, f(y) = { y}, so ⋂_k P^m_k, f(x) = { x}.
Hence, we have a sequence of angles s_k, t_k colanding at boundary points of some puzzle pieces with s_k → s and t_k → t.
Since d(f, g) = 0, the angles s_k, t_k define a leaf for the rational lamination of g, for all k.
As rational laminations are closed, the angles s, t also define a leaf of the rational lamination for g.
Therefore, the rational lamination of f is contained in the rational lamination of g.
The other inclusion can be proved similarly.






 §.§ Combinatorial continuity

In this section, we will prove a combinatorial continuity statement for maps in 𝒞^±_d, r.
The proof is divided into two cases depending on whether there are critical points on boundaries of puzzle pieces or not.




  §.§.§ No boundary critical points

We first present the proof where no critical points are on the boundaries of puzzle pieces.

Let f ∈𝒞^±_d, r. Suppose that no critical point is on the boundary of any puzzle piece. 


  * If f_n → f in 𝒞^±_d, r, then d(f, f_n) → 0.

  * If d(f, f_n) → 0, then for any accumulation point g∈𝒞^±_d, r of f_n, we have that d(f, g) = 0.




Suppose that for some fixed level m, the distance d_m(f, f_n) → 0 as n→∞. We remark that it is not true that any accumulation point g∈𝒞^±_d, r of f_n satisfies d_m(f, g) = 0.

For example, let f be the quadratic Feigenbaum polynomial, and f_n be the tuning of the basilica polynomial with a sequence of real polynomials g_n that goes to the Chebyshev polynomial.
The limit g = lim_n f_n is the basilica polynomial tuned with the Chebyshev polynomial.
Then d_0(f, f_n) = 0 for all n, but d_0(f, g) ≠ 0 since two rays (lying on the boundary of level zero puzzles) that land at distinct Julia points for f coland at the critical point of g.





Let f ∈𝒞^±_d, r. Suppose that no critical point is on the boundary of any puzzle piece.
Let c be a critical point for f, and P^m_k(c) be the level m puzzle piece at depth k that contains c.
Then there exist l, j so that P^l_j(c) is compactly contained in (P^m_k(c)).










Suppose for contradiction that there exists a Julia point x on the boundary of P^m_k(c) so that x ∈ P^l_j(c) for all j, l.
Let q be the pre-period and p be the period of the pre-periodic point x. The assumption x ∈ P^l_j(c) for all j, l implies that f^q+np(x)=f^q(x)∈ P^l_j(f^q+np(c)) for all j, l, and n. It follows that P^l_j(f^q(c)) = P^l_j(f^q+rp(c)), where r is the valence of f^q(x) as a branch point of J(f).


If for some l, P^l_j(f^q(c)) contains no critical points for all large j, then ⋂_j P^l_j(f^q(c)) = f^q(x); i.e., f^q(c)=f^q(x).
This is a contradiction to the fact that c is not on the boundary of any puzzle piece.
Otherwise, there exists some critical point c' (potentially equal to c) which is infinitely renormalizable, and f^q(x) is in the small Julia set J^l(c') for all l. But this is not possible as the period of f^q(x) is a fixed integer.


We are now ready to prove Theorem <ref>.

To prove the first statement, it suffices to show d_m(f, f_n) → 0 for all m.
Let P be a puzzle piece of level m and depth k.
Since there are no critical points of f on the boundaries of any puzzle pieces, any points in ∂ P is eventually repelling and its forward iterate avoids any critical points.
Thus, by stability of landing external rays (see <cit.>), there exists a small neighborhood U_m,k of f so that any map g ∈ U_m,k is aligned with f at depth k of level m.
Therefore d_m(f, f_n) → 0 as n →∞.

For the second statement, by passing to a subsequence, we may assume that f_n → g ∈𝒞^±_d, r. 
It suffices to show that d_m(f, g) = 0 for all m.
Let m be the minimal level where d_m(f, g) ≠ 0, and let k:= k_m(f, g).

We claim that k > 0.
By way of contradiction, suppose that k=0.
Let {t_i} be the set of angles of external rays landing at boundaries of the puzzle pieces at depth 0 of 𝔓^m(f).
Note that the Julia points on boundaries of puzzle pieces at depth 0 for f are periodic points, so t_i are periodic under the map m_d(t) = dt  or m_-d(t)=-dt.
Since d(f, f_n) → 0, the external rays with angles in {t_i} have the same landing patterns in the dynamical planes of f_n and f for all sufficiently large n.
By our assumption, the landing patterns of the external rays with angles in {t_i} in the dynamical plane of g are different from the common landing patterns of these rays in the dynamical planes of f and f_n (with sufficiently large n).
Thus, by stability of landing external rays, some angle must land at critical points.
But this is not possible as each t_i is periodic.

Let P_k, f^m be a puzzle piece for f which does not correspond to an induced puzzle piece for g.
Since k > 0, P_k-1, f^m:=f(P_k, f^m) is a puzzle piece at depth k-1.
It corresponds to some induced puzzle piece P_k-1, g|f^m for g.
We claim that some critical value v of g must lie on ∂P_k-1, g|f^m.
Indeed, otherwise, the Julia points on the boundary of g^-1(P_k-1, g|f^m) are pre-repelling and avoid critical points. Since P_k, f^m does not correspond to a puzzle piece for g, this is a contradiction by stability of landing rays.

Let c_g be the critical point of g that is mapped to v.
Since d(f, f_n) → 0, the puzzle pieces of f_n at depth k and level m align with f for all large n.
Since no critical point of f is on the boundary of puzzle pieces, for all large n, no critical point of f_n is on the boundary of puzzle pieces of f_n at depth k and level m.
Since f_n → g, there exists a sequence of critical points c_n for f_n with c_n → c_g.
Let P_k, f_n^m(c_n) be the puzzle piece that contains c_n, which corresponds to some puzzle piece P_k, f^m(n) for f.
Since there are only finitely many puzzle pieces at depth k and level m, we may assume that P_k, f^m(n) is eventually constant.
Denote this puzzle piece by P_k, f^m. Note that it contains some critical point c of f.
One can verify that P_k, f^m does not correspond to an induced puzzle piece for g.
Since f(P_k, f^m) corresponds to an induced puzzle piece for g, there exist two angles α, β landing at distinct points a, b ∈∂ P_k, f^m for f, but they land at a critical point for g.
By Lemma <ref>, there exists some puzzle piece P^m_1_k_1, f(c) that is compactly contained in (P_k, f^m) = (P_k, f^m(c)).
Inductively, we obtain a sequence 

    ⋯⋐(P^m_2_k_2, f(c)) ⋐(P^m_1_k_1, f(c)) ⋐(P^m_k, f(c)).

Therefore, we can find a sequence of pairs of angles t_i, s_i so that 


  * t_i, s_i coland at a point on ∂ P^m_i_k_i, f(c);

  * the union of external rays ℛ_f(t_i)∪ℛ_f(s_i) separates the two points a, b ∈∂ P_k, f^m(c).

Therefore, ℛ_f_n(t_i) and ℛ_f_n(s_i) land at the same point for all sufficiently large n. 
On the other hand, since α, β coland for g, the union ℛ_g(α)∪ℛ_g(β) separates the external rays ℛ_g(t_i) and ℛ_g(s_i).
It follows that both ℛ_g(t_i) and ℛ_g(s_i) land at c_g (see <cit.>).
This is a contradiction, as there can be at most finitely many external rays landing at a pre-periodic point.














  §.§.§ With boundary critical points

To deal with the situation where critical points lie on boundaries of puzzle pieces, we need to modify the definition of combinatorial distances.


Let f ∈𝒞^±_d, r with puzzles 𝔓^m.
Let c be a critical point on the boundary of some puzzle piece.
Then c is pre-periodic.
Denote by 𝒪 the periodic cycle associated to c.
We define a modification of 𝔓^m by removing all external rays landing on 𝒪 and their iterated preimages in the definition of puzzle pieces.

As the first step of the modification, we denote by 𝔓^m the modified puzzle after removing all external rays associated with all critical points on the boundaries of puzzle pieces.





We say that c is statically renormalizable with respect to the modified puzzle 𝔓^m_0 if c is renormalizable with respect to the modified puzzle 𝔓^m_0 and

    ⋂_k P^m_k(c) = ⋂_k P^m_0_k(c)
 
for all m ≥ m_0.
We denote by

    J^∞(c) = J^m_0(c) := ⋂_k P^m_0_k(c) = ⋂_m ⋂_k P^m_k(c)

the static small Julia set for c.
Similarly, we say that a critical point c is statically pre-renormalizable if c is eventually mapped to some static small Julia set.



    We remark that if c is statically renormalizable, then c is renormalizable with respect to 𝔓^m for all m.
    However, for all m ≥ m_0, the renormalization with respect to 𝔓^m stays the same, as the condition ⋂_k P^m_k(c) = ⋂_k P^m_0_k(c) implies that all higher level puzzle pieces do not cut through the small Julia set J^m_0(c).
    This is a consequence of the fact we generate the puzzle by rays landing at the fixed points of renormalizations.





Let f ∈𝒞^±_d, r.
Let c be a statically renormalizable critical point with respect to 𝔓^m_0.
Then the corresponding polynomial-like map

    f^p_m_0: U^m_0⟶ V^m_0

is finitely renormalizable.


If f^p_m_0: U^m_0⟶ V^m_0 is infinitely renormalizable, then as m →∞, there are infinitely many puzzle pieces of 𝔓^m at depth 0 that cut through the small Julia set.
The modified puzzles can only omit finitely many of them.
Therefore, as m →∞, there exist an unbounded number of modified puzzle pieces of 𝔓^m at depth 0 that cut through the small Julia set J^∞(c). 
Since c is a statically renormalizable, there are no modified puzzle pieces of 𝔓^m at depth 0 that cut through J^∞(c).
This is a contradiction.



























The second step of the modification is not canonical.
Let c be a statically renormalizable critical point for 𝔓^m_0, with static small Julia set J^∞(c).
We choose periodic cut cycles in J^∞(c) disjoint from the critical orbit of f.
We add puzzle pieces induced by the external rays landing at these cut points as depth zero puzzles in 𝔓^m for all m ≥ m_0.
Pulling back by dynamics, we add depth k puzzles in 𝔓^m.
By Lemma <ref>, we can choose these periodic cut cycles so that no critical point in J^∞(c) is renormalizable with respect to the new modified puzzle.
By making such modifications for each static renormalization critical point, we obtain a new puzzle system.
We denote it by 𝔓.
We denote by d(f, g) the distance from g to f with respect to the modified puzzle 𝔓.

By our construction, any static critical point is only finitely renormalizable with respect to the modified puzzle 𝔓. 
A similar argument as Lemma <ref> gives

Let f, g∈𝒞^±_d, r. 
Then d(f, g) = 0 if and only if f, g have the same rational lamination.



By construction, no critical point is on the boundary of puzzle pieces in 𝔓. Thus, the proof of Theorem <ref> applies to this setting as well, and yields:

Let f, f_n ∈𝒞^±_d, r.


  * If f_n → f in 𝒞^±_d, r, then d(f, f_n) → 0.


  * If d(f, f_n) → 0, 
then for any accumulation point g∈𝒞^±_d, r of f_n, we have that d(f, g) = 0.





















































 §.§ Approximation by post-critically finite maps

In this subsection, we will prove the following theorem about approximating any combinatorial class using post-critically finite maps.

Let f∈𝒞^±_d, r. Then there exists a sequence of post-critically finite polynomials f_n ∈𝒞^±_d, r so that d(f, f_n) → 0.




    Let λ(f) be the rational lamination of f.
    By <cit.>, there exists a sequence of post-critically finite f_n converging to a periodically repelling map g so that λ(g) = λ(f).
    Since f is periodically repelling, f_n can be chosen to be periodically repelling as well.
    Then by Theorem <ref>, we have  d(g, f_n) → 0. Since λ(g) = λ(f) (and the construction of the puzzles can be uniquely recovered from the lamination), it follows that d(f, f_n) → 0.




































§ COMBINATORIAL CLASSES FOR POLYGONAL SCHWARZ REFLECTIONS


Recall that for a Schwarz reflection map S:→, the set 𝔖 stands for the singular (fixed) points on the boundary ∂. The local dynamics near a singular point is parabolic in nature. In fact, for S∈𝒮__d, these singular fixed points have attracting directions in the tiling set. Indeed, the points in 𝔖 correspond, under the conformal map ψ:→ T^∞(S) that conjugates _d to S, to the (parabolic) fixed points of _d. The set of (parabolic) fixed points of _d, denoted by Fix(_d), consists of the (d+1)-st roots of unity. It is easy to see that points in Fix(_d) attract nearby points in  under iterations of _d.

We define the space 𝒮__d,r⊂𝒮__d as the collection of those maps S for which 


  * no point in 𝔖 has an attracting direction in K(S), and 

  * each periodic point of S not in 𝔖 is repelling.





 §.§ Renormalization puzzles

Recall that T^0(S) is the fundamental tile for the Schwarz reflection map S, and connected components of S^-n(T^0(S)) are called tiles of rank n. As in Proposition <ref>, we denote the union of the tiles of rank 0 through k by E^k. Clearly, E^0=T^0(S).
We define the depth zero equipotential for S as ∂ E^0.
Similarly, the depth k equipotential is defined as ∂ E^k.

Note that these equipotentials enclose K(S); however, unlike the polynomial or anti-polynomial setting, an equipotential for a Schwarz reflection always intersects the limit set Λ(S).
We also remark that any intersection point of an equipotential of depth zero with the limit set Λ(S) is a fixed point of S.

To define level zero renormalization puzzles for a Schwarz reflection S ∈𝒮__d,r, we consider the union ⋃ of closures of all dynamical rays landing at all fixed points of S (see Section <ref> for the definition of dynamical rays for Schwarz reflections in 𝒮__d).
A similar counting argument as in Lemma <ref> shows that this closure cuts the limit set Λ(S).
We define puzzle pieces at depth zero for level zero renormalization as closures of the connected components of 

    - E^0 - ⋃

intersecting Λ(S).
Similarly as in the polynomial case, the puzzle pieces at depth k+1 are defined inductively as closures of the components of S^-1((P)), where P ranges over all puzzle pieces at depth k.
We will use a similar notation as in the polynomial case, where P_k(z) represents the depth k puzzle piece that contains z in its interior.

Renormalization can be defined similarly.

Let c be a critical point for S.
We say that c is level zero renormalizable if there exists an integer p ≥ 1 so that 

    P_k(c) = P_k(S^p(c))  for all  k≥ 0.

The smallest such integer p is called the renormalization period.

We say that S is level zero renormalizable if any some critical point of S is level zero renormalizable, and level zero non-renormalizable otherwise.


Similar to the polynomial setting, we have the following lemma (c.f. Lemma <ref>).

If c is level zero renormalizable with renormalization period p>2, then there exist open sets U∋ c, V∋ c with U compactly contained in V so that 

    S^p: U ⟶ V

is a polynomial/anti-polynomial-like map with small Julia set J(c) := ⋂_k P_k(c).


We comment on the nuance of the assumption that the renormalization period p is larger than two.
Indeed, the same argument as in Lemma <ref> gives a branched covering between puzzles P_k+p(c) ⊆ P_k(S^p(c)) = P_k(c),

    S^p: P_k+p(c) ⟶  P_k(c).

The subtlety here is to thicken the puzzle pieces slightly to ensure compact containment between them.
Since S is assumed to be periodically repelling, this can be done if there are no singular points on the boundaries of the puzzle pieces.

Now let x be a singular point. Then there are at most two different nested sequences of puzzle pieces P_n with x ∈∂ P_n (recall that at most two quadrature domains can touch at a point).
Thus, we have S^2: P_n+2⟶ P_n.
Therefore, if the renormalization period p is larger than two, no singular point can lie on the boundary, and the lemma follows.


A modification of the proof of Corollary <ref> also gives the following lemma.

If c is level zero renormalizable, then its renormalization period p is at least 2.


By way of contradiction, let us assume that the renormalization period p is equal to 1.
The proof is identical to Corollary <ref> if there are no singular points on J(c) := ⋂_k P_k(c).
Thus, we may assume that there is a singular point x ∈ J(c).
Then for all large n, we have a map S: P_n+1(c) ⟶ P_n(c) of degree e>1.
Since we assume that the map is periodically repelling, we can extend the map S: P_n+1(c) ⟶ P_n(c) to a topological degree e branched covering F of the sphere so that x has (Lefschetz fixed point) index -1 and there is a unique fixed point in - P_n(c) with index +1.
Thus, by Lefschetz–Hopf fixed point theorem, there exists another fixed point z ≠ x in P_n(c).
The arguments used in the proof of Corollary <ref> now show that the dynamical rays landing at the fixed
points of S: P_n+1(c) ⟶ P_n(c) cut through the small Julia set J(c).


We can define level m renormalization in this setting as in the polynomial or anti-polynomial case.
The following lemma says that except possibly level zero renormalizations, all subsequent renormalizations are actually polynomial/anti-polynomial-like maps.


If c is level m renormalizable with m ≥ 1, then there exist open sets U∋ c, V∋ c with U compactly contained in V so that 

    S^p_0p_1...p_m: U ⟶ V

is a polynomial/anti-polynomial-like map with small Julia set J^m(c):= ⋂_k P^m_k(c).


Note that the arguments of Lemma <ref> imply that p_0, p_1 ≥ 2, so p_0p_1...p_m>2.
Thus the lemma follows from arguments used in the proof of Lemma <ref>.


Recall that 𝒮__d, r consists of combinatorial classes of periodically repelling Schwarz reflections in 𝒮__d.
The same proof as Lemma <ref> also gives the following lemma.

Let S, R ∈𝒮__d,r. Then d(S, R) = 0 if and only if S, R have the same rational lamination; i.e., [S] = [R] ∈𝒮__d, r.






Let S ∈𝒮__d,r. Suppose that no critical point is on the boundary of any puzzle piece. 


  * If S_n → S in 𝒮__d,r, then d(S, S_n) → 0.

  * If d(S, S_n) → 0, then for any accumulation point R∈𝒮__d,r of S_n, we have that d(S, R) = 0.



    The first statement follows from the same argument as in Theorem <ref>, as an external ray landing at a repelling periodic point, or a repelling pre-periodic point whose orbit does not contain a critical point is stable under perturbation (see Proposition <ref>).

    By Lemma <ref>, all renormalization of level m ≥ 1 are (anti-)polynomial-like maps.
    Thus, the argument of the Separation Lemma <ref> applies.
    Now the same argument in Theorem <ref> applies to show the second statement.


If there are critical points on the boundary of a puzzle piece, then we can define modified puzzles 𝔓^m for S exactly as in the (anti-)polynomial setting.
Then, no critical points lie on the boundary of puzzle pieces for 𝔓. This gives the following combinatorial continuity result, where d(S_1, S_2) stands for the distance from S_2 to S_1 with respect to the modified puzzle 𝔓.




Let S, S_n ∈𝒮__d,r.


  * If S_n → S in 𝒮__d,r, then d(S, S_n) → 0.


  * If d(S, S_n) → 0 
then for any accumulation point R∈𝒮__d,r of S_n, we have that d(S, R) = 0.








§ COMBINATORIAL STRAIGHTENING

In this section, we shall assemble all the ingredients to prove Theorem <ref>.

To construct the map Φ, we first note that if f is post-critically finite, then there exists unique conformal mating S ∈𝒮__d,r of f with _d (by Theorem <ref>).
Let [f] and [S] be the combinatorial class of f and S respectively.
We define Φ([f]) = [S].

Let f ∈𝒞^-_d, r be a periodically repelling anti-polynomial.
By Theorem <ref>, there exists a sequence of post-critically finite anti-polynomials f_n with d(f, f_n) → 0. Let  S_n ∈𝒮__d,r be the unique conformal mating of f_n with _d.
By the compactness result Theorem <ref>, after passing to a subsequence, we may assume that S_n converges.
Denote this limit by S.
Using a similar argument in <cit.>, one can show that S cannot have parabolic cycles or irrationally neutral cycles.
Thus S ∈𝒮__d,r.
We define Φ([f]) = [S].

Since S_n → S, by Theorem <ref>, d(S, S_n) → 0.

We claim that the rational lamination (ℰ_d)_*(λ(S)) = λ(f).
Indeed, since (ℰ_d)_*(λ(S)) is m_-d-invariant, there exists a periodically repelling anti-polynomial g with λ(g) = (ℰ_d)_*(λ(S)) (see <cit.>).
Since (ℰ_d)_*(λ(S_n)) = λ(f_n), and the construction of the puzzles can be uniquely recovered from the lamination, we conclude that d(g, f_n) → 0.
Therefore, by Theorem <ref>, d(f, g) = 0, and hence λ(f) = λ(g) by Lemma <ref>.
This also shows that the map Φ is well-defined.












We now show that Φ is continuous.
Suppose f_n → f. 
Then d(f, f_n) → 0.
Let S_n ∈Φ([f_n]) and let S∈𝒮__d,r be an accumulation point of S_n.
By Theorem <ref>, d(S, S_n) → 0.
The same argument as above shows that (ℰ_d)_*(λ(S)) = λ(f).
Therefore S ∈Φ([f]), proving continuity of Φ.

A similar argument can be applied to construct a well-defined continuous inverse Φ^-1, and the theorem follows.




§ RIGIDITY OF AT MOST FINITELY RENORMALIZABLE SCHWARZ REFLECTIONS

In this section, we will show that a periodically repelling, finitely renormalizable Schwarz reflection is rigid, and deduce Theorem <ref>.
More precisely, we have

Let S ∈𝒮__d,r be finitely renormalizable.
Then


  * the limit set Λ(S) is locally connected;

  * S has no invariant line field on Λ(S);

  * if S∈𝒮__d,r has the same rational lamination as S, then S = S. In particular, the combinatorial class of S is a singleton.



The proof uses rigidity of complex box mappings, which is a refinement of the notion of generalized polynomial-like maps (cf. <cit.>, <cit.>). Although the proof is almost identical to the polynomial setting (see <cit.>, cf. <cit.>), there is a subtle difference from the polynomial setting, as the complement - 𝒟 of the domain of a Schwarz reflection S: 𝒟⟶ may be pinched (i.e., not Jordan), which requires some additional treatment.
(<cit.>)
A holomorphic (or anti-holomorphic) map F: U ⟶ V between open sets U ⊆ V in  is a complex box mapping if


  * F has finitely many critical points;

  * V is a union of finitely many pairwise disjoint Jordan disks;

  * let V' be a component of V, then either V' is a component of U or V' ∩ U is a union of Jordan disks with pairwise disjoint closures which are compactly contained in V';

  * let U' be a component of U, then F(U') is a component of V.





Since S is finitely renormalizable, we can choose the level m puzzle 𝔓 = 𝔓^m so that S is non-renormalizable with respect to 𝔓.
The same argument as in <cit.> allows us to construct a finite collection of open puzzle pieces V that contains all critical points, such that the first return map into V defines a complex box mapping F:U ⟶ V.

Let {a_1,..., a_r} be the collection of pinching points for - 𝒟 where 𝒟 is the domain for S.
Note that each a_i is a fixed point for S.
We may assume that {a_1,..., a_r}∩V = ∅.

Note that for each depth k, there are at most two puzzle pieces attached at a_i.
Since S is non-renormalizable with respect to 𝔓, the puzzle pieces attached at a_i contain no critical points at all sufficiently large depth k.
Since the local dynamics on Λ(S) near these fixed points are parabolic repelling, these puzzle pieces shrink to points as the depth k →∞.

Now choose k_0 large enough so that these puzzle pieces have pairwise disjoint closure except at a_i and their closure are also disjoint from V.
Let W be the union of all depth k_0 puzzle pieces attached to a_i for some i. 

Then rigidity of complex box mappings <cit.> and the above definition of W imply that all puzzle pieces in V∪ W shrink to points, and it carries no measurable invariant line field on its Julia set.
Note that the set of points which are never mapped into V ∪ W is hyperbolic, so all puzzle pieces shrink to points and its area is 0.
This proves (1) and (2) in Theorem <ref>.

Let S∈𝒮__d,r be a normalized polygonal Schwarz reflection that has the same lamination as S.
Then S is also finitely renormalizable.
Since the limit sets Λ(S) and Λ(S) are locally connected, and all cycles of S, S are repelling, the maps S|_K(S) and S|_K(S) are topologically conjugate.
By the same argument as in <cit.>, the corresponding complex box mappings F:U ⟶ V and F: U⟶V are quasiconformally conjugate.
By spreading around the conjugacy (see the Spreading Principle <cit.>), we can promote this local quasiconformal conjugacy to a quasiconformal map Ψ: ⟶ that conjugates the dynamics of S and S.
Since the external dynamics of S, S are conformally conjugate, and the Julia set Λ(S) is a dendrite that does not support any invariant line field, Ψ is a conformal map.
This proves (3) in Theorem <ref>.



The theorem now follows from Theorem <ref> and Theorem <ref>.









§ CORRESPONDENCES ASSOCIATED WITH POLYGONAL SCHWARZ REFLECTIONS


Let S:𝒟→ be a conformal mating of a degree d anti-polynomial f with connected Julia set and the Nielsen map _d. Recall that 

    = ⋃_i=1^k Ω_i,

where Ω_i are Jordan quadrature domains, and S|_Ω_i is the Schwarz reflection map of Ω_i.
In this section, we will associate anti-holomorphic correspondences with such matings, and show that they can also be interpreted as matings of anti-polynomials and ideal (d+1)-gon reflection groups. 

Let R_i:→Ω_i be uniformizing rational maps. To make book-keeping easier, we denote the domain of R_i (which is a copy of the Riemann sphere) by _i and the unit disk in _i by _i. We also set ℐ:={1,⋯,k}.

We recall that ∂ meets the limit set Λ(S) in the finite set 𝔖, which is the set of singular points on ∂.
We define 

    𝔖_i := (R_i|_∂_i)^-1(𝔖∩∂Ω_i).
 





 §.§ Construction of the correspondence




Let us now consider the disjoint union 

    𝔘 := _i∈ℐ_i ≅ ×ℐ,

and define the maps

    R: 𝔘⟶,   (z,i)↦ R_i(z),

and

    η^- : 𝔘⟶𝔘,   (z,i)↦ (η^-(z),i).

Note that R is a branched covering of degree d+1, and η^- is a homeomorphism.

The conformal mating S gives rise to an anti-holomorphic correspondence on 𝔘 as follows. For i,j∈ℐ, define

    ℭ_i,j:= {(z,w)∈_i×_j: R_j(w)-R_i(η^-(z))/w-η^-(z)=0},    if i=j,

and

    ℭ_i,j:= {(z,w)∈_i×_j: R_j(w)-R_i(η^-(z))=0},    if i≠ j.

The union of the real-algebraic curves ℭ_i,j can be written succinctly as

    {(𝔲_1,𝔲_2)∈𝔘×𝔘: R(𝔲_2)-R(η^-(𝔲_1))/𝔲_2-η^-(𝔲_1)=0}.

(The division in Equation (<ref>) makes sense since the numerator and the denominator can be viewed as points of .)
The first and second coordinate projection maps π_1^i and π_2^j from ℭ_i,j onto _i and _j define an anti-holomorphic correspondence (i.e., a multi-valued map with anti-holomorphic local branches) from _i onto _j (cf. <cit.>):

    ℭ_i,jdl[swap]π_1^idrπ_2^j
    _i       _j.

Combining all these anti-holomorphic correspondences for various i,j∈ℐ, we obtain an anti-holomorphic correspondence on 𝔘 defined by the (in general reducible) curve ∑_i,jℭ_i,j. We denote this correspondence by ℭ^⊛.

The following observations show that ℭ^⊛ is obtained by lifting S and its appropriate backward branches via the rational maps R_i. 


∙ Fix z∈_i. Then, S(R_i(z))=R_i(η^-(z)), and hence,

    ((z,,i),(w,j))∈ℭ^⊛ R_j(w)=R_i(η^-(z))=S(R_i(z)), (w,j)≠η^-(z,i).



∙ Now fix z∈_i∖_i. Then, S(R_i(η^-(z)))=R_i(z); i.e.,

    ((z,,i),(w,j))∈ℭ^⊛ R_j(w)=R_i(η^-(z))=S^-1(R_i(z)), (w,j)≠η^-(z,i),

where S^-1 is a suitable backward branch of S. 


To capture our desired mating phenomenon, we need to pass to a correspondence induced by ℭ^⊛ on a quotient of 𝔘. To this end, we endow 𝔘 with the following finite equivalence relation:


For z∈𝔖_i⊂_i and w∈𝔖_j⊂_j,


(z,i)∼_w (w,j) R_i(z)=R_j(w).




The fact that  is the quotient of  by a finite lamination and that R_i|_∂_i is injective (for all i∈ℐ) imply that 

    𝔚 := 𝔘∼_w

has a trivial fundamental group (it can be thought of as a compact, simply connected, noded Riemann surface). 
By definition, the map R descends to a map from 𝔚⟶. This induced map, which we denote by R, is also a branched covering. Abusing notation, we denote the image of a set X⊂𝔘 (respectively, a point p∈𝔘) under the quotient map 𝔘⟶𝔚 by X (respectively, p). We further set 

    𝔇:=⋃_i∈ℐ_i,  𝔖:=⋃_i∈ℐ𝔖_i.
 



The homeomorphism η^-:𝔘→𝔘 descends to a homeomorphism η:𝔚⟶𝔚.


Let us suppose that (z,i)∼_w (w,j); i.e., z∈𝔖_i, w∈𝔖_j, and R_i(z)=R_j(w). We need to show that η^-(z,i)∼_wη^-(w,j). However, this is obvious since η^-(z,i)=(η^-(z),i)=(z,i)∈𝔖_i and η^-(w,j)=(η^-(w),j)=(w,j) ∈ 𝔖_j. 
 

Thus, the correspondence ℭ^⊛ on 𝔘 also descends to a correspondence on 𝔚. We denote this correspondence by ℭ.



 §.§ Dynamical partition for ℭ



The invariant partition of the dynamical plane of S, given by K(S) and T^∞(S), can be pulled back by R to produce an invariant partition of the dynamical plane of the correspondence ℭ.
We define

    𝒦:=R^-1(K(S)),  𝒯:= R^-1(T^∞(S)).




  * η(𝒯)=𝒯, and η(𝒦)=𝒦.


  * Let (𝔲_1,𝔲_2)∈ℭ. Then 𝔲_1∈𝒯 (respectively, 𝔲_1∈𝒦) if and only if 𝔲_2∈𝒯 (respectively, 𝔲_2∈𝒦).



1) It suffices to show that η(𝒦)=𝒦. 

Let us first assume that (z,i)∈_i∩𝒦. Then, R_i(z)∈ K(S) and R_i(η^-(z))=S(R_i(z)). As K(S) is invariant under S, it follows that R_i(η^-(z))∈ K(S). We conclude that η(z,i)=(η^-(z),i)∈𝒦.

Next let (z,i)∈𝒦∖_i. Then, R_i(z)∈ K(S) and S(R_i(η^-(z)))=R_i(z). As K(S) is also backward invariant under S, it follows that R_i(η^-(z))∈ K(S), and hence η(z,i)=(η^-(z),i)∈𝒦.

2) Let (𝔲_1,𝔲_2)∈ℭ, where 𝔲_1=(z,i) and 𝔲_2=(w,j). It suffices to show that if (z,i)∈𝒯 (respectively, if (z,i)∈𝒦), then (w,j)∈𝒯 (respectively, (w,j)∈𝒦). 

To this end, first suppose that z∈_i, which implies that R_j(w)=R_i(η^-(z))=S(R_i(z)). Now let (z,i)∈𝒯 (respectively, (z,i)∈𝒦). The S-invariance of T^∞(S) (respectively, of K(S)) implies that R(w,j)=R_j(w)∈ T^∞(S) (respectively, R(w,j)=R_j(w)∈ K(S)). Hence, (w,j)∈𝒯 (respectively, (w,j)∈𝒦). 

Next suppose that z∈_i∖_i, which implies that S(R_j(w))=R_i(z). Now let (z,i)∈𝒯 (respectively, (z,i)∈𝒦). The backward invariance of T^∞(S) (respectively, of K(S)) under S implies that R(w,j)=R_j(w)∈ T^∞(S) (respectively, R(w,j)=R_j(w)∈ K(S)). Hence, (w,j)∈𝒯 (respectively, (w,j)∈𝒦). 





 §.§ Polynomial structure of ℭ


Note that η maps 𝔇 onto 𝔚∖𝔇, and preserves 𝔖. 
We set 𝒦_1:=𝒦∩𝔇 and 𝒦_2:=𝒦∖𝔇.





We have that 𝒦_2=η(𝒦_1), and 𝒦_1∩𝒦_2=𝔖.


The first statement follows from the fact that η(𝒯)=𝒯. 

For the second statement, first observe that

    𝒦_1∩𝒦_2={𝔲∈∂𝔇: R(𝔲)∈ K(S)}.

The result is now a consequence of the fact that R(𝔖)=𝔖⊂ K(S) and R(∂𝔇∖ 𝔖)=∂∖𝔖⊂ T^∞(S).






  * 𝒦_2 is forward invariant, and hence, 𝒦_1 is backward invariant under ℭ.


  * ℭ has a forward branch carrying 𝒦_1 onto itself with degree d, and this branch is conformally conjugate to S:K(S)→ K(S). 


  * ℭ has a backward branch carrying 𝒦_2 onto itself with degree d, and this branch is topologically conjugate to S:K(S)→ K(S).



1) This follows immediately from the definition of ℭ and Lemma <ref>. 
 
2) The forward branch of ℭ carrying 𝒦_1 onto itself (with degree d) acts as 

    𝔟: (z,i)↦(R|_𝔇)^-1(R(η(z,i)))= ((R_j|__j)^-1(R_i(η^-(z))),j),

where (z,i)∈𝒦_1, and R_i(η^-(z))∈Ω_j.
It is easy to see from the construction that R:𝒦_1⟶ K(S) is a homeomorphism. We claim that R|_𝒦_1 is the desired conjugating map. To this end, note that 

    S(R(z,i))=S(R_i(z))=R_i(η^-(z)),

and hence

    R(𝔟(z,i))=R((R|_𝔇)^-1(R(η(z,i))))=R_i(η^-(z))=S(R(z,i)).


It follows that R|_𝒦_1∘𝔟=S∘R|_𝒦_1.

3) This follows from the previous part and the fact that η conjugates a forward branch of the correspondence on 𝒦_1 to a backward branch of the correspondence on 𝒦_2.






 §.§ Group structure of ℭ


The fact that S: T^∞(S)∩⟶ T^∞(S) is conformally conjugate to the Nielsen map _d implies that S has no critical value in T^∞(S). Hence, in light of the description of S in terms of η^- and R_i, we conclude that no R_i has a critical value in 𝒯.
Thus, R:𝒯→ T^∞(S) is a covering map. Simple connectedness of T^∞(S) now implies that 𝒯 is the union of d+1 many disjoint open topological disks U_0,⋯,U_d.

We define a conformal automorphism

    τ:𝒯→𝒯
 
satisfying the conditions


  * τ(U_j)=U_j+1, j∈/(d+1),


  * R∘τ=R, and


  * τ^d+1=id,


  * R^-1(R(z))={z,τ(z),⋯,τ^d(z)} ∀ z∈𝒯.

By construction, the forward branches of ℭ on 𝒯 are given by the anti-conformal automorphisms τ∘η,⋯,τ^d∘η. 

Recall the notation T^0≡ T^0(S)=∖(∪𝔖). We call T^0 the rank zero tile of T^∞(S), and connected components of S^-n(T^0) tiles of rank n. This yields a natural tessellation of the lifted tiling set 𝒯.  A connected component of the pre-image of a rank n tile of T^∞(S) under R is called a rank n tile of 𝒯.




  * The grand orbits of the correspondence ℭ on 𝒯 are equal to the orbits of ⟨η⟩∗⟨τ⟩. 

  * The group ⟨η⟩∗⟨τ⟩ acts properly discontinuously on , and /ℭ is biholomorphic to /G_d (where G_d is the regular ideal (d+1)-gon reflection group).



1) We have already observed that the forward branches of ℭ on 𝒯 are given by τ∘η,⋯,τ^d∘η. Furthermore, τ=(τ^2∘η)∘(τ∘η)^-1, and hence ⟨τ∘η,⋯,τ^d∘η⟩=⟨η,τ⟩. It now remains to justify that ⟨η,τ⟩ is the free product of the cyclic groups ⟨η⟩ and ⟨τ⟩.

To this end, first note that any relation in ⟨η,τ⟩ other than η^2=id and τ^d+1=id can be reduced to one of the form 

    (τ^k_1∘η)∘⋯∘(τ^k_r∘η)=id
 
or 

    (τ^k_1∘η)∘⋯∘(τ^k_r∘η)=η,

where k_1,⋯,k_r∈{1,⋯,n}.


Case 1: Let us first assume that there exists a relation of the form (<ref>) in ⟨η,τ⟩. It is readily seen using Relation (<ref>) that (τ^k_p∘η) maps the interior of a tile of rank s in 𝒯∖𝔇 to the interior of a tile of rank (s+1) in 𝒯∖𝔇. Hence, the group element on the left of Relation (<ref>) maps the tile of rank 0 to a tile of rank r. Clearly, such an element cannot be the identity map.


Case 2: Now we consider a relation of the form (<ref>) in ⟨τ,η⟩. Each (τ^k_p∘η) maps 𝒯∖𝔇 to itself. Hence, the group element on the left of Relation (<ref>) maps 𝒯∖𝔇 to itself, while η maps 𝒯∖𝔇 to 𝒯∩𝔇. This shows that there cannot exist a relation of the form (<ref>) in ⟨τ,η⟩.

We conclude that η^2=id and τ^d+1=id are the only relations in ⟨η,τ⟩, and hence ⟨η,τ⟩=⟨η⟩∗⟨τ⟩≅/2∗/(d+1). 


2) Recall that ⟨τ⟩⩽⟨η⟩∗⟨τ⟩ acts transitively on the components U_0,⋯,U_d of . Hence it suffices to study the action of the stabilizer subgroup

    Stab(U_0):={γ∈⟨η⟩∗⟨τ⟩: γ(U_0)=U_0}

of U_0 in ⟨η⟩∗⟨τ⟩. It is easily seen that Stab(U_0) is generated by {τ^j∘η∘τ^-j, j∈{0,⋯,d}}. 

We denote the rank zero tile R^-1(T^0)∩ U_0 of  in the component U_0 by T^0_0. As R:U_0→ T^∞(S) is a conformal isomorphism, it follows that T^0_0 is a regular ideal (d+1)-gon in the conformal disk U_0. Further, each of the above generators of Stab(U_0) is an anti-conformal involution of U_0 which fixes a side of T^0_0 pointwise. This implies that the Stab(U_0)-action on U_0 is conformally conjugate to the G_d-action on . The result now follows.


We are now ready to prove a part of Theorem <ref>.


Let f be a degree d geometrically finite or finitely renormalizable, periodically repelling anti-polynomial with connected Julia set. By Theorem <ref>, there exists a polygonal Schwarz reflection S:→ that is a conformal mating of f with the Nielsen map _d. 

We claim that the desired correspondence ℭ on 𝔚 is the one constructed in Section <ref>. To this end, note that the required ℭ-invariant partition of 𝔚 is given in Proposition <ref>. Finally, the desired dynamical properties of ℭ on  follow from Proposition <ref>, and those on  follow from Lemma <ref>, Proposition <ref>, and the fact that S|_K(S) is topologically conjugate (conformally on the interior) to f|_(f).






§ ANTI-FAREY SCHWARZ REFLECTIONS


We now look at degenerate anti-polynomial-like maps having the anti-Farey map _d (see Section <ref>) as their external map. Since _d:∖ℋ→ fixes ∂ℋ pointwise and since ∂ℋ is a Jordan curve touching the unit circle at exactly one point, it follows that any such degenerate anti-polynomial-like map is the restriction of a Schwarz reflection map S:→ of a simply connected quadrature domain . Moreover, the tiling set dynamics of this Schwarz reflection map is conformally conjugate to _d. 

Following <cit.>, we will denote by 𝒮__d the space of Schwarz reflection maps S:→ such that  is a Jordan quadrature domain and the tiling set dynamics of S is conformally conjugate to _d. We denote by 𝔘_d+1 the space of degree d+1 polynomials p such that p|_ is injective and p has a unique (non-degenerate) critical point on 𝕊^1.

<cit.>
Let R be a rational map of degree d+1 that is injective on , :=R(), and S the Schwarz reflection map associated with . Then the following are equivalent.

  * There exists a conformal conjugacy ψ between 

    _d:∖ℋ⟶  and   S:T^∞(S)∖T^0(S)⟶ T^∞(S).

In particular, T^∞(S) is simply connected.


  * After possibly conjugating S by a Möbius map and pre-composing R with an element of Aut(), the uniformizing map R can be chosen to be a polynomial with a unique critical point on 𝕊^1; i.e., R can be chosen to be a member of 𝔘_d+1. Moreover, K(S) is connected. 


  *  is a Jordan domain with a unique conformal cusp on its boundary. Moreover, S has a unique critical point in its tiling set T^∞(S), and this critical point maps to T^0(S) with local degree d+1.




That fact that rational uniformizing map R of the quadrature domain of a Schwarz reflection S∈𝒮__d can be chosen to be a polynomial follows from the existence of a multiplicity d critical point of S in its tiling set (recall that _d has such a critical point). 


<cit.>

The moduli space [𝒮__d]:= 𝒮__dPSL_2() is compact.


We normalize S∈𝒮__d so that the conformal map ψ:→ T^∞(S) that conjugates _d to S sends 0 to ∞, and has asymptotics z↦ 1/z+O(z) as z→ 0.

We say two maps S_1, S_2∈𝒮__d are combinatorially equivalent, denoted by S_1∼ S_2, if they have the same rational lamination (cf. Definition <ref>). As in the case for the space 𝒮__d, we denote by 𝒮__d, r⊂𝒮__d the subset of periodically repelling Schwarz reflections, and by 𝒮__d, r := 𝒮__d, r/∼ the space of combinatorial classes.
Let 𝒮__d, fr⊂𝒮__d (respectively, 𝒮__d, gf⊂𝒮__d) be the subset of at most finitely renormalizable periodically repelling Schwarz reflections (respectively, geometrically finite Schwarz reflections).



The proofs of Theorems <ref>, <ref>, and <ref> work mutatis mutandis for the space 𝒮__d of Schwarz reflections and yield the following result.




  * There is a dynamically natural homeomorphism 

    Φ: 𝒞^-_d, r⟶𝒮__d, r,

where `dynamically natural' means that the circle homeomorphism conjugating _d to z^d carries the rational lamination of an anti-polynomial to the rational lamination of a Schwarz reflection.


  * There is a dynamically natural homeomorphism 

    Φ: 𝒞^-_d, fr⟶𝒮__d, fr.

Here, dynamically natural means that for any f∈𝒞^-_d, fr, Φ(f) is the unique conformal mating of f with the anti-Farey map _d.


  * There is a dynamically natural bijection 

    Φ: 𝒞^-_d, gf⟶𝒮__d, gf.
 
Here, dynamically natural has the same meaning as in item (2) of this theorem.





We believe that the bijection Φ: 𝒞^-_d, gf⟶𝒮__d, gf is not continuous in general. Although the proof of discontinuity given in Subsection <ref> does not apply to the family 𝒮__d (essentially because all Schwarz reflections in 𝒮__d are defined by a single quadrature domain), one possible way of proving discontinuity in the current setting is to use the arguments of <cit.>.



We end this section with the proof of Theorem <ref> in the case of the anti-Hecke group.


Let f a degree d anti-polynomial as in the statement of the theorem. By Theorem <ref>, there exists a Schwarz reflection S=Φ(f):→ in 𝒮__d such that the non-escaping set dynamics of S is topologically conjugate (conformally on the interior) to f|_(f) and the tiling set dynamics of S is conformally conjugate to _d.

By Propositions <ref> and <ref>, after possibly Möbius conjugating S (which amounts to replacing  by a Möbius image of it), we can assume that p()=, for some polynomial p∈𝔘_d+1. We note that S≡ p∘η^-∘ (p|_)^-1.

We now define the antiholomorphic correspondence ℭ on  by the relation

    (z,w)∈ℭp(w)-p(η^-(z))/w-η^-(z)=0,
 
and set :=p^-1(K(S)), :=p^-1(T^∞(S)). By <cit.>, the sets  and  are completely invariant under the action of the correspondence. 

Note that by construction of S, the tiling set T^∞(S) is simply connected, and it contains exactly one critical point of S. More precisely, this critical point lies at p(0), and it maps to ∞ with local degree d+1 under S. Thus, by <cit.>, the dynamics of ℭ on  is conformally orbit-equivalent to the action of the anti-Hecke group G_d on .

Next, we set _1:=∩ and _2:=∖. Since p carries 𝕊^1∖{1} onto ∂∖{s}⊂ T^∞(S) (where s is the unique cusp of ∂), it follows that 𝕊^1∖{1} is disjoint from . As s∈ K(S), it follows that (p|_𝕊^1)^-1(s) is the unique point of intersection of _1 and _2. 

By <cit.>, ℭ has a forward branch carrying _1 onto itself with degree d, and this branch is topologically conjugate to S|_K(S) such that the conjugacy is conformal on the interior. The result now follows from the fact that S|_K(S) is topologically conjugate (conformally on the interior) to f|_(f).











PART: 
Holomorphic world




§ FACTOR BOWEN-SERIES MAPS FOR GENUS ZERO ORBIFOLDS


We will now describe a collection of circle coverings that were introduced in <cit.>.
Let us first define a class of orbifolds that give rise to these external maps:

 𝔉:= hyperbolic orbifolds Σ of genus zero with
	
	
  * at least one puncture,
	
  * at most one order two orbifold point,
	
  * at most one order ν≥ 3 orbifold point.
	
We set

    n=
    ν  if Σ∈𝔉  has an order ν≥ 3  orbifold point,
    
    1   otherwise.

Each Σ∈𝔉 has an n-fold cyclic cover Σ. Specifically, if Σ has δ_1≥1 punctures and δ_2∈{0,1} order two orbifold points, then Σ is a genus zero orbifold with n(δ_1-1)+1 punctures and nδ_2 order two orbifold points. Topologically, Σ is obtained by skewering the surface Σ along an infinite geodesic (in Σ) connecting the order ν orbifold point and a cusp, and gluing n copies of it cyclically.

The surface Σ is uniformized by a Fuchsian group Γ which admits a (closed) ideal m-gon Π as a fundamental domain. This fundamental domain Π can be chosen so that it has ideal vertices at 1 and exp(2π i/n) which are identified, and such that Π is symmetric under rotation by 2π/n around the origin. Here, 

    m=
    
    2n(δ_1-1)    when  δ_2=0,
     
    n(2δ_1-1)    when  δ_2=1.







The Bowen-Series map A^BS_Σ≡ A^BS_Γ:∖Π→ of Γ equipped with the fundamental domain Π has jump discontinuities at the n-th roots of unity. However, the 2π/n rotation symmetry of Π and of the associated side-pairing transformations imply that A^BS_Σ commutes with rotation by 2π/n. This allows one to pass to a factor of the above Bowen-Series map on the quotient cone /⟨ M_ω⟩ (where M_ω(z)=ω z, and ω=e^2π i/n). The resulting map is denoted by 

    A^BS_Σ: (∖Π)/⟨ M_ω⟩→/⟨ M_ω⟩.
 
We uniformize the cone /⟨ M_ω⟩ by the closed disk  and call this uniformizing map ξ:/⟨ M_ω⟩→ (note that ξ is induced by z↦ z^n). Finally, we define the factor Bowen-Series map associated with Σ as

    A^f_Σ:=ξ∘A^BS_Σ∘ξ^-1: ∖ℋ→,
 
where ℋ:=ξ(Π/⟨ M_ω⟩). The set ℋ has p:=m/n ideal boundary points on 𝕊^1, and the map A_Σ^f restricts to a self-homeomorphism of order two on ∂ℋ.


1) In the anti-holomorphic world, a Nielsen map or an anti-Farey map has a unique domain of definition which is bounded by the unit circle and curves comprising fixed points of the map. However, this is not the case for factor Bowen-Series maps since there is no unique collection of curves in the disk on which a factor Bowen-series map acts as an involution. As a convention, we will fix ∖ℋ as the canonical domain of definition of the factor Bowen-Series map A_Σ^f.

2) The superscript `f' in A^f_Σ stands for `factor'.



The factor Bowen-Series map A^f_Σ turns out to be an expansive covering map of 𝕊^1 of degree  

    d≡ d(Σ)= m-1 = 
    2n(δ_1-1)-1   when  δ_2=0,
    
    n(2δ_1-1)-1   when  δ_2=1.
 
Moreover, when n≥ 3, the map A_Σ^f has p=m/n critical points, each of multiplicity n-1.
All these critical points are mapped to the same critical value. 


Two special classes of examples of factor Bowen-Series maps are given by the cases 

    n=1, δ_1≥ 2, δ_2∈{0,1},
 and 

    n≥ 3, δ_1=1, δ_2=1.
 
In the former case, Σ is a punctured sphere with possibly an order two orbifold point, and Σ=Σ. In particular, the factor Bowen-Series map of Σ is simply the Bowen-Series map of Σ (see Figure <ref>). In the latter case, Σ is a Hecke surface, and Σ is a sphere with one puncture and ν≥ 3 order two orbifold points (see Figure <ref>).



 §.§ Conformal mating of factor Bowen-Series maps with polynomials


We continue to use the notation used above.
Let f be a monic, centered complex polynomial of degree d with a connected and locally connected Julia set. We denote the Böttcher coordinate of f by

    ψ_f:∖→ℬ_∞(f):=∖𝒦(f).
 
This map, which conjugates z^d to f, is normalized to be tangent to the identity map near infinity.
As 𝒥(f) is locally connected, ψ_f extends continuously to 𝕊^1 to yield a semi-conjugacy between z^d|_𝕊^1 and f|_𝒥(f). 
By <cit.>, there exists a homeomorphism 𝔥:𝕊^1→𝕊^1 that conjugates z^d to A_Σ^f. We normalize 𝔥 so that it sends the fixed point 1 of z^d to the fixed point 1 of A_Σ^f.

We define an equivalence relation ∼ on 𝒦(f) generated by 

    ψ_f(ζ)∼𝔥(ζ), for all z∈𝕊^1.
 




The maps f and A_Σ^f are said to be conformally mateable if there exist a continuous map S: Dom(S)⊊→ (called a conformal mating of A_Σ^f and f) that is complex-analytic in the interior of Dom(S) and continuous maps 
	
    𝔛_f:𝒦(f)→ and 𝔛_Σ: →,

	conformal on 𝒦(f) and  (respectively), satisfying
	
		
  *  𝔛_f(𝒦(f))∪𝔛_Σ() =,
		
		
  *  Dom(S)= 𝔛_f(𝒦(f))∪𝔛_Σ(∖ℋ),
		
		
  * 𝔛_f∘ f(z) = S∘𝔛_f(z),  for z∈𝒦(f),
		
		
  * 𝔛_Σ∘ A_Σ^f(w) = F∘𝔛_Σ(w),  for w∈∖ℋ,  and 
		
		
		
  *  𝔛_f(z)=𝔛_Σ(w) if and only if z∼ w where ∼ is the equivalence relation on 𝒦(f)⊔ defined by Relation (<ref>).
	



<cit.>
Let f be a hyperbolic complex polynomial of degree d with connected Julia set.
Then, there exists a conformal mating S of f:𝒦(f)→𝒦(f) and A_Σ^f:∖ℋ→. Moreover, S is a unique up to Möbius conjugacy.








§ B-INVOLUTIONS


We now introduce a holomorphic analogue of Schwarz reflections.




Let {Ω_1,⋯,Ω_k} be a disjoint collection of proper simply connected sub-domains of  such that Ω_j=Ω_j, j∈{1,⋯, k}, and let :=_j=1^kΩ_j. Further, let 𝔖⊂∂ be a (possibly empty) finite set such that ∂^0:=∂∖𝔖 is a finite union of disjoint non-singular real-analytic curves.

The set  is called an inversive multi-domain if it admits a continuous map S:→ satisfying the properties:


  *  S is meromorphic on ,

  *  S(∂Ω_j)=∂Ω_j', for some j'∈{1,⋯,k}, and

  *  S:∂→∂ is an orientation-reversing involution preserving 𝔖; i.e., S(𝔖)=𝔖.

The map S is called a B-involution of the inversive multi-domain .

When k=1, the domain  is called an inversive domain.


For a B-involution S:→, we set T:=∖, and T^0:= T∖𝔖.


1) If each Ω_j is a Jordan domain, then Condition <ref> of Definiion <ref> follows from Condition <ref>.

2) The nomenclature `B-involution' is chosen to emphasize the property that such a map induces an involution on the boundary of an inversive multi-domain .





The main result of this section is an explicit algebraic description of B-involutions, which can be thought of as a holomorphic counterpart of Proposition <ref>. We first prove this characterization for an inversive domain.
Recall that η^+(z)=1/z.


Let  be an inversive domain with associated B-involution S:→. 
Then, there exists a rational map R of degree d+1 and a Jordan disk 𝔇 such that the following hold.

  * 1, -1∈∂𝔇.

  * η^+:𝔇→∖𝔇 is a homeomorphism.

  * ∂𝔇 is a piecewise non-singular real-analytic curve.

  * R:𝔇→ is a conformal isomorphism.

  * S|_:= R∘η^+∘ (R|_𝔇)^-1.

  * S:S^-1()→ is a proper map of degree d.

  * S:S^-1(T)→T is a proper map of degree d+1.




Step 1: Extending S to a conformal involution around ∂^0.
By assumption, S:∂→∂ is an order two homeomorphism preserving 𝔖. Hence, it carries each component of ∂^0 (which is a non-singular real-analytic curve) onto some component of ∂^0. Depending on the parity of the number of components of ∂^0, no component or exactly one component of ∂^0 is mapped onto itself by S. Since S is meromorphic on  and continuous up to ∂, it now follows by the Schwarz Reflection Principle that S extends to a holomorphic map (which we call S) in a neighborhood of ∂^0. Since S carries ∂^0 onto itself, the map S^∘ 2 is also well-defined in a neighborhood of ∂^0. Moreover, as S^∘ 2=id on ∂^0, the Identity Principle for holomorphic maps implies that S is a conformal involution in a neighborhood of ∂^0.











Step 2: Constructing a welded Riemann surface Σ with S as the welding map.
We now weld the bordered Riemann surface ∪∂^0 with itself using the conformal involution S:∂^0→∂^0 as the boundary identification map. The resulting (non-compact) Riemann surface is denoted by Σ. Note that Σ is a genus zero surface with finitely many boundary components.

By definition of Σ, there exist homeomorphisms ψ_1,ψ_2:∪∂^0→Σ such that


  * ψ_i, i∈{1,2}, is conformal on ,

  * ψ_1()∩ψ_2()=∅,

  * ψ_1(∂^0)=ψ_2(∂^0) is a union of non-singular real-analytic curves (which we denote by 𝔍),

  * ψ_1(w)=ψ_2(S(w)), for w∈∂^0, and

  * Σ=ψ_1(∪∂^0)∪ψ_2().



Step 3: Defining a conformal involution M on Σ.
Let us define a map 

    M:Σ→Σ,  
    M=
    ψ_2∘ψ_1^-1  on  ψ_1(∪∂^0),
    ψ_1∘ψ_2^-1  on  ψ_2().
 
The relations ψ_1=ψ_2∘ S (on ∂^0) and S^∘ 2=id (on ∂^0) imply that the piecewise definitions of M match continuously along 𝔍. Moreover, M is easily seen to be holomorphic, and hence a conformal involution on Σ. We also note that ψ_1 conjugates the involution S|_∂^0 to M|_𝔍. 


Step 4: A finite degree branched cover R and the type of Σ. We define a holomorphic map R:Σ∖ψ_2(S^-1(𝔖))⟶∖𝔖 as follows:

    R=
    ψ_1^-1  on  ψ_1(∪∂^0),
    
    S∘ψ_1^-1∘ M = S∘ψ_2^-1  on  ψ_2(∖ S^-1(𝔖)).
 
The assumption that S extends continuously to an involution of ∂ fixing 𝔖 implies that R is a proper holomorphic map. Hence, R is a finite degree branched covering map. Suppose that the degree of R is d+1, for some non-negative integer d.

As the punctured sphere ∖𝔖 is a finite type Riemann surface, it follows that Σ∖ψ_2(S^-1(𝔖)) must also be a finite type Riemann surface. Hence, Σ∖ψ_2(S^-1(𝔖)) is also a punctured sphere. This means that S^-1(𝔖) is a finite set and the boundary components of Σ arising from the welding procedure are points (i.e., not holes). 


Step 5: Extending R to a rational map and M to a Möbius involution. Since R is of finite degree, it extends holomorphically to the punctures of Σ∖ψ_2(S^-1(𝔖)). This produces a holomorphic map R of the Riemann sphere, and hence R is a rational map of degree d+1.

The fact that Σ is a punctured sphere also implies that the conformal involution M extends to the entire Riemann sphere, and yields a Möbius involution. Note that M preserves the set of punctures of Σ.
After possibly a change of coordinates, we can assume that M=η^+.


Step 6: Putting everything together.
Since Σ is a punctured sphere, the curve 𝔍 can also extended to a Jordan curve 𝔍⊂ by throwing in the punctures. Moreover, the assumption that S:∂→∂ is orientation-reversing implies that η^+ is an orientation-reversing self-homeomorphism of 𝔍. Hence, η^+ must have exactly two fixed points (namely, ± 1) on 𝔍.

We set 𝔇:=ψ_1() and note that 𝔇 is one of components of ∖𝔍. Finally, the definition of R and the above discussion imply that 

    R|_∖𝔇=S|_∘ R|_𝔇∘η^+|_∖𝔇  S|_= R∘η^+∘ (R|_𝔇)^-1.

Finally, the degrees of the branched coverings S:S^-1()→ and S:S^-1(T)→T follow directly from Expression (<ref>).




The next result gives a similar description for inversive multi-domains with two components such that the boundary components are swapped by the B-involution.


Let =Ω_0⊔Ω_1 be an inversive multi-domain with associated B-involution S:→. 
Assume further that S(∂Ω_0)=∂Ω_1.
Then, there exist rational maps R_0, R_1 and Jordan disks 𝔇_0, 𝔇_1 such that the following hold for j∈{0,1}.

  * η^+:𝔇_j→∖𝔇_1-j is a homeomorphism.

  * ∂𝔇_j is a piecewise non-singular real-analytic curve.

  * R_j:𝔇_j→Ω_j is a conformal isomorphism.

  * S|_Ω_j:= R_1-j∘η^+∘ (R_j|_𝔇_j)^-1.

  * S:S^-1(Ω_1-j)∩Ω_j⟶Ω_1-j is a branched covering of degree (R_1-j)-1.

  * S:S^-1(Ω_1-j^c)∩Ω_j⟶Ω_1-j^c is a branched covering of degree (R_1-j).



The proof essentially follows the ideas used in the proof of Lemma <ref>.

As in the case of inversive domains, the map S extends to a conformal involution in a neighborhood of ∂^0. 
We set 𝔖_j:=𝔖∩∂Ω_j and ∂^0Ω_j:=∂Ω_j∖𝔖_j=∂^0∩∂Ω_j, j∈{0,1}. The map S:∂Ω_j→∂Ω_1-j carries 𝔖_j (respectively, ∂^0Ω_j) onto 𝔖_1-j (respectively, onto ∂Ω^0_1-j). We now weld the bordered Riemann surfaces Ω_0∪∂^0Ω_0 and Ω_1∪∂^0Ω_1 using the real-analytic homeomorphism S:∂^0Ω_0→∂^0Ω_1, and call the resulting Riemann surface Σ. 

By construction of Σ, there exist homeomorphisms ψ_0:Ω_0∪∂^0Ω_0→Σ and ψ_1:Ω_1∪∂^0Ω_1→Σ such that


  * ψ_j is conformal on Ω_j, j∈{0,1}, 

  * ψ_0(Ω_0)∩ψ_1(Ω_1)=∅,

  * 𝔍:=ψ_0(∂^0Ω_0)=ψ_1(∂^0Ω_1) is a union of non-singular real-analytic curves,

  *  ψ_0=ψ_1∘ S on ∂^0Ω_0, and

  * Σ=ψ_0(Ω_0)⊔𝔍⊔ψ_1(Ω_1).

We now define a holomorphic map 

    R_0:Σ∖ψ_1((S|_Ω_1)^-1(𝔖_0))⟶∖𝔖_0


    R_0=
    ψ_0^-1  on  ψ_0(Ω_0∪∂^0Ω_0),
    
    S|_Ω_1∘ψ_1^-1  on  ψ_1(Ω_1 ∖(S|_Ω_1)^-1(𝔖_0)).
 
Condition (<ref>) and the relation S|_∂Ω_1=(S|_∂Ω_0)^-1 together guarantee that the piecewise definitions of R_0 match continuously along the piecewise real-analytic curve 𝔍.
Moreover, the assumption that the continuous extension of S to ∂ fixes 𝔖 implies that R_0 is a proper holomorphic map, and hence a finite degree branched covering map. As in the proof of Lemma <ref>, this implies that Σ∖ψ_1((S|_Ω_1)^-1(𝔖_0)) is biholomorphic to a punctured sphere. This allows us to extend R_0 to a rational map of the Riemann sphere. Moreover, the closure of 𝔍 in the Riemann sphere (obtained by adding finitely many points to 𝔍) is a Jordan curve.


Let us now set 𝔇_0:=ψ_0(Ω_0) and 𝔇_1:=η^+(∖𝔇_0). We further define a rational map R_1 as

    R_1=
    ψ_1^-1∘η^+   on  𝔇_1,
    
    S|_Ω_0∘ψ_0^-1∘η^+   on  ∖𝔇_1.
 
Once again, Condition (<ref>) shows that the piecewise definitions of R_1 match continuously along the Jordan curve 𝔍 (where the closure is taken in ).

By construction, each R_j:𝔇_j→Ω_j is a conformal isomorphism, j ∈ {0,1}. To justify that R_0, R_1 are the desired rational maps, it only remains to check that S|_Ω_j:= R_1-j∘η^+∘ (R_j|_𝔇_j)^-1, j∈{0,1}. To this end, first note that

    S|_Ω_1∘ψ_1^-1 = R_0    on   ∖𝔇_0
        S|_Ω_1 = R_0∘ψ_1 = R_0∘η^+∘η^+∘ψ_1
        S|_Ω_1 = R_0∘η^+∘(ψ_1^-1∘η^+|_𝔇_1)^-1 = R_0∘η^+∘(R_1|_𝔇_1)^-1.

On the other hand,

    S|_Ω_0∘ψ_0^-1∘η^+ = R_1    on   ∖𝔇_1
        S|_Ω_0 = R_1∘η^+∘ψ_0
        S|_Ω_0 = R_1∘η^+∘(ψ_0^-1|_𝔇_0)^-1 = R_1∘η^+∘(R_0|_𝔇_0)^-1.

Finally, the statements about the degrees of S follow from the relations S|_Ω_j≡ R_1-j∘η^+∘(R_j|_𝔇_j)^-1, j∈{0,1}.
The proof is now complete.






We are now ready to establish the promised algebraic description of a general B-involution.


Let S:→ be a B-involution of the inversive multi-domain =_j=1^kΩ_j.
Then, there exist Jordan domains 𝔇_j and rational maps R_j, j∈{1,⋯,k}, such that the following hold.


  * η^+:𝔇_j→∖𝔇_κ(j) is a homeomorphism.

  * ∂𝔇_j is a piecewise non-singular real-analytic curve.

  * R_j:𝔇_j→Ω_j is a conformal isomorphism.

  * S|_Ω_j≡ R_κ(j)∘η^+∘(R_j|_𝔇_j)^-1.

Here, κ is a permutation of the set {1,⋯,k} of order two.


The assumption that S:∂→∂ is an involution which maps each ∂Ω_j homeomorphically onto some ∂Ω_j' implies that S induces an order two permutation on the set {1,⋯,k}. We denote this permutation by κ.
Since κ is of order two, it can be factored as a product of disjoint 1-cycles and 2-cycles; i.e., S either preserves ∂Ω_j, or swaps ∂Ω_j and ∂Ω_κ(j).

If κ(j)=j, then Ω_j is an inversive domain with associated B-involution S|_Ω_j. If κ(j)≠ j, then Ω_j⊔Ω_κ(j) is an inversive multi-domain with associated B-involution given by the restriction of S. The existence of the desired Jordan domains 𝔇_j and rational maps R_j now follow from Lemmas <ref> and <ref>.




§ B-INVOLUTIONS WITH FACTOR BOWEN-SERIES EXTERNAL MAPS


Throughout this section, we will work with a fixed Σ∈𝔉. Let m,n∈ be as in Section <ref>.


We define the space ℬ_Σ to be the collection of all degenerate polynomial-like maps with connected non-escaping set that admit the factor Bowen-Series map A_Σ^f as their external map.




We equip every degenerate polynomial-like map (with connected filled Julia set) with an invariant external access to its filled Julia set (or equivalently, by a fixed point on the ideal boundary of its escaping set). Two marked degenerate polynomial-like maps are identified if they are conjugate via a Möbius map that preserves the marking.      



By definition, for a map (S,P_1,P_2)∈ℬ_Σ, there exists a conformal isomorphism 

    ψ:→ T^∞(S):=∖ K(S)
 
that conjugates A_Σ^f to S, wherever the maps are defined. We set 

    :=∖ψ(ℋ)
 
and remark that  is a pinched polygon (note that the image of ∂ℋ under the conformal isomorphism ψ consists of finitely many non-singular real-analytic curves). The map S extends to a continuous map on  which is meromorphic on . We call  the canonical domain of definition of S (see Remark <ref>). 


We normalize S∈ℬ_Σ so that the conformal map ψ:→ T^∞(S) that conjugates A_Σ^f to S sends 0 to ∞, and has asymptotics z↦ 1/z+O(z) as z→ 0.

We also remark that since S:∂ S^-1()→∂ has degree d, the map S:S^-1()→ can be extended to a degree d branched cover of the sphere with a fixed critical point of multiplicity d-1 at ∞. In fact, this can be done in such a way that there is no other critical point of the branched cover outside K(S). The fact that a degree d branched cover of the sphere has 2d-2 critical points (counted with multiplicities), it follows that S has d-1 critical points (counted with multiplicities) in K(S).



 §.§ Coarse lamination

We now proceed to give an explicit description of the pinched polygon . 
Let us denote the set of p=m/n ideal boundary points of ℋ on 𝕊^1 by S_Σ. These points are periodic of period at most two under A_Σ^f; in fact, at most two points of S_Σ are fixed by A_Σ^f.
Under the circle homeomorphism 𝔥 that conjugates z^d to A_Σ^f, the set S_Σ is pulled back to the set 

    𝒜:={i/p:i∈{0,⋯,p-1}}.

(Here we identify 𝕊^1 with =/.) If p is even, A_Σ^f has two fixed points in S_Σ and these correspond to 0,1/2∈𝒜. For odd p, the map A_Σ^f has a unique fixed point in S_Σ and this fixed point corresponds to 0∈𝒜. On the other hand, the 2-cycles of A_Σ^f in S_Σ correspond to the 2-cycles ±r/p of m_d:/→/, θ↦ dθ.

Following Section <ref>, we can associate a coarse lamination ℒ with S∈ℬ_Σ that records which points of 𝒜 are identified on the boundary of K(S). Then,  is homoemorphic to the pinched disk /ℒ (cf. <cit.>.

As a slight abuse of terminology, we will call the components of ∖ of infinite (respectively, finite) hyperbolic area gaps (respectively, polygons) of .
For a gap 𝒢, the `boundary at infinity' ∂𝒢∩ has length equal to k/p for some integer k≥ 1.
We shall call this number k the degree of the gap, denoted by (𝒢). 
We also denote the number of points of 𝒜 in (∂𝒢∩) by (𝒢), and call it the cusp number of the gap.

The next result is immediate from the definition of coarse laminations.


Let S∈ℬ_Σ with canonical domain of definition .
The components of  are in bijective correspondence with the gaps of ℒ. Two such components share at most one boundary point; moreover, they share a boundary point if and only if the corresponding gaps are cobordant on a leaf or a polygon of .


We enumerate the gaps of as 𝒢_1,⋯,𝒢_k such that 𝒢_j corresponds to Ω_j.

Note that no leaf of ℒ connects the fixed points 0 or 1/2 to the rest of (2-peridoic) points of 𝒜 (cf. <cit.>).
If r/p, s/p∈𝒜 lie in the same -equivalence class, then so do -r/p=m_d(r/p) and -s/p=m_d(s/p). Hence, the gaps of the lamination ℒ either intersect the real line, or come in complex conjugate pairs. It is easily seen that


∙ if 𝒢_j intersects the real line, then S(∂Ω_j)=∂Ω_j; otherwise,


∙ 𝒢_j and 𝒢_j' (for some j'∈{1,⋯,k}) form a complex conjugate pair of gaps, and S(∂Ω_j)=∂Ω_j'.


The next result demonstrates that degenerate polynomial-like maps having factor Bowen-Series maps as their external maps are B-involutions.


Let S∈ℬ_Σ with canonical domain of definition .
The map S:→ is a B-involution of an inversive multi-domain. In particular, the conformal mating S described in Theorem <ref> is a B-involution of an inversive multi-domain. 


We have already observed that  is a pinched polygon with. Let us set 𝔖=ψ(S_Σ), where ψ:→ T^∞(S) is the conformal conjugacy between A_Σ^f to S. The first statement is now an immediate consequence of the facts that a factor Bowen-Series map A_Σ^f:∖ℋ→ restricts to a self-homeomorphism of order two on ∂ℋ and the discussion above (or from the fact that each component of  is a Jordan domain, see Remark <ref>). The second statement is now immediate as the conformal matings of Theorem <ref> lie in the class ℬ_Σ.





 §.§ Degrees and critical points of uniformizing rational maps

Throughout this subsection, we fix an S∈ℬ_Σ with canonical domain of definition .
By Theorem <ref> and Proposition <ref>, there exist Jordan domains 𝔇_j and rational maps R_j, j∈{1,⋯,k}, such that 

  * η^+:𝔇_j→∖𝔇_κ(j) is a homeomorphism,

  * R_j:𝔇_j→Ω_j is a conformal isomorphism, and

  * S|_Ω_j≡ R_κ(j)∘η^+∘(R_j|_𝔇_j)^-1,

where κ is a permutation of the set {1,⋯,k} of order two. We denote the domain of R_j by _j.

Let d_j:=R_j. The relation S|_Ω_j≡ R_κ(j)∘η^+∘(R_j|_𝔇_j)^-1 implies that S:S^-1(Ω_κ(j))∩Ω_j⟶Ω_κ(j) is a branched covering of degree d_κ(j)-1, and
S:S^-1(Ω_κ(j)^c)∩Ω_j⟶Ω_κ(j)^c is a branched covering of degree d_κ(j). As S:S^-1()→ has degree d, we conclude that

    ∑_j=1^k d_j = d+1.



For each j∈{1,⋯,k},

  * (𝒢_j)=(𝒢_κ(j)), and

  * d_j=n·(𝒢_j), and hence d_j=d_κ(j).



1) This follows from the structure of the gaps of  described in Section <ref>.

2) Recall that S:S^-1(Ω_κ(j))∩Ω_j⟶Ω_κ(j) is a branched covering of degree d_j-1.

Note that under the map m_d, each arc of ∖𝒜 is wrapped onto the whole circle (n-1) times and onto the complement of the closure of its complex conjugate arc one more time. Hence, m_d(∂𝒢_j∩) covers ∂𝒢_κ(j)∩ exactly 

    (n-1)·(𝒢_j)+((𝒢_j)-1)=n·(𝒢_j)-1
 
times. It follows that ∂ K(S)∩Ω_j covers ∂ K(S)∩Ω_κ(j) exactly n·(𝒢_j)-1 times under the map S. Since ∂ K(S) is completely invariant under S, we conclude that S:S^-1(Ω_κ(j))∩Ω_j⟶Ω_κ(j) is a degree n·(𝒢_j)-1 branched covering. Therefore, d_j=n·(𝒢_j).



To describe the structure of critical points of the rational maps R_j concisely, we employ the following notation. 
Consider the disjoint union 

    𝔘 := _j=1^k _j ≅ ×{1,⋯,k},

and define the maps

    R: 𝔘⟶,   (z,j)↦ R_j(z),

and

    η^+ : 𝔘⟶𝔘,   (z,j)↦ (η^+(z),κ(j)).

Note that R is a branched covering of degree d+1, and η^+ is a homeomorphism. We also set 

    𝔇:= _j=1^k 𝔇_j⊂𝔘.

Recall the notation 𝔖=ψ(S_Σ). We denote the set of points in 𝔖 that do not disconnect ∂ by 𝔖^cusp. Note that |𝔖^∩∂Ω_j| = (𝒢_j).

The following result was proved in <cit.> in certain special cases. We include the proof for completeness (and since the current setting is more general).




  * R has no critical points in 𝔇.


  * R has d-1=np-2 critical points in R^-1()∖𝔇. On the other hand, for n≥ 3, the map R has p distinct critical points, each of multiplicity n-1, in R^-1(T^∞(S))∖𝔇, and all these critical points are mapped by R to the same point in T^∞(S). 


  * crit(R)∩∂𝔇=(R|_∂𝔇)^-1(𝔖^cusp).



1) This follows from injectivity of R_j on 𝔇_j.

2) Recall that S has d-1=np-2 critical points in K(S), and p critical points, each of multiplicity n-1 (when n≥ 3), in T^∞(S) (coming from the p critical points of A_Σ^f). Moreover, S maps all the p(n-1) critical points in T^∞(S) to the same critical value since A_Σ^f sends all of its p(n-1) critical points to the origin.

The result now follows from the relation S|_Ω_κ(j)≡ R_j∘η^+∘(R_κ(j)|_𝔇_κ(j))^-1. 




3) Since A_Σ^f does not admit an analytic continuation in a neighborhood of any point of S_Σ, the map S does not admit an analytic continuation in a neighborhood of any point of 𝔖. Suppose that x∈𝔖^cusp lies on the boundary of Ω_j.
Then, the relation S|_Ω_j=R_κ(j)∘η^+∘ (R_j|_𝔇_j)^-1 implies that (R_j|_𝔇_j)^-1 does not extend complex-analytically to a neighborhood of x. This forces x to be a critical value of R_j with a corresponding critical point on ∂𝔇_j. Hence, (R|_∂𝔇)^-1(𝔖^cusp)⊂crit(R)∩∂𝔇.

It remains to show that R has no other critical point on ∂𝔇. This will be derived from the following counting argument.

    |crit(R)| =∑_j=1^k (2d_j-2)≥ (np-2)+p(n-1)+∑_j=1^k (𝒢_j)
        2np-2k ≥ 2np-p-2+ ∑_j=1^k (𝒢_j)
          ( recall that  d_j=n·(𝒢_j) and ∑_j=1^k(𝒢_j)=p ) 
        p+2≥ 2k+ ∑_j=1^k (𝒢_j). ()


Claim: The lamination  contains no polygon.
 If the lamination  is empty; i.e., if k=1, then (𝒢_1) consists of p points, and the two sides of Inequality () coincide. Every time an r-gon is introduced in , the number k (i.e., the number of gaps) increases by r-1 and the number ∑_j=1^k (𝒢_j) drops by r. Thus, if r>2, the introduction of an r-gon in  would increase the right side of Inequality () by 2(r-1)-r=r-2>0. But this would violate the inequality, proving that  contains no polygon.


Thanks to the above claim, the leaves of  have disjoint closures in . It is now easily seen that Inequality () is actually an equality, and hence |crit(R)∩∂𝔇|=|𝔖^cusp|. We conclude that (R|_∂𝔇)^-1(𝔖^cusp)= crit(R) ∩ ∂𝔇.



The coarse lamination  associated with S∈ℬ_Σ has no polygon.




 §.§ Coarse partition and topology of ℬ_Σ

As in Section <ref>, the above coarse laminations lead to a coarse partition of ℬ_Σ:

    ℬ_Σ = ⋃_ℒℬ_Σ,ℒ,

where ℬ_Σ,ℒ consists of all maps in ℬ_Σ with coarse lamination ℒ, and the union is taken over all possible coarse laminations. This coarse partition can be used to topologize ℬ_Σ as in Definition <ref>. Further the proof of Theorem <ref> also applies mutatis mutandis to show compactness of ℬ_Σ.




 §.§ Parameter space homeomorphisms


The local dynamics of the B-involution S near the singular points of ∂ is similar to that of a Schwarz reflection near the singular points of the boundary of its domain of definition, thanks to the algebraic description of B-involutions given in Theorem <ref>. This fact allows one to give a classification of Fatou components of S (i.e., connected components of K(S)) and to establish the classical relations between Fatou components/non-repelling cycles and critical points. Moreover, the tessellation of  given by the Fuchsian group Γ (uniformizing the cyclic cover Σ of Σ) and the preferred fundamental domain Π can be used to construct external dynamical rays for S (following Definition <ref>). The proof of Proposiion <ref> can also be applied to current setting to prove landing of pre-periodic dynamical rays, and this gives rise to the notion of the rational lamination of a B-involution S (see Definition <ref>).

One can now define the spaces ℬ_Σ, r, ℬ_Σ, r, ℬ_Σ, fr, and ℬ_Σ, gf as in the case of polygonal (or anti-Farey) Schwarz reflections.

Let d≥ 2 be the degree of A_Σ^f:𝕊^1→𝕊^1.



  * There is a dynamically natural homeomorphism 

    Φ: 𝒞^+_d, r⟶ℬ_Σ, r,

where `dynamically natural' means that the circle homeomorphism conjugating A_Σ^f to z^d carries the rational lamination of a polynomial to the rational lamination of a B-involution.


  * There is a dynamically natural homeomorphism 

    Φ: 𝒞^+_d, fr⟶ℬ_Σ, fr.

Here, dynamically natural means that for any f∈𝒞^+_d, fr, the B-involution Φ(f) is the unique conformal mating of f with the factor Bowen-Series map A_Σ^f.


  * There is a dynamically natural bijection 

    Φ: 𝒞^+_d, gf⟶ℬ_Σ, gf.
 
Here, dynamically natural has the same meaning as in item (2) of this theorem.





As in the antiholomorphic setting, the bijection Φ: 𝒞^+_d, gf⟶ℬ_Σ, gf should be discontinuous at infinitely many parameters; however, we refrain from justifying this statement here (see <cit.>).



The David surgery technique used in Theorem <ref> works mutatis mutandis in the current setting, and yields a dynamically natural bijection between geometrically finite parameters as asserted in (3).
To construct puzzles for B-involutions, we need a slight modification at level 0; namely, we consider all external rays landing at fixed and period 2 cycles.
This is to accommodate the coarse lamination coming from the degenerate polynomial-like structure of the B-involution (see <ref>).
With this modification, the proofs of (1) and (2) are the same as the proofs for Theorem <ref> and Theorem <ref>.






 §.§ Correspondences associated with conformal matings


As in Section <ref>, the map S∈ℬ_Σ can be lifted by the degree d+1 branched cover R:𝔘→ to obtain a bi-degree d:d correspondence ℭ^⊛ on 𝔘. Specifically,  ℭ^⊛ is defined the algebraic equation:

    {(𝔲_1,𝔲_2)∈𝔘×𝔘: R(𝔲_2)-R(η^+(𝔲_1))/𝔲_2-η^+(𝔲_1)=0}.

We then pass to the quotient 

    𝔚 := 𝔘∼_w,

where ∼_w is the finite equivalence relation defined as


For z∈∂𝔇_i⊂_i and w∈∂𝔇_j⊂_j,


(z,i)∼_w (w,j) R_i(z)=R_j(w).


The space 𝔚 can be viewed as a compact, simply connected, (possibly) noded Riemann surface.

As in Section <ref>, the maps R, η^+ descend to 𝔚, defining a bi-degree d:d correspondence ℭ on 𝔚.

 


Let f be a degree d=d(Σ)


  * geometrically finite; or 

  * finitely renormalizable and periodically repelling

polynomial with connected Julia set. By Theorem <ref>, there exists S∈𝒮_Σ that is a conformal mating of f with the factor Bowen-Series map A_Σ^f. The above discussion allows us to associate with S a bi-degree d:d correspondence ℭ on a compact, simply connected, (possibly) noded Riemann surface 𝔚. The proof of <cit.> applies verbatim to the current setting to show that ℭ is a mating of f with the Fuchsian group G uniformizing Σ.






§ TEICHMÜLLER SPACES AND CONNECTEDNESS LOCI IN THE SPACE OF DEGENERATE POLYNOMIAL-LIKE MAPS




 §.§ Hybrid classes and vertical fibers


Recall that the triple (g, P_1, P_2) is a degenerate (anti-)polynomial-like map of degree d if g: P_1 ⟶ P_2 is a degree d holomorphic/anti-holomorphic map between two pinched polygons P_1⊂ P_2 satisfying conditions in Definition <ref>.


Let (g_1,P_1, P_2) and (g_2,Q_1,Q_2) be two degenerate (anti-)polynomial-like maps. We say that g_1 and g_2 are hybrid conjugate if there exists a quasiconformal map Φ→ such that:


	
  * Φ sends the pinched points (respectively, corners) of P_2 to the pinched points (respectively, corners) of Q_2,
	
  * Φ conjugates g_1 to g_2, restricted to the closure of a neighborhood of K(g_i) pinched at the singular points of ∂ P_1, ∂ Q_1,
	
  * ∂Φ≡ 0 almost everywhere on K(g_1).



Hybrid conjugacy defines an an equivalence relation on the space of degenerate (anti-)polynomial-like maps, and we call the corresponding equivalence classes hybrid classes of degenerate (anti-)polynomial-like maps.

Recall that for a degenerate (anti-)polynomial-like map, the external map can be defined in a similar way as external maps of polynomial-like maps, which is a piecewise real-analytic, degree d, covering map of 𝕊^1.
As in the classical setting of polynomial-like maps, equivalence classes of piecewise real-analytic degree d coverings of the circle under the equivalence relation induced by piecewise real-analytic circle conjugacy are called external classes (cf. <cit.>). Two (piecewise) conformally conjugate degenerate (anti-)polynomial-like maps have the same external class. For our purposes, however, it is more convenient to follow the approach of <cit.> and consider the space of degenerate (anti-)polynomial-like maps up to Möbius equivalence. Analogously, we will identify two external maps when they are conjugate by a Möbius automorphism of the disk.
Finally, we define a vertical fiber in the space of degree d degenerate (anti-)polynomial-like maps with connected Julia set as the collection of all such maps with a fixed external map.




 §.§ Total space as union of vertical fibers


The space 𝒮_G_d≡𝒮__d introduced in Section <ref> consists of degree d degenerate anti-polynomial-like maps with connected Julia set having the Nielsen map _d of the regular ideal (d+1)-gon reflection group G_d as their external map. The Teichmüller space Teich(G_d) of the group G_d is real (d-2)-dimensional. Specifically,
the space Teich(G_d) consists of Möbius conjugacy classes of discrete, faithful, strongly type-preserving representations of G_d into Aut^±(). Any such representation ρ with image G is induced by a quasiconformal homeomorphism ψ_ρ of  that preserves the disk  and carries the preferred fundamental domain Π of G_d to a preferred fundamental domain Π(G) of G:

    ρ(g)=ψ_ρ∘ g∘ψ_ρ^-1, g∈G_d

(cf. <cit.>).
Thus, the Nielsen map _G of G associated with the fundamental domain Π(G) is simply the conjugate of _d via ψ_ρ. We denote by 𝒮_G≡𝒮__G the space of degree d degenerate anti-polynomial-like maps with connected Julia set having the Nielsen map _G as their external map. Note that each 𝒮_G is a vertical fiber in the space of degree d degenerate anti-polynomial-like maps with connected Julia set (see Subsection <ref>).
Putting all these vertical fibers together, we get the total space

    𝒯:=_G∈Teich(G_d) 𝒮_G.



On the holomorphic side, for a marked surface Σ_0∈𝔉 with associated Fuchsian model G_0 (see Section <ref>), the slice ℬ_G_0≡ℬ_Σ_0 consists of all degree d≡ d(Σ_0) degenerate polynomial-like maps with connected Julia set that have the factor Bowen-Series map A^f_Σ_0 as their external map. It follows from the construction of factor Bowen-Series maps that for Σ∈Teich(Σ_0) (or equivalently, for G∈Teich(G_0)), the map A_Σ^f is quasiconformally conjugate to A_Σ_0^f. Specifically, the quasiconformal conjugacy between A_Σ^f, A_Σ_0^f is induced by the quasiconformal conjugacy between the Fuchsian models G, G_0 of Σ, Σ_0 (cf. <cit.>). We will use the notation ℬ_Σ and ℬ_G interchangeably.
As in the antiholomorphic case, we consider the total space

    𝒯:=_G∈Teich(G_0) ℬ_G,

which is the disjoint union of all the vertical fibers ℬ_G where G runs over the Teichmüller space of G_0. 



    To make things interesting, we will assume that Teich(G_d) (respectively, Teich(G_0)) is not a singleton set. This is equivalent to saying that the ideal polygon defining G_d has at least four sides (in the antiholomorphic case) or the surface Σ_0 has at least four marked points (in the holomorphic case). In both cases, this implies that d>2.




 §.§ Cartesian product structure of the total space


In what follows, the notation _G will stand for the vertical fiber 𝒮_G or ℬ_G.
It is easy to see that maps in _G are completely determined by their hybrid class. This can be proved using the arguments of <cit.> (cf. <cit.>).




Let G_0 stand for the regular ideal polygon reflection group G_d or the Fuchsian model of a surface Σ_0∈𝔉.


  * For each S∈_G_0, there exists an injective continuous map 

    Ψ_S:(Teich(G_0),G_0)↪(𝒯,S)

such that Ψ_S(G) is hybrid conjugate to S and has A_G^f (respectively, _G) as its external map.

  * For G∈Teich(G_0), the map

    Ψ_G:_G_0→_G 
     
    S↦Ψ_S(G)

is a bijection.

  * The space 𝒯 has a Cartesian product structure; i.e., 
the map

    Ψ:Teich(G_0)×_G_0⟶𝒯
    
    (G,S)↦Ψ_S(G)=Ψ_G(S)

is a bijection.




1) The factor Bowen-Series maps A_G^f (respectively, the Nielsen maps _G), G∈Teich(G_0), are quasiconformally conjugate to A_G_0^f (respectively, to _G_0), and these conjugacies can be chosen to depend continuously on Teich(G_0). With these quasiconformal conjugacies at hand, one can employ a standard quasiconformal surgery argument to replace the external map of S with A_G^f (respectively, _G), for any given G∈Teich(G_0). By construction, this quasiconformal deformation of S is supported on its escaping set (i.e., it is conformal on K(S)), and hence, any such deformed map is hybrid conjugate to S and lies in the vertical fiber _G. It follows that the deformed map is unique up to affine conjugacy (preserving marking).
This produces a well-defined map Ψ_S:Teich(G_0)↪𝒯.

Finally, injectivity of Ψ_S is a consequence of the fact that for G≠ G'∈Teich(G_0), the maps Ψ_S(G), Ψ_S(G') lie in distinct vertical fibers.


2) Injectivity of Ψ_G follows from the definition of Ψ_G and the fact that maps in the vertical fiber _G_0 are completely determined by their hybrid class. That the map Ψ_G is surjective can be seen from the observation that the quasiconformal deformation of the previous part can be reversed. More precisely, the
external map of any map S_1∈_G can be replaced with the external map A_G_0^f (respectively, _G_0) producing a degenerate (anti-)polynomial-like map S∈_G_0 with Ψ_G(S)=S_1. 


3) By definition of Ψ and part (2) of this theorem, the map Ψ induces a bijection between {G}×_G_0 and the vertical fiber _G. Since 𝒯 is the disjoint union of all vertical fibers _G as G runs over Teich(G_0), the result follows.



The upshot of Theorem <ref> can be summarized as follows. The image Ψ_S(Teich(G_0))⊂𝒯 passes through any map S∈_G_0, and it is a single hybrid class [S]. We call this, which is defined by quasiconformal deformation of the external map, the horizontal leaf through S in 𝒯. Such a horizontal leaf intersects each vertical fiber _G at a unique point. This defines a bijective holonomy map Ψ_G between the vertical fibers _G_0 and _G. These holonomy maps piece together to yield a product structure 𝒯≅Teich(G_0)×_G_0.


It is desirable to replace _G_0 with 𝒞_d^± in the product structure of 𝒯 given in Theorem <ref>. However, our techniques do not allow us to deduce this result due to analytic limitations of David surgery and combinatorial rigidity issues. However, thanks to Theorems <ref>, <ref>, and <ref>, we have the following weaker substitute. Here, 𝒯_gf (respectively, 𝒯_fr) denotes the collection of geometrically finite (respectively, periodically repelling and finitely renormalizable) maps in 𝒯.


There exists a natural bijection between the restricted total space 𝒯_gf∪𝒯_fr and the product space Teich(G_0)×(𝒞_d,gf^±∪𝒞_d,fr^±).





 §.§ A topological product structure for a restricted total space

The Cartesian product structure of Theorem <ref> may not be upgradable to a topological product structure due to quasiconformally non-rigid parameters. 
On the other hand, if we restrict to the smaller space of periodically repelling, finitely renormalizable maps, we have transverse continuity and hence the desired product structure.

There exists a dynamically natural homeomorphism

    𝒯_fr≅Teich(G_0) ×𝒞_d,fr^±.



For simplicity of the presentation, we will present the sketch of the proof for the holomorphic case.
Let Φ:𝒞_d,fr^+→ℬ_G_0,fr be the homeomorphism of Theorem <ref>. We will show that the map

    Ψ: Teich(G_0) ×𝒞_d,fr^+⟶𝒯_fr
    
    (G,f)↦Ψ(G,Φ(f))

is the desired homeomorphism, where Ψ:Teich(G_0)×ℬ_G_0⟶𝒯 is the bijection of Theorem <ref>.
For this, we only need to justify that Ψ is a homeomorphism.

We proceed to demonstrate continuity of the map Ψ. Let (G_n,S_n)→ (G_∞,S_∞) in Teich(G_0) ×ℬ_G_0. We need to verify that the sequence of maps Ψ(G_n,S_n)=Ψ_S_n(G_n) converges to the map Ψ(G_∞,S_∞)=Ψ_S_∞(G_∞).
Since {G_n} is contained in a compact set of Teich(G_0), it is easy to see that Ψ(G_n,S_n) is pre-compact (cf. Theorem <ref>.
Let 𝒮 be a subsequential limit of Ψ(G_n,S_n).
By Theorem <ref>, 𝒮 = Ψ(G_∞',S_∞') for some G_∞' ∈Teich(G_0) and S_∞' ∈ℬ_G_0.
Consider the uniformization map ϕ_n of the tiling set U_n of Ψ(G_n,S_n).
Then ϕ_n conjugates the dynamics of the factor Bowen-Series maps A_G_n^f to Ψ(G_n,S_n)|_U_n.
It is easy to see that after appropriate normalization, ϕ_n is pre-compact.
Thus we can take a subsequential limit, and conclude that G_∞' = G_∞.
Since Φ(G_n,S_n) is hybrid conjugate to S_n = Φ(G_0,S_n), and the hybrid conjugacy is uniformly quasiconformal, after passing to a subsequence, we conclude that 𝒮 is quasiconformally conjugate to S_∞ = Φ(G_0,S_∞).
As S_∞ is periodically repelling and finitely renormalizable,the non-escaping set K(S_∞) does not support any non-trivial quasiconformal deformation. Hence, the quasiconformal deformation space of the map S_∞ agrees with the horizontal leaf (or hybrid class) Ψ_S_∞(Teich(G_0)).
It follows that 𝒮 = Ψ(G_∞,S_∞).

 Continuity of the inverse Ψ^-1 can be proved using similar arguments. This completes the proof of the result.



By the discussion preceding this theorem, the sequence of holomorphic maps 
{Ψ_S_n:Teich(G_0)→𝒯} is pre-compact, and any subsequential limit Ψ_∞:Teich(G_0)→𝒯 satisfies the properties

    
  * Ψ_∞(G)∈ℬ_G, for each G∈Teich(G_0), and
    
  * Ψ_∞(Teich(G_0)) is contained in the quasiconformal deformation space of the map S_∞.

The fact that S_∞ is periodically repelling and finitely renormalizable implies that K(S_∞) does not support any non-trivial quasiconformal deformation. Hence, the quasiconformal deformation space of the map S_∞ agrees with the horizontal leaf (or hybrid class) Ψ_∞(Teich(G_0)). It now follows that Ψ_∞=Ψ_S_∞; i.e., the maps Ψ_S_n converge to Ψ_S_∞ locally uniformly.

Let 𝒳⊂Teich(G_0) be a compact set containing the sequence {G_n} and its limit G_∞. Since Ψ_S_n→Ψ_S_∞ uniformly on 𝒳, it follows that

    Ψ(G_n,S_n)=Ψ_S_n(G_n)→Ψ_S_∞(G_∞)=Ψ(G_∞,S_∞),  as n→+∞.


 Continuity of the inverse Ψ^-1 can be proved using similar arguments. This completes the proof of the result.






 §.§ A holomorphic motion and source of discontinuity

We will end our discussion with a brief interpretation of the picture in terms of holomorphic motions. The account given below has roots in the analogous description of the space of quadratic-like maps given in <cit.>.




  §.§.§ The holomorphic case




We note that the domain of definition of a map in 𝒯 is a finite tree of closed Jordan disks (a tree-like inversive multi-domain), and these Jordan domains can be uniformized by rational maps. 
One can regard  each strata

    𝒯_ℒ:=_G∈Teich(G_0)ℬ_G,ℒ

of the total space as a subset of a certain quotient of the product space ∏_j=1^k Rat_d_j(), where d_j are the degrees of the uniformizing rational maps and the quotients come from normalizations of the rational maps (cf. <cit.>). Here, ℬ_G,ℒ consists of maps in ℬ_G with coarse lamination ℒ, see Subsection <ref>.

The map Ψ of Theorem <ref> defines a `holomorphic motion' of ℬ_G_0 over the Teichmüller space Teich(G_0) in the above ambient space; indeed, the maps Ψ_S can be shown to be holomorphic (cf. <cit.>). However, the dimension of ℬ_G_0 is larger than one as d> 2. Hence, there is no λ-lemma in this setting that guarantees transverse continuity of the holomorphic motion. Note that transverse continuity of the motion is equivalent to continuity of the holonomy maps Ψ_G:ℬ_G_0→ℬ_G, G∈Teich(G_0).

We will now analyze the source of possible discontinuity of the above holomorphic motion in the transverse direction. To this end, let us first fix G_1∈Teich(G_0). For G close to G_1 in Teich(G_0), the conjugacies between G and G_0 can be chosen to be K-quasiconformal, for some K>1. This implies that the hybrid conjugacies between S∈ℬ_G_0 and Ψ_S(G) are given by global K-quasiconformal homeomorphisms, for G close to G_1 (note that K does not depend on S∈ℬ_G_0). Consequently, the family of holomorphic maps 

    {Ψ_S:Teich(G_0)→𝒯_ℒ}_S∈ℬ_G_0,ℒ

is locally bounded and hence normal (recall that ℬ_G_0 is compact).
Therefore, for a sequence S_n→ S_∞ in ℬ_G_0,ℒ, the holomorphic maps Ψ_S_n have (injective) holomorphic subsequential limits that send each G∈Teich(G_0) into the vertical fiber ℬ_G. However, although each Ψ_S_n takes values in the hybrid class [S_n], a priori, the images of the holomorphic subsequential limits are contained in the quasiconformal deformation space of the map S_∞, which is possibly larger than the hybrid class [S_∞] (cf. <cit.>). In other words, if the filled Julia set K(S_∞) supports non-trivial quasiconformal deformations, then the map Ψ_S_∞ may not be the local uniform limit of {Ψ_S_n}. This is exactly the reason why the holonomy maps Ψ_G may fail to be continuous at quasiconformally non-rigid parameters. (This is also the same phenomenon that is responsible for discontinuity in Theorem <ref> and Theorem <ref>, and c.f. Theorem <ref>.)




    The situation discussed above is analogous to the holomorphic motion of the Mandelbrot set over  in the space of quadratic rational maps (where one varies the multiplier of the fixed point at ∞ throughout , cf. <cit.>). Since the Mandelbrot set lives in , this motion enjoys transverse continuity (in fact, quasiconformality).




  §.§.§ The antiholomorphic case


Much of the discussion of the previous subsection can be applied to Schwarz reflections after passing to the second iterate. Nevertheless, since the Teichmüller space of the reflection group G_d only admits a real-analytic structure (as opposed to complex structures on Teichmüller spaces of Fuchsian groups), the motion of vertical fibers is only real-analytic in this context. 


To set up the framework, let us consider the second iterates _G^∘ 2 of Nielsel maps of the reflection groups G∈Teich(G_d). Such a map is defined on the complement of the interior of an ideal d(d+1)-gon Π(G) in . In Figure <ref> (where d=3), the polygon Π(G) is bounded by the red geodesics, while the blue geodesics bound the original polygon Π(G) whose boundary reflections generate G. If r_1,⋯, r_d+1 are the reflections in the sides of Π(G), then for each j∈{1,⋯,d+1}, 
    Ξ_j:={r_j∘ r_k: k∈{1,⋯,d+1}, k≠ j}
 
is a minimal generating set of the index two Fuchsian subgroup G of G. In fact, the elements of Ξ_j are the side-pairing transformations for the fundamental domain Π(G)∪ r_j(Π(G)) of G. It is also easily seen that 

    Π(G)= Π(G)∪⋃_j=1^d+1r_j(Π(G)),

and /G is a (d+1)-times punctured sphere that admits an antiholomorphic involution (induced by r_j). It can now be readily verified that the maps _G^∘ 2, G∈Teich(G_d), are higher Bowen-Series maps associated with (d+1)-times punctured spheres in the sense of <cit.> (cf. <cit.>).

According to <cit.>, one can associate a higher Bowen-Series map to each member Γ∈Teich(G_d) (where G_d is the index two Fuchsian subgroup of G_d). Further, a quasiconformal conjugation between G_d and Γ induces a quasiconformal conjugation between their higher Bowen-Series maps. We denote the higher Bowen-Series map by A_Γ^h. We denote the vertical fiber of A_Γ^h in the space of degree d^2 degenerate polynomial-like maps with connected Julia set by _Γ^h. We define the total space

    𝒯^h:=_Γ∈Teich(G_d) _Γ^h.

The arguments of the previous subsection now implies that the `base' vertical fiber _G_d^h moves holomorphically over the Teichmüller space of G_d. Here, we embed the total space 𝒯^h into the product space of holomorphic maps on finitely many domains in .

The operation of passing to the second iterate embeds the vertical fiber 𝒮_G_d (in the space of degree d degenerate anti-polynomial-like maps with connected Julia set) into the vertical fiber _G_d^h (in the space of degree d^2 degenerate polynomial-like maps with connected Julia set). Thus, this copy of 𝒮_G_d in _G_d^h moves holomorphically over Teich(G_d). This holomorphic motion can be used to explain the topological product structure in the antiholomorphic case of Theorem <ref>.

Finally, note that Teich(G_d) is the fixed-point set of an antiholomorphic involution on Teich(G_d) (elements of Teich(G_d) correspond to `symmetric' (d+1)-times punctured spheres, cf. <cit.>), and hence Teich(G_d) can be regarded as a real slice of the complex manifold Teich(G_d). It follows that the above copy of 𝒮_G_d moves real-analytically over Teich(G_d).


LLMM23a


[AS76]AS76
D. Aharonov and H. S. Shapiro.
Domains on which analytic functions satisfy quadrature identities.
J. Analyse Math., 30:39–73, 1976.

[BB09]BB09
G. Bassanelli and F. Berteloot.
Lyapunov exponents, bifurcation currents and laminations in bifurcation loci.
Math. Ann., 345:1–23, 2009.

[BF03]BF03
S. Bullett and M. Freiberger.
Hecke groups, polynomial maps and matings.
Internat. J. Modern Phys. B, 17:3922–3931, 2003.

[BF05]BF05
S. Bullett and M. Freiberger.
Holomorphic correspondences mating Chebyshev-like maps with Hecke groups.
Ergodic Theory Dynam. Systems, 25:1057–1090, 2005.

[BH07]BH07
S. Bullett and P. Haïssinsky.
Pinching holomorphic correspondences. 
Conform. Geom. Dyn., 11:65–89, 2007.

[BL20]BL20
S. Bullett and L. Lomonaco.
Mating quadratic maps with the modular group II.
Invent. Math., 220:185–210, 2020.

[BL22]BL22
S. Bullett and L. Lomonaco.
Dynamics of Modular Matings.
Adv. Math., 410, Part B, 108758, 2022.

[BLLM24]BLLM
S. Bullett, L. Lomonaco, M. Lyubich, and S. Mukherjee.
Mating parabolic rational maps with Hecke groups.
<https://arxiv.org/abs/2407.14780>, 2024.

[BP94]BP
S. Bullett and C. Penrose.
Mating quadratic maps with the modular group.
Invent. Math., 115:483–511, 1994.

[CDKvS22]CDKvS22
T. Clark, K. Drach, O. Kozlovski, and S. van Strien.
The dynamics of complex box mappings. 
Arnold Math. J., 8:319–-410, 2022.

[DS06]DS06
T.-C. Dinh and N. Sibony.
Distribution des valeurs de transformations méromorphes et applications.
Comment. Math. Helv. 81:221–258, 2006.

[DH85a]DH1
A. Douady and J. H. Hubbard.
Étude dynamique des polynômes complexes I, II.
Publications Mathématiques d'Orsay. Université de Paris-Sud, Département de Mathématiques, Orsay, 1984 - 1985.
  
[DH85b]DH2
A. Douady and J. H. Hubbard.
On the dynamics of polynomial-like mappings.
Ann. Sci. Éc. Norm. Sup., 18:287–343, 1985.

[Dud11]Dud11
D. Dudko.
Matings with laminations.
<https://arxiv.org/abs/1112.4780>, 2011.

[Eps93]Eps93
A. L. Epstein.
Towers of finite type complex analytic maps.
PhD thesis, The City University of New York, 1993.

[Fat29]Fatou29
P. Fatou.
Notice sur les travaux scientifiques de M. P. Fatou.
Astronome titulaire de l'observatoire de Paris, 5–29, 1929, <https://www.math.purdue.edu/ eremenko/dvi/fatou-b.pdf>.

[Gey20]Gey20
L. Geyer.
Classification of critically fixed anti-rational maps.
<https://arxiv.org/abs/2006.10788v3>, 2020.

[GM93]GM93
L. Goldberg and J. Milnor.
Fixed points of polynomial maps, part II: fixed point portraits.
Ann. Sci. Éc. Norm. Supér. (4), 26(1):51–98, 1993.

[GT21]GT21
Y. Gao and G. Tiozzo.
The core entropy for polynomials of higher degree.
J. Eur. Math. Soc. (JEMS), 24:2555–2603, 2022.

[HS14]HS
J. H. Hubbard and D. Schleicher.
Multicorns are not path connected.
Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday, pages 73–102, 2014.

[Ino09]Ino09
H. Inou.
Combinatorics and topology of straightening maps II: Discontinuity.
<http://arxiv.org/abs/0903.4289>, 2009.

[IK12]IK
H. Inou and J. Kiwi.
Combinatorics and topology of straightening maps, I: Compactness and bijectivity.
Adv. Math., 231:2666–2733, 2012.

[IM21]IM21
H. Inou and S. Mukherjee.
Discontinuity of straightening in anti-holomorphic dynamics: I.
Trans. Amer. Math. Soc., 374:6445–6481, 2021.
  
[KvS09]KvS09
O. Kozlovski, S. van Strien. 
Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials. 
Proc. Lond. Math. Soc. (3), 99:275–296, 2009.

[Kiw97]Kiw97
J. Kiwi.
Rational rays and critical portraits of complex polynomials. 
Thesis, SUNY at Stony Brook, 1997.

[Kiw01]Kiw01
J. Kiwi.
Rational laminations of complex polynomials.
In Mikhail Lyubich, John W. Milnor, and Yair N. Minsky, editors, Laminations and Foliations in Dynamics, Geometry and Topology, volume 269 of
  Contemporary Mathematics, pages 111–154. American Mathematical Society, 2001.
  
[Kiw04]Kiw04
J. Kiwi.
ℝeal laminations and the topological dynamics of complex polynomials.
Adv. Math., 184:207–267, 2004.
 
[LMM22]LMM2
K. Lazebnik, N. G. Makarov, and S. Mukherjee.
Bers slices in families of univalent maps.
Math. Z., 300:2771–2808, 2022.


[LLMM21]LLMM3
S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee.
Schwarz reflections and anti-holomorphic correspondences.
Adv. Math., 385:Paper No. 107766, 88, 2021.

[LLMM23a]LLMM1
S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee.
Dynamics of Schwarz reflections: the mating phenomena.
Ann. Sci. Éc. Norm. Supér. (4), 56:1825–1881, 2023.
  
  
[LLMM23b]LLMM2
S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee.
Schwarz reflections and the Tricorn.
To appear in Ann. Inst. Fourier (Grenoble), <https://arxiv.org/abs/1812.01573v2>, 2023.

[LLMM23]LLMM4
R. Lodge, M. Lyubich, S. Merenkov, and S. Mukherjee.
On dynamical gaskets generated by rational Maps, Kleinian groups, and Schwarz reflections.
Conform. Geom. Dyn., 27:1–54, 2023. 

[LLM22a]LLM1
R. Lodge, Y. Luo, and S. Mukherjee.
Circle packings, kissing reflection groups and critically fixed anti-rational maps.
Forum Math. Sigma, 10, paper no. e3, 38 pp., 2022.

[LM14]LM14
S. Y. Lee and N. G. Makarov.
Sharpness of connectivity bounds for quadrature domains.
<https://arxiv.org/abs/1411.3415v1>, 2016.

[LM16]LM16
S. Y. Lee and N. G. Makarov.
Topology of quadrature domains.
J. Amer. Math. Soc., 29(2):333–369, 2016. 

[Lom15]Lom15
L. Lomonaco.
Parabolic-like mappings.
Ergodic Theory Dynam. Systems, 35:2171–2197, 2015.

[Luo23]L23
Y. Luo.
On geometrically finite degenerations I: boundaries of main hyperbolic components.
to appear in J. Eur. Math. Soc., 2023.

[Lyu97]Lyu97
M. Lyubich.
Dynamics of quadratic polynomials. I, II.
Acta Math., 178:185–297, 1997.

[Lyu99]Lyu99
M. Lyubich.
Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture. 
Ann. of Math. (2), 149:319–420, 1999.

[Lyu24]Lyu23
M. Lyubich.
Conformal geometry and dynamics of quadratic polynomials, vol i-ii.
<http://www.math.stonybrook.edu/ mlyubich/book.pdf>, 2023.

[LMM23]LMM23
M. Lyubich, J. Mazor, and S. Mukherjee.
Antiholomorphic correspondences and mating I: realization theorems.
To appear in Comm. Amer. Math. Soc., <https://arxiv.org/abs/2303.02459>, 2023.

[LMMN20]LMMN
M. Lyubich, S. Merenkov, S. Mukherjee, and D. Ntalampekos.
David extension of circle homeomorphisms, welding, mating, and removability.
To appear in Mem. Amer. Math. Soc., <https://arxiv.org/abs/2010.11256v2>, 2020.

[LM93]LM93
M. Lyubich and J. Milnor.
The Fibonacci unimodal map.
J. Amer. Math. Soc., 6:425–457, 1993.

[Mak93]Mak93
P. Makienko.
Pinching and plumbing deformations of  quadratic rational maps.
IC/93/32, International Centre for Theoretical Physics, Trieste, Italy, 1993, <https://inis.iaea.org/collection/NCLCollectionStore/_Public/24/036/24036519.pdf>.

[Man93]Man93
R. Mañé.
On a theorem of Fatou.
Bol. Soc. Brasil. Mat. (N.S.), 24:1–11, 1993.

[McM88]McM
C. McMullen.
Automorphisms of rational maps.
In Holomorphic functions and moduli, vol. I (Berkeley, CA, 1986), 31–60, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988. 

[McM94]McM1
C. McMullen.
Complex dynamics and renormalization.
Annals of Mathematics Studies, 135. Princeton University Press, Princeton, NJ, 1994. x+214 pp. 
         
[Mil00]Mil00
J. Milnor.
Periodic orbits, external rays and the Mandelbrot set.
Astérisque, 261:277–333, 2000.


[Mil06]Mil06
J. Milnor.
Dynamics in one complex variable.
Princeton University Press, New Jersey, 3rd edition, 2006.         

[MM23a]MM1
M. Mj and S. Mukherjee.
Combining rational maps and Kleinian groups via orbit equivalence.
Proc. Lond. Math. Soc (3), 126:1740–1809, 2023.

[MM23b]MM2
M. Mj and S. Mukherjee.
Matings, holomorphic correspondences, and a Bers slice. 
<https://arxiv.org/abs/2304.12699>, 2023.

[MM23c]MM3
M. Mj and S. Mukherjee.
The Sullivan dictionary and Bowen-Series maps.
EMS Surv. Math. Sci., 10:179–221, 2023.

[MNS17]MNS
S. Mukherjee, S. Nakane, and D. Schleicher.
On Multicorns and Unicorns II: bifurcations in spaces of antiholomorphic polynomials.
Ergodic Theory Dynam. Systems, 37:859–899, 2017.

[PR21]PR21
C. L. Petersen and P. Roesch.
The Parabolic Mandelbrot Set.
<https://arxiv.org/abs/2107.09407>, 2021.

[Sak91]Sak91
M. Sakai.
Regularity of a boundary having a Schwarz function.
Acta Math., 166:263–297, 1991.

[Sul85]Sul
D. Sullivan.
Quasiconformal homeomorphisms and dynamics I. solution of the
  Fatou-Julia problem on wandering domains.
Ann. of Math. (2), 122(2):401–418, 1985.

[Sul92]Sul1
D. Sullivan.
Bounds, quadratic differentials, and renormalization conjectures. 
American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988), 417–466, Amer. Math. Soc., Providence, RI, 1992.

[SM98]SulMc 
D. Sullivan and C. T. McMullen.
Quasiconformal homeomorphisms and dynamics III: The Teichmüller space of a holomorphic dynamical system.
Adv. Math., 135:351–395, 1998.

[Urb03]Urb1
M. Urbanski.
Measures and dimensions in conformal dynamics.
Bull. Amer. Math. Soc. (N.S.), 40:281–321, 2003.





 Paper Title: The Aldous--Lyons Conjecture I: Subgroup Tests.












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L. Bowen]Lewis Bowen
Lewis BowenDepartment of Mathematics 1 University Station C1200 University of Texas at AustinAustin, TX, 78712 USA.
lpbowen@math.utexas.edu

M. Chapman]Michael Chapman
Michael ChapmanCourant Institute of Mathematical SciencesNew York University,251 Mercer St, New York, NY 10012, USA.
mc9578@nyu.edu

A. Lubotzky]Alexander Lubotzky
Alexander LubotzkyWeizmann institute of ScienceRehovot, Israel.
alex.lubotzky@mail.huji.ac.il

T. Vidick]Thomas Vidick
Thomas VidickWeizmann institute of ScienceRehovot, Israel.
thomas.vidick@weizmann.ac.il


§ ABSTRACT 
    This paper, and its companion <cit.>, are devoted to  a negative resolution of the Aldous–Lyons Conjecture Aldous_Lyons_Conj,Aldous–Lyons_conj_blogpost. This conjecture, originated in probability theory, is well known (cf. <cit.>) to be equivalent to the statement that every invariant random subgroup of the free group is co-sofic. We disprove this last statement.
    

    
   In this part we introduce subgroup tests. These tests are finite distributions over continuous functions from the space of subgroups of the free group to {0,1}. Subgroup tests provide a general framework in which one can study invariant random subgroups of the free group. Classical notions such as group soficity and  group stability arise naturally in this framework. By the correspondence between subgroups of the free group and Schreier graphs, one can view subgroup tests as a property testing model for certain edge-labeled graphs. This correspondence  also provides the connection to random networks.
    
    Subgroup tests have values, which are their asymptotic optimal expectations when integrated against co-sofic invariant random subgroups.  Our first main result is that, if every invariant random subgroup of the free group is co-sofic, then one can  approximate the value of a subgroup test up to any positive additive constant. 

    Our second main result is an essentially value preserving correspondence between certain non-local games and subgroup tests. By composing this correspondence with a stronger variant of the reduction in ^*= <cit.>,  proved in the companion paper <cit.>, we deduce that approximating the sofic value of a subgroup test  is as hard as the Halting Problem, and in particular, undecidable. The combination of our two main results proves the existence of non co-sofic invariant random subgroups of the free group. 

    

The Aldous–Lyons Conjecture I: Subgroup Tests
    [
    August 5, 2024
=============================================





§ INTRODUCTION

In their seminal paper <cit.>, Aldous and Lyons ask whether every unimodular network is sofic. Conditional on a certain result which will be addressed in a companion paper <cit.>, we prove the answer is `no'.
This conjecture, which originated in probability theory, can also be expressed in the language of invariant random subgroups, and this is the one we will use in this paper.
We explain and motivate this conjecture before discussing our approach. The formal definitions and results will start in  Section <ref>.



 §.§ The Aldous–Lyons Conjecture




  §.§.§ What is soficity?

Roughly speaking, a countable group Γ is sofic if it admits a sequence of partial actions on finite sets which approximates the action of Γ on itself by left-translations. If Γ=⟨ S | R⟩ is finitely presented then this is equivalent to the existence of a sequence of finite graphs which converge to the Cayley graph of Γ in the sense of Benjamini–Schramm, which is a kind of local-on-average convergence.

To be more precise, a rooted graph is a pair (G,o) where G=(V,E) is a graph and o ∈ V is a distinguished vertex called the root. A random rooted graph (G,o) is sofic if there exists a sequence (G_i)_i ∈ of finite graphs such that, if o_i is a uniformly random vertex of G_i, then (G_i,o_i) converges to (G,o) in distribution (this means: for every r>0, the radius r neighborhood of the root in G_i converges in distribution to the radius r neighborhood of the root in G). This notion of convergence is due to Benjamini and Schramm <cit.>: They proved that if each finite graph G_i is planar and there is a uniform degree bound, then the limit (G,o) is almost surely recurrent. 

The notions of Benjamini–Schramm convergence and soficity naturally generalize from graphs to labeled graphs or networks <cit.>, simplicial complexes <cit.>, manifolds <cit.>, measured equivalence relations <cit.>, measured groupoids 
<cit.> and most generally, measured metric spaces <cit.>. 

Again, a finitely presented group is sofic if one of its Cayley graphs is sofic.[If one of the Cayley graphs is sofic, then all those that use finitely many generators are.]  For example, amenable groups are sofic. In fact, the Cayley graph of an amenable group admits a sequence of finite sub-graphs in which the isoperimetric ratio (number of boundary vertices to number of vertices) tends to zero, which implies this sequence Benjamini–Schramm converges to the Cayley graph.  Also, residually finite groups are sofic because they admit sequences of finite quotient groups whose Cayley graphs approximate the Cayley graph of the given group. 

Gromov implicitly introduced sofic groups in  <cit.>. He was motivated by Gottshalk's Conjecture: If Γ is a countable group and k ∈, then any Γ-equivariant continuous map ϕ:[k]^Γ→ [k]^Γ which is injective is necessarily surjective. This is obviously true when Γ is finite. Gromov proved that it holds when Γ is sofic. Benjy Weiss gave another proof and coined the term `sofic' from the Hebrew word for finite <cit.>.



      The soficity property has been useful in obtaining positive results about L^2 and spectral theory invariants <cit.>, group rings MR2089244, MR2417890, invariant couplings of random fields <cit.>, and in constructing dynamical invariants of groups actions MR2552252, MR2854085, MR3077882, MR3132735, MR3993930. See Cap_Lup_Sofic_Hyperlinear_book, MR3821628 for more background on sofic groups. The basic idea is roughly the same in each case: One transfers properties of the finite approximating objects to properties of the limit.  

      It is a major open problem to determine whether all countable groups are sofic. We do not directly address this problem because the class of objects we study, described next, is more general than the class of groups.

 





  §.§.§ What is unimodularity?






The main tool for analyzing Benjamini–Schramm limits is called the Mass Transport Principle which makes precise the intuitive notion of statistical homogeneity. To explain, a doubly-rooted graph is an ordered triple (G,o_1,o_2), where o_1,o_2 are vertices of G. A transport function is a function f, satisfying some measurability condition, which takes as input a doubly rooted graph and outputs a non-negative real number. The interpretation is that f(G,o_1,o_2) is the amount of mass the first root o_1 sends to the second root o_2. So if (G,o) is a random rooted graph then _(G,o)[∑_x ∈ V(G) f(G,o,x)] is the average amount of mass sent out of the root. Symmetrically, _(G,o)[∑_x ∈ V(G) f(G,x,o)] is the average amount of mass sent into the root. If these two quantities are equal for every transport function f, then we say (G,o) satisfies the Mass Transport Principle. Equivalently, we say it is unimodular (this is the term used by Aldous and Lyons in <cit.>). Alternatively, unimodularity can be formulated in terms of graphings or as invariance with respect to the root-changing equivalence relation on the space of rooted graphs. The former arises from the ergodic theory of measured equivalence relations (e.g. <cit.>). The latter point-of-view was introduced in <cit.> and further developed in <cit.>.


The term unimodular comes from the following special case. Suppose G is a connected transitive graph; this means that its automorphism group Aut(G) acts transitively on its vertex set. Then (G,o) is unimodular if and only if the automorphism group Aut(G) is unimodular in the sense that its left and right Haar measures agree.











The Mass Transport Principle arose in Häggström's study of percolation clusters on trees <cit.>. In fact, it is an exercise to show that if (G,o) is unimodular and p ∈ [0,1] is given, then the Bernoulli p-percolation cluster containing o is also unimodular. Unimodularity was further developed in <cit.> (see also <cit.> where it was used to construct a version of the Euler characteristic for unimodular planar maps).

It is straightforward to verify that (1) unimodularity is closed under weak limits and (2) if G is a finite graph and o is a uniformly random vertex of G then (G,o) is unimodular. It follows from these observations that soficity implies unimodularity. Question 10.1 of <cit.> asks whether unimodularity implies soficity.

As above, unimodularity has been generalized to random rooted labeled graphs, simplicial complexes, manifolds and so on. Benjamini and Schramm note that many results which are known to hold in the deterministic setting of unimodular transitive graphs can be generalized to  unimodular random graphs <cit.>. Aldous and Lyons demonstrate this with results about random walks, amenability, ends of graphs, spanning forests, percolation, and so on <cit.>.



  §.§.§ What is an invariant random subgroup?


As a first step towards our goal, let us describe the algebraic formulation of the Aldous–Lyons Conjecture in terms of invariant random subgroups, which were introduced in <cit.>, <cit.>, <cit.> (and implicitly in <cit.>). The concept of invariant random subgroups has been profitably studied in the context of locally compact groups (e.g. stuck1994stabilizers, MR3664810,Gelander_ICM2018). 
However, we will restrict our focus to the case of most relevance to this paper: finitely generated groups. 
Already in this discrete setup, invariant random subgroups play an important role in graph convergence <cit.>, stability properties of groups <cit.> and the analysis of dynamical systems stuck1994stabilizers, MR3193754 for example.

So, fix a finitely generated group Γ. An invariant random subgroup (or IRS) of Γ is a random variable taking values in the space of all subgroups of Γ, whose law is invariant under the conjugation action of Γ. For example, every normal subgroup of Γ is an IRS. More generally, if H ≤Γ has only finitely many conjugates, then a uniformly random sample of its conjugates is an IRS. As finite index subgroups have only finitely many conjugates, they induce  IRSs which are called elementary. To generate more examples, one can take convex combinations and weak* limits of elementary IRSs, and the IRSs that arise this way are called co-sofic. It turns out that for some groups every IRS is co-sofic, and for others there are non co-sofic IRSs (cf. <cit.>). The Aldous–Lyons Conjecture can be formulated as follows: 


    Are all IRSs of a (non-commutative) free group co-sofic?

 
For the rest of the paper, this is the formulation that we tackle, and it is known — and we explain it in the following paragraphs — to be equivalent to the probability theoretic formulation on Benjamini–Schramm limits of finite graphs.

The law (or distribution) of an IRS is a Borel probability measure μ on 𝔰𝔲𝔟(Γ), the space of subgroups of Γ. We let (Γ) denote the space of all such measures. By abuse of language, we say that μ is an IRS if μ∈(Γ).
To see the relation between the two formulations of the Aldous–Lyons Conjecture, note that if H is a subgroup of Γ, then the Schreier coset graph of _H\^Γ with respect to some generating set S is a rooted (directed, edge-labeled) graph, with the root being the coset of the identity.  This gives a map from the space  of subgroups 𝔰𝔲𝔟(Γ) to the space of isomorphism classes of rooted (edge-labeled, directed) graphs. 

If μ∈(Γ) then μ pushes forward under this map to a unimodular measure <cit.>. 
 Moreover, if μ is elementary, then its pushforward is induced by uniformly choosing a root of some finite connected graph. Therefore, if μ is co-sofic, then its pushforward is sofic.

Now, consider the special case in which Γ=(S) is the free group generated by a finite set S. Suppose μ is an invariant random subgroup of Γ, and that the (random) Schreier coset graph of μ is sofic (in the directed, edge-labeled rooted graph category). This means there exists a sequence of finite directed, edge-labeled graphs G_i which Benjamini–Schramm converge to the (random) Schreier coset graph of μ. After perturbing the given sequence if necessary, we may assume that each G_i is itself a Schreier coset graph with respect to some finite-index subgroup H_i of (S) — the existence of such perturbations is exactly where the freeness of the group (S) plays a role. Moreover, if H_i is a uniformly random conjugate of H_i, then the distribution of H_i converges to μ. Therefore, a positive solution to the Aldous–Lyons Conjecture implies that every IRS of (S) is co-sofic.[There is a standard “decoration of graphs” technique that allows one to show that the directed, edge labeled (with finitely many labels) version of the Aldous–Lyons conjecture is equivalent to the non-directed, non-edge labeled version. As Bálint Virág pointed to us, the same technique is used, e.g., to show that every group is the automorphism group of some graph. As this is standard, we do not elaborate on it anymore.]







 §.§ Our approach



  §.§.§ What is a subgroup test?

The set () of all IRSs of the free group and the set _() of co-sofic IRSs of the free group are both convex. Thus, the Aldous–Lyons Conjecture is asking whether certain convex sets can be separated. A natural way to distinguish between two convex sets is to use (continuous) linear functionals. So, in order to separate the above, one should search for a rich enough collection of functionals. Subgroup tests and their values will play this exact role.  

A challenge is a continuous map from () to {0,1}. A subgroup test  is a probability distribution on a finite set of challenges. One can integrate any subgroup test against any probability measure over (). If π is in (), then this integral is called the value of the strategy π against the test .

 Many natural and well studied properties of groups, specifically soficity and pointwise permutation stability, can be formulated as properties of subgroup tests. 

The terminology of “tests” and “values” originates in the field of interactive proofs in computer science. 
An interactive proof designates an interaction between two entities, a “verifier” and a “prover”. The goal of the prover is to convince the verifier of the validity of a certain claim, and the goal of the verifier is to test the prover so that it  accepts the interaction only if the claim  is indeed correct. 
Examples of “claims” studied in computer science are the 3-colorability of a given input graph, or that a Turing machine whose description is passed as input halts.
In our approach, the methods of the verifier and the prover are restricted. The prover is required to sample an element of (). 

The sampling method itself is restricted to some sub-class of IRSs  (e.g., elementary IRSs); the choice of distribution according which to sample is the only degree of freedom the prover has, and is called its strategy. The verifier gets access to (the indicator function of) the prover's sampled subgroup H, and it makes a decision whether to accept or reject this subgroup. 
To decide, the verifier holds some finite set K ⊆ — known beforehand to the prover — and according to which combinations of elements of K are contained in H, it accepts or rejects; the rules that control which combinations of elements result in acceptance and which combinations result in rejection are also know  to the prover beforehand. The value of a strategy in a game is the probability, taken over the prover's sampling and the verifier's probabilistic choices, that the verifier makes the decision to accept when interacting with a prover using this strategy. 

In spite of these restrictions, it is possible to verify hard computational problems in this setup.[For the expert, we note that restrictions on the  prover and verifier can affect both the completeness and soundness properties of a proof system. Hence, a priori, they may either lower or raise the complexity of the associated class of interactive proofs compared to classical single-prover interactive proofs.] For example, it is not difficult to design a subgroup test such that there is a value-1 strategy for this test if and only if a given graph is 3-colorable. Jumping ahead, the correct analogy is with the theory of multiprover interactive proofs, and even more specifically interactive proofs with quantum provers sharing entanglement, also known as non-local games. We explain below this connection, which plays an essential role in the second part of this paper.





  §.§.§ What is a non-local game?

Morally, non-local games originated in Bell's resolution <cit.> of the Einstein–Podolsky–Rosen paradox <cit.>. Einstein was famously concerned by the ability of space-time separated particles to correlate in ways that might  violate relativity, a possibility that seemed to be suggested by the mathematical modeling of quantum mechanics. Einstein argued for a local hidden variable theory — namely, that the particles did share some information beforehand,  which allowed them to correlate — and that physicists should search for this hidden information.
Bell provided a thought experiment, that in modern jargon is an instance of a non-local game, which proves that the kinds of correlations that arise from isolated quantum mechanical systems are intrinsically different from those that could be generated in any hidden variable model. 
Bell's work lay the foundation for the subsequent design, and execution, of experiments which verify the existence of quantum entanglement (and thus refute Einstein's approach; see also <cit.>). The 2022 Nobel prize in physics was awarded to Aspect, Clauser and Zeilinger, partly for performing non-local games as experiments and verifying that the winning statistics in the games exceeds what local hidden variable models allow.


A correlation is a function p A× A× X× X→ [0,1], where X and A are finite sets, such that for every pair x,y∈ X the function p(·,·|x,y) is a probability distribution over A× A — the quantity p(a,b|x,y) should be thought of as the answer to “what is the probability a and b are provided as answers given that x and y were asked as questions?”; this cryptic phrase will soon be clarified.  A correlation  p is said to be deterministic if there are two functions f_1,f_2 X→ A such that p(a,b|x,y)=1 only if f_1(x)=a and f_2(y)=b.
The convex hull   C_c of  deterministic correlations is the set of local or  classical correlations.[An example of a non-local correlation is p(a,b|x,y)=1 only when a=y and b=x. Intuitively, this is because this correlation implies signaling from one system to the other.] 
On the other hand, the  quantum correlations  C_q are those that can be generated by performing quantum measurements on a (finite-dimensional) bipartite physical system, where the measurement on one part depends only on x and generates a, while  the measurement on the other part depends only on y and generates b. For a formal description, see Remark <ref>. The set C_q is convex, but interestingly it is not closed <cit.>.

A non-local game  is specified by a probability distribution μ on the set X×X, and a decision function D:X× X× A× A→{0,1}.[As can be seen in Section <ref>, our definition of non-local games is slightly different. This difference is mainly cosmetic, and is driven by our motivation to relate non-local games to subgroup tests.] 
The interpretation of this combinatorial object  as a “game” comes from thinking of it (similarly to subgroup tests) as an interactive proof, but this time with two provers, a la <cit.>. 
The common dramatization goes as follows: Two provers are spatially separated. The verifier samples a pair of “questions” x,y∼μ, and “asks”  one prover x and   the other y. The provers then apply some local procedure to choose their “answers” a and b respectively. Finally, the verifier accepts if D(x,y,a,b)=1 and rejects otherwise.  

Since the process by which the provers generate their answers is hidden to us, we can only “observe” (samples from) the distribution of answers given questions, namely, the underlying correlation. So, the correlation can be thought of as the “strategy” used by the provers (similar to the way IRSs are seen as strategies for the prover in a subgroup test).  Bell's separation of C_c from C_q amounts to devising a game that is “easy” for players that use quantum correlations, yet “hard” for players that use only classical ones. 

Now, there is more than one candidate mathematical model for entanglement in quantum mechanics. A slightly generalized model, suggested by Tsirelson <cit.>,  gives birth to a set of correlations known as quantum commuting correlations C_qc. This set is closed and contains the quantum correlations.  Tsirelson famously asked whether the closure of C_q is equal to C_qc <cit.>. At this point, we hope the type of problem already resonates with the reader, as this is again a separation of convex sets type of problem.[Connes' embedding problem (CEP) is also a separation of convex sets type of problem, specifically, separating all  characters of the free group from those that are limits of finite dimensional characters. This problem was shown <cit.> to be equivalent to Tsirelson's problem, and also has close connections with invariant random subgroups — as every IRS induces a character. We omit discussions on the relations between this work and CEP from the text, as they are not particularly helpful for understanding our approach.]



  §.§.§ Harnessing undecidability

It was known that if  the quantum correlations are dense in the quantum commuting correlations, then the complexity class of multiprover interactive proofs with entangled provers (^*) contains only decidable languages.
By proving that the Halting Problem, which is undecidable, is in ^*, the authors of <cit.> were able to resolve Tsirelson's problem in the negative.


In this paper a similar path is followed: Rather than directly find a functional that separates the co-sofic and general IRSs of the free group, we describe a computational problem — approximating the sofic value of a subgroup test — whose undecidability implies a negative resolution of the Aldous–Lyons conjecture.  Our approach can be described in two steps: First, show that a positive solution to the Aldous–Lyons Conjecture implies that approximating the sofic value of a subgroup test is decidable. Second, prove that this computational task is undecidable.

Elaborating on the first,  given a subgroup test , we prove that its optimal value against co-sofic IRSs — which we call the sofic value of  — can be approximated from below, and its optimal value against any IRS — which we call the ergodic value of  — can be approximated from above.  Thus, if the Aldous–Lyons Conjecture has a positive answer, then the sofic and ergodic value of a test always agree, and using the aforementioned approximation procedures they can be calculated to any  predetermined accuracy. 

Elaborating on the second, we provide a mechanism for translating certain non-local games (see Section <ref>), which we call tailored non-local games, to subgroup tests. This translation is essentially value preserving. Namely, if the non-local game had a (certain kind) of perfect quantum strategy — which we call a Z-aligned permutation strategy that commutes along edges — then the analogous subgroup test has a perfect (co-sofic) strategy. In the other direction, if the subgroup test has an almost perfect (co-sofic) strategy, then the non-local game has an almost perfect strategy. Hence, a certain strengthening of the reduction in <cit.>, which states that small additive constant approximations to the value of tailored non-local game are undecidable (Theorem <ref>), implies that the sofic value of a game cannot be approximated to arbitrary precision. Thus, the Aldous–Lyons Conjecture has a negative solution.




The rest of the introduction provides a deeper dive into the  definitions, results and ideas   of this paper.




    

 §.§ Our approach





§ INTRODUCTION

Probabilistic methods for studying algebraic, geometric and combinatorial objects have proved themselves extremely powerful (cf. alon2016probabilistic,tenenbaum2015introduction). To better understand a collection of objects, you define a sampling scheme on it and analyze the expected behaviours of the samples. Invariant random subgroups, defined in <cit.>, are a flexible tool of this exact kind: They allow one to study lattices in Lie groups <cit.>, graph convergence <cit.>, stability properties of groups <cit.> and various dynamical systems <cit.>.

Given a countable group Γ, let (Γ) be the collection of all subgroups of Γ. An invariant random subgroup (IRS) is a Borel probability distribution over (Γ) which is invariant under conjugation. The most straightforward example of an IRS is the Dirac measure concentrated on a normal subgroup. Another basic example is the uniform distribution over the conjugation orbit of a finite index subgroup. We call such an IRS simple. To generate more examples, one can take convex combinations and weak* limits of the simple IRSs. The IRSs that arise this way are called co-sofic. It turns out that for some groups every IRS is co-sofic, and for others there are non co-sofic IRSs (cf. <cit.>). A major open problem in this field, due to Aldous and Lyons (Conjecture <ref>), asks  which is true for a (non-commutative) free group? In this paper (and its companion <cit.>), we prove that non-commutative free  groups have non co-sofic IRSs. 

The set of all IRSs of the free group, and the set of co-sofic IRSs of the free group, are both convex. Thus, the Aldous–Lyons conjecture is asking whether certain convex sets can be separated. The natural way to distinguish between convex sets is by using linear functionals. Thus, one should search for a rich collection of functionals if they seek to separate the above. Subgroup tests and their values will play this role.

Though it is tempting to try and find a specific functional that separates the co-sofic and general IRSs  of a free group, we suggest another approach in the spirit of the recent resolution of Tsirelson's problem <cit.>.  Tsirelson's problem is also a separation problem of convex sets, the quantum and quantum commuting correlations. 
It was known that if  the quantum correlations are dense in the quantum commuting correlations, then the complexity class of Multi-prover Interactive Proofs with Entangled Provers (MIP^*) contains only decidable languages.
Thus, by proving that the Halting Problem is in MIP*, they were able to resolve Tsirelson's problem.
In this paper we construct a similar setup, namely, define a complexity class whose containment of undecidable languages will imply the existence of non co-sofic IRSs in the free group. 
For those familiar with multi prover interactive proofs, we remark that  subgroup tests  are analogous to non-local games. They are combinatorial objects that test probability distributions over subsets of the free group. Many natural and well studied properties of groups, specifically soficity and pointwise permutation stability, can be formulated as properties of subgroup tests. 


In the rest of the introduction we give brief definitions of the relevant notions, and overview the main results of this paper.





 §.§ Subgroup Tests
 
Let S be a finite set, =(S) be the free group with basis S, () the collection of subgroups of  and Prob(sub(ℱ)) the set of all Borel probability measures on sub(ℱ). 
A challenge  is a pair (K;D) such that K is a finite subset of , and D is a function from subsets of K to {0,1}. A subset A⊆ passes the challenge (K;D) if D(A∩ K)=1, and fails it otherwise.
A test  is a finite collection of challenges {(K_i;D_i)}_i∈ Q, where Q is a finite index set, together with a probability distribution μ over the set Q. 
A strategy is a (Borel) probability distribution over subgroups of the free group, namely an element π∈( ()). 
One can run the test  against π, as follows. 

First the verifier samples i∈ Q according to μ. Next, the prover samples a subgroup  H according to π. Finally, the verifier makes the decision to accept if H passes the i^ th challenge (K_i;D_i), and reject if H fails the i^ th challenge.






The value of using the strategy π against the test  is its acceptance probability, namely

    (,π)=_H∼π_i∼μ[D_i(H∩ K_i)].

For a fixed , the value is a continuous linear functional from probability distributions over subgroups of  to ℝ. 
Hence, it enables us to study convex subsets of (()) by their optimal value against a given test. 


    As mentioned earlier, a challenge is, essentially, a continuous map from the space of subgroups of  to {0,1}. One can extract from any such continuous map  a challenge (K;D) that describes it and vice versa. Therefore, a test is a convex combination of continuous maps from () to {0,1}, and the value is integration of this map against a (probability) measure. 







 §.§ Invariant random subgroups


As sub(ℱ) inherits the product topology
from the power set {0,1}^ℱ,  it is compact.
Every action of a group Γ on a set X extends naturally to an action of Γ on the power set {0,1}^X by 

    ∀γ∈Γ,A⊆ X   γ.A={γ.a | a∈ A} .

Hence, the conjugation action of ℱ on itself is inherited by {0,1}^. Also, ()⊆{0,1}^ is preserved by this action. Hence, the conjugation action extends  to its power  set {0,1}^() by 

    ∀ w∈ℱ, A⊆sub(ℱ)    w. A={wHw^-1| H∈ A} .

 The conjugation action of ℱ further extends to Prob(sub(ℱ)):
For every Borel set B⊆() and μ∈ Prob(sub(ℱ)), we have

    (w.μ)(B)=μ(w^-1.B) .
 



The set of invariant random subgroups IRS(ℱ)
consists of all probability measures in Prob(sub(ℱ))
that are invariant under the action of ℱ. 


The ergodic value of a test  is the best performance of a strategy π∈() against it, namely

    _()=sup{(,π) |π∈()} .


The extremal points in the convex set  (Γ) are commonly referred to as ergodic IRSs.
    As we take the supremum of a linear functional over a convex compact set, the optimum is obtained on an extremal point, which explains the phrase ergodic value. 




 §.§ Finitely described invariant random subgroups
 
Let X be a finite set, and let (X) be the symmetric group acting on X. Every map σ S→(X) extends uniquely to an action of =ℱ(S) on X. 
For a vertex x∈ X, we can define its stabilizer to be 
    Stab(σ,x)={ w∈ℱ|σ(w).x=x} ,

which is a subgroup of ℱ.
Hence, we can associate with σ S→(X) an IRS of  via the following sampling procedure: 
(1) Choose x∈ X uniformly at random. (2)
  Output Stab(σ,x).
We denote by Φ the map from finite actions of  to () defined by this procedure, namely

    Φ(σ)=_x∈ X[ 1_ Stab(σ,x)] ,

where 1_H is the Dirac measure concentrated on the subgroup H and _x∈ X is the expectation according to the uniform measure over X. The set of finitely described invariant random subgroups _() is the image of the map Φ in (). 

The sofic value of a test  is the (asymptotic) best performance of a strategy π∈_() against it, namely

    _()=sup {(,Φ(σ)) | |X|<∞, σ S→(X)} .




The name sofic value is appropriate, as demonstrated in Section <ref>. Furthermore, we often abuse notation and use (,σ) instead of (,Φ(σ)), even though the action itself is not an IRS.



Since _()⊆(), we have _()≤_() for every test . The Aldous–Lyons conjecture <cit.> can be formulated as follows:
  Is  IRS_(ℱ)
 weak* dense in IRS(ℱ)?


In their paper <cit.>, Aldous and Lyons ask this as a question. In the passing of time, the positive form of this problem, namely that _() is dense in (), received the name The Aldous–Lyons Conjecture (cf. <cit.>). Thus, we use  the more common “Conjecture” phrasing instead of Problem. 

The algebraic formulation we  provided earlier, which can also be found  in Section 6 of <cit.>, may seem different from the above. The question was  whether every invariant random subgroup of  is co-sofic, where co-sofic means an IRS which is the weak* limit of convex combinations of uniform distributions over finite index subgroups. Since the finitely described IRSs are dense in the co-sofic ones, the formulation in Conjecture <ref> is equivalent to this standard (algebraic) formulation.



Lastly, the term co-sofic is closely related to the better known term of a sofic group (cf. Definition <ref>).
A finitely presented group ℱ/N is sofic if and only if the Dirac measure concentrated on N is a co-sofic IRS (cf. Proposition 6.1 in <cit.>).
      Thus, a positive solution to the Aldous–Lyons Conjecture <ref> implies in particular that every group is sofic.[The way we formulate it, a positive solution to the Aldous–Lyons Conjecture only implies that all finitely presented groups are sofic. It is standard to deduce it for all groups from the finitely presented ones <cit.>.  ] 


      The soficity property has been useful in obtaining positive results about group rings <cit.>, invariant couplings of random fields <cit.>, and invariants of groups actions <cit.>. See <cit.> for more background on sofic groups.

    




 
Since (,·) is a continuous functional, a positive solution to the Aldous–Lyons Conjecture <ref> implies that  for every test ,  _()=_().



    Due to the relation between subgroups of the free group and Schreier graphs, one can view subgroup tests as a new property testing model for (edge-labeled) graphs:
Let X be a finite set and σ S→ X  a transitive action (the non-transitive case is a convex combination of transitive ones). For x∈ X, the stabilizer Stab(σ,x) is exactly the set of labeled paths in the Schreier graph of (S) with respect to Stab(σ,x) and S that begin and end at x, namely, closed paths originating from x. Thus, when the   finitely described IRS Φ(σ) is put against a test , we actually test the aforementioned Schreier graph  in the following way:
First, choose a vertex x∈ X uniformly at random and sample i∼μ.
  List the paths from K_i and for each one check whether it is closed or open.  According to this check,  decide using D_i whether to accept or reject.





 §.§ Decidability of approximating the sofic value
 

    We briefly recall basic notions from computability and complexity theory.
Let Σ be a finite set that will play the role of an alphabet.
Let Σ^* be the free monoid generated by Σ, namely
the set of all finite words in Σ. 
A language is a subset L⊆Σ^*. Every language
L defines a decision problem: given w∈Σ^*, decide
whether w∈ L.
A promise language is a pair L_yes,L_no⊆Σ^*
such that L_yes∩ L_no=∅. A promise language (L_yes,L_no)
defines a promise decision problem: given w∈ L_yes∪ L_no,
decide whether w∈ L_yes. A promise language (L_yes,L_no) is decidable if there exists a Turing Machine (TM for short) M satisfying the following: If w∈ L_yes∪ L_no is given to M as input, then M halts and outputs a single bit M(w) satisfying

    M(w)=1     w∈ L_yes,
    
        M(w)=0     w∈ L_no.

Let L^1 and L^2 be two (promise) languages.  We say that L^1 reduces to L^2 if there exists a time Turing machine M for which: If σ∈ L^1_yes (respectively,  σ∈ L^1_no) is taken as the input of M, then M halts and the output M(σ) is in L^2_yes (respectively, M halts and the output  M(σ) in L^2_no).
Thus, if L^2 is decidable then L^1 is also decidable.







If the  challenge distribution μ in a subgroup test  is rational, then  can be encoded as a finite bit string. There are many (non-equivalent) ways of doing so.  
Assume we fixed some encoding for tests with rational challenge distributions such that, given the encoding, all underlying combinatorial data of the test can be calculated in finite time. 
Namely, the  set of generators S, the indexing set of challenges Q, the collection of challenges {(K_i;D_i)}_i∈ Q and the distribution μ can all be read from the encoding in finite time.
 

For every fixed encoding of tests such that all combinatorial data of the test can be read from it in finite time:

    
  * There is a Turing machine M_1 that takes as an input (the encoding of) a test , and outputs an infinite sequence of non-decreasing numbers (α_t)_t=1^∞⊆ [0,1]  such that lim_t→∞α_t=_().
    
  * There is a Turing machine M_2 that takes as an input (the encoding of) a test , and outputs an infinite sequence of non-increasing numbers (β_t)_t=1^∞⊆ [0,1]  such that lim_t→∞β_t=_().



    Clause (2) of Theorem <ref> should be compared to the Navascués–Pironio–Acín (NPA) hierarchy in the study of non-local games <cit.>. See also the introduction of <cit.>.



    If the Aldous–Lyons Conjecture <ref> has a positive solution, then for every θ>0, there is a Turing machine which accepts a subgroup test  as input and outputs its sofic value _() up to an additive error of at most θ.






























 §.§ Tailored non-local games and their associated subgroup tests





As described before, the standard definition of  a (non-local) game consists of two finite sets, a questions set X and an answers set A, together with a probability distribution μ over X× X and a decision function D X× X× A× A→{0,1}. In this paper, we use a slightly modified definition, that makes the connection with subgroup tests clearer. Before diving into our alternative definition, note first that the distribution μ defines a graph structure on X, by associating edges with its support. Furthermore, as the exact choice of A is irrelevant, we can assume it is the set of all bit strings up to some length Λ.

A (synchronous, non-local) game consists of a graph  G=(V,E), a distribution μ over E,
a length function ℓ V→ℕ,  formal sets of generators S_x={^x,i| 1≤ i≤ℓ(x)} — the union of which is denoted by S=⋃_x∈ VS_x — and decision functions D_xy{0,1}^S_xy→{0,1} for every edge xy∈ E, where S_xy=S_x∪ S_y.[So, the main difference in our definition, is that the decision function D rejects automatically answers that are not of the appropriate length according to ℓ.]
     A (synchronous, quantum) strategy for  is a map ρ⋃_x∈ VS_x→ U(n), where the images are involutions, and ρ(S_x) commute for every fixed x∈ V.[This is again not the standard definition of a quantum strategy. Unpacking our definition, ρ associates an observable (Hermitian operator on a finite-dimensional Hilbert space that squares to the identity) with each bit of the answer when asked for x∈ V. Since the answer must be totally measured, all the observables associated with a specific vertex must commute. The fact there is a single  map ρ, and not one for each prover, is the synchronicity of this strategy. ]
Such a strategy defines for every xy∈ E a probability distribution over maps γ S_xy→{0,1} in the following way: 
For every ∈ S, let  ^_0 be the projection on the (+1)-eigenspace of ρ(), and ^_1  the projection on its (-1)-eigenspace. 
Furthermore, let τ be the dimension normalized trace on n× n matrices.  
Then, the probability that γ S_xy→{0,1} is sampled according to ρ is by definition 

    τ(∏_∈ S_x^_γ()∏_∈ S_y^_γ()) .

A map γ sampled that way is said to be sampled according to ρ, and we denote it by γ∼ρ.[Note that this sampling depends on the chosen edge xy∈ E. Namely, though implicit in the notation, γ is defined from S_xy for some edge xy∈ E. It is important to note that γ is not globally defined on all of S.]
    Relating to standard notations, we remark that by letting a=γ|_S_x and b=γ|_S_y, the correlation p(a,b|x,y) induced by the quantum strategy  ρ (as in Remark <ref>) agrees with   formula (<ref>).

    In a similar manner to subgroup tests, one can run the game  against the strategy ρ: 
    
    First, the verifier samples an edge xy∈ E according to μ. Then, the prover samples γ S_xy→{0,1} according to ρ. Finally, the verifier accepts if D_xy(γ)=1, and rejects otherwise.
   
   
   
   
   
 
  As for subgroup tests, the  value of the strategy ρ against the non-local game  is its acceptance probability when ran against the game. Namely,

    (,ρ)=_xy∼μ_γ∼ρ[D_xy(γ)] .

 The (synchronous) quantum value of , ^*(), is the supremum of (,ρ) over all possible quantum strategies against .


The main result in <cit.> (recalled in detail in Theorem <ref> herein) is a reduction from the Halting Problem to approximating  the quantum value of a game.
Our main contribution in the second half of this paper (Sections <ref>,<ref> and <ref>) is a mapping from a specific sub-class of non-local games — which we term tailored non-local games

and describe further below — to subgroup tests, that  (almost) preserves their values.


    There is a transformation that takes as input a tailored non-local game (see Definition <ref>) and outputs a subgroup test (which we call the associated synchronous subgroup test, see Definition  <ref>), such that:
    
        
  * If the game had a perfect (i.e., a value 1) Z-aligned, commuting along edges, permutation strategy (see Definitions <ref> and <ref>), then the associated subgroup test has a perfect strategy. 
        
  * Every almost perfect finitely described strategy for the associated subgroup test can be transformed into an almost perfect quantum strategy to the original tailored non-local game. 
    




Informally, a tailored game is one where the decision function D_xy takes the following form. First, D_xy examines the value taken by γ on a subset of the generators S_x∪ S_y (we refer to these as readable variables). Based on this, D_xy determines a collection of linear equations on the entire string γ. Finally, D_xy accepts if and only if all these equations are satisfied. This form for the game can be seen as a generalization of binary linear constraint system games (Example <ref>), which corresponds to the case where the set of readable variables is empty. 

A permutation strategy, as its name suggests, is a quantum strategy which is induced by permutations. Such a strategy is said Z-aligned when the observables associated with the readable variables (i.e. the bits of the answer γ that control the parity checks in the tailored game) are all diagonal matrices (with respect to some natural basis). The restriction to Z-aligned strategies is crucial for item (1) in Theorem <ref> above to hold. 

In a companion paper <cit.>,  we prove that the reduction of the Halting Problem to approximating the quantum value of a game presented in <cit.> can be modified such that the resulting non-local game is tailored and satisfies the required stronger assumptions (Theorem <ref>): If the input Turing machine halts, then the game has a perfect Z-aligned permutation strategy that commutes along edges, and if the input Turing machine does not halt, then the value of the game is bounded from above by 1/2. Therefore, together with our mapping from non-local games to subgroup tests, we deduce that approximating the sofic value of a game is undecidable. In turn, by Corollary <ref>, 
  the Aldous–Lyons Conjecture <ref> has a negative solution.


    In <cit.>, the authors were able to  specify a non-local game whose quantum value is bounded above by 1/2 but has a perfect quantum commuting strategy. Using the results of <cit.>, which we elaborate on further in the next remark,  this implies the existence of a finitely presented *-algebra which can be endowed with a tracial state and cannot be embedded into an ultrapower of the hyperfinite II_1 factor. This is a somewhat constructive counter example, but the state itself is only proven to exist and does not have a constructive form (at this point). By the various formulations of Connes' embedding problem, this proves that there is a non-hyperlinear character of the free group, which is the analogue of a non co-sofic IRS of the free group, but it is provided in a non-constructive form.

By applying the “same” construction as in <cit.> with the algorithm constructed in Theorem <ref> replacing the NPA hierarchy algorithm,  one can pin down an explicit subgroup test with perfect ergodic strategy but all sofic strategies bounded above by some constant (which can be approximated by following the proofs). This is the most constructive we can get in this case, namely, the specific non co-sofic IRS which is the perfect strategy for this subgroup test is proven only to exist, and is not constructed. Thus, we decided not to specify this test. 
    
    It would be very interesting to find an explicit non co-sofic IRS, namely separate between () and _() in a constructive manner.


  We comment on a natural attempt to extend the construction of <cit.> so as to directly deduce the existence of a non-sofic group, a result which would imply a negative answer to the Aldous–Lyons Conjecture. 

The idea (which is folklore, and explicitly mentioned in <cit.>) is as follows. It is well-known that to every synchronous game 𝒢 one can associate an algebra 𝒜(𝒢) such that there is a tight connection between the C^*-representation theory of this algebra and perfect  values for the game in various correlation sets <cit.>. In the special case where 𝒢 is a linear constraint system (LCS) game, there is a group  Γ such that 𝒜()  is (isomorphic to the completion of) the complex group algebra ℂ[Γ] <cit.>.[Actually, it is isomorphic to the completion of  ℂ[Γ]/⟨+1⟩, where  is a special generator in Γ and ⟨+1⟩ is the two-sided ideal generated by +1 in ℂ[Γ].] 

Now, if the reduction given in <cit.> returned only LCS games, then one would obtain the following result: there exists an LCS game with a perfect quantum commuting strategy, but no (asymptotically) perfect quantum strategy. By the connection mentioned above this implies that the group Γ associated with 𝒜()  is non-hyperlinear. As every sofic group is hyperlinear, one would thus obtain an example of a non-sofic group.

Since the reduction in <cit.> does not return LCS games, a workaround could be to find an embedding from the algebra 𝒜(𝒢), where 𝒢 is returned by <cit.>, into 𝒜(𝒢') where 𝒢' is an LCS game. Obstacles to this “black-box” approach are discussed in <cit.>. 

One can view our approach as finding a “sweet spot” that is in some sense half-way between synchronous games, which were used in <cit.>, and LCS games. This is because tailored games are a natural generalization of LCS games. In the companion paper we show that <cit.> can be adapted to return tailored games, and in the present paper we show that this suffices to resolve the Aldous–Lyons Conjecture. The existence of non-sofic groups (and non-hyperlinear groups) remains an  elusive open problem. 


  

    
Recall that a (synchronous 2-player) non-local game  consists of a (finite) graph G=(V,E), a length function ℓ V→ℕ, formal sets of generators S_x={^x,i| 1≤ i≤ℓ(x)} for every vertex x∈ V, a distribution μ over the edges E, and decision functions D_xy{0,1}^S_xy→{0,1} for every edge xy∈ E, where S_xy=S_x∪ S_y. Namely, the decision function D_xy takes as input a pair of functions α S_x→{0,1} and β S_y→{0,1} and outputs a single bit. A (synchronous quantum) strategy for  is a map ρ⋃_x∈ VS_x→ U(n), where the images are of order 2, and ρ(S_x) commute for every fixed x∈ V.   Since ρ() is of order 2, we denote by ^_0 the projection on the (+1)-eigenspace of ρ(), while ^_1 the projection on its (-1)-eigenspace. Then, the  value of  ρ against  is

    (,ρ)=_xy∼μ∑_α S_x→_2 
    β S_y→_2 D_xy(α,β)τ(∏_∈ S_x∏_∈ S_y^_α()^_β()),

where τ is the dimension normalized trace on n× n matrices. The (synchronous) quantum value of , ^*(), is the supremum over all possible strategies against . The main result of MIP*=RE <cit.> is a reduction from the halting problem to approximating  the quantum value of a game. Given a Turing machine M, they construct a game _M such that: If M halts, then _M has a strategy ρ with (_M,ρ)=1 (and in particular, ^*(_M)=1).
    And if M never halts, then ^*(_M)≤1/2.
We now define a family of tests that will allow us to leverage this result, up to some caveats, to our needs.



A synchronous test  consists of a (finite) graph G=(V,E), a length function ℓ V→ℕ, a distribution μ over the edges E, and a challenge (K_xy;D_xy) associated to every edge xy∈ E. While keeping the S_x notation from before, we assume K_xy consists of only words in S_xy∪, where  is a global generator which plays the role of negation. Regardless of the specific test, the challenges always include the following checks:

    
  * All generators are order 2, namely ^2∈ H and
    
    ∀∈ S_xy   ^2∈ H.

    
  * The special generator  is non-trivial and central, namely ∉ H and 
    
    ∀∈ S_xy   [,]∈ H,

    where [·,·] is the (multiplicative) commutator.

The similarities between the two definitions are clear. One would hope that there is a straightforward way of transforming non-local games into synchronous tests, such that the values are preserved, or can  at least be bounded by one another in an appropriate manner. If this is possible, then one can take the transformation from Turing machines M to games _M a la MIP*=RE,  transform _M to the associated synchronous test (_M), and obtain a reduction from the halting problem to SOFVAL_c,s. While this is not true in general, by restricting the types of non-local games we focus on, we are able to do just that. The terms tailored form, perfect Z-aligned permutation strategy and augmented synchronoys test will be explained in Section <ref>.

There is a map from non-local games  in tailored form to (augmented) synchronous tests (()) such that:

    
  * If  has a perfect Z-aligned permutation strategy, then the associated  test (()) has a perfect strategy.
    
  * If ^*()≤1/2, then _((()))≤ 1-δ, for some universal constant δ>0.
 


    If the reduction from Turing machines to non-local games in MIP*=RE can be done such that all games are in tailored form, and in the completeness case the outcoming game has a perfect Z-aligned permutation strategy, then SOFVAL_1,1-δ is undecidable, and thus Conjecture <ref> is false.




 §.§ Structure of the paper

 In Section <ref>, we prove  structural results about invariant random subgroups, some of which are standard and well known to experts. In Section <ref>, we  prove our first main theorem, 
 Theorem <ref>,  and draw corollaries from it. 
 In Section <ref>, we provide  several examples of subgroup tests,  inspired  by  both group theory and complexity theory.  In Section <ref>, we discuss a rigidity property of tests that generalizes group stability and which is analogous to robustness of non-local games. 
In Section <ref>, we define tailored non-local games and analyze their properties. 
In Section <ref>,  we describe the (almost) value preserving correspondence between tailored non-local games and synchronous subgroup tests, informally outlined in Theorem <ref>. This allows us in Corollary <ref> to provide a negative solution to the Aldous–Lyons Conjecture <ref>. The proof of the aforementioned value preserving correspondence, Theorem <ref>, is proved in Section <ref>. Theorem <ref>, which is the other main ingredient in our refutation, is proved in  a companion paper <cit.>. There is no open problems section, but  further research directions can be found in Remarks  <ref>, <ref> and <ref>.

For a reader who seeks the main ideas of the paper without diving too deep into the proofs, we suggest reading, in addition to the introduction, only Sections <ref> and <ref>.



 §.§ Acknowledgements



We thank Alon Dogon for reading an early draft of this paper and providing many useful improvements to the presentation. We would also like to thank Mikael de la Salle for various discussions along the way. Finally, we want to thank Peter Sarnak for organizing a seminar on “strong convergence” in Princeton and asking the second author to give a talk, as it organized his thoughts  and led to the current presentation of ideas.

Lewis Bowen is supported by NSF grant DMS-2154680.
Michael Chapman acknowledges with gratitude the Simons Society of Fellows and is supported by a grant from the Simons Foundation (N. 965535).
Alex Lubotzky is supported by the European Research Council (ERC)
under the European Union's Horizon 2020 (N. 882751), and by a research grant from the Center for New Scientists at the Weizmann Institute of Science.
Thomas Vidick is supported by a research grant from the Center for New Scientists at the
Weizmann Institute of Science, a Simons Investigator award, AFOSR Grant No. FA9550-22-1-0391, and
ERC Consolidator Grant VerNisQDevS (101086733).
 


§ GROUP THEORETIC PRELIMINARIES




 §.§ Topology (and other properties) of invariant random subgroups
[The content of this section is standard. For example, it appears in Section 3 of <cit.>.]
We first go through some properties of the space {0,1}^ (induced with the product topology). For every  K,L⊆ℱ, let 

    𝒞(K,L)={A⊆ℱ| K⊆ A, L∩ A=∅}⊆{0,1}^.
 
Then,

    
  * When K and L are finite, the set (K,L) is open and closed. Moreover, the collection 
    {(K,L)| K,L⊆, |K|,|L|<∞}
 is a basis for the topology of {0,1}^. In addition, {0,1}^ is compact.
    
  * For every w∈ℱ and subsets K,L⊆ℱ we have
    
    w.𝒞(K,L)=𝒞(wKw^-1,wLw^-1).


The following is a standard fact. A proof is added because elements of it are later used in Lemma <ref>.
 
    The set of subgroups () is closed in the collection of all subsets {0,1}^. 



    A subset A⊆ is a subgroup if it satisfies the following three conditions: 
  
    (1) _∈ A ,  
           (2)  w∈ A w^-1∈ A ,  
           (3) w,w'∈ A w· w'∈ A.

For every w,w'∈, define the following open sets:  The subsets not containing _, C_=(∅,{_}). The subsets containing w∈ but not w^-1, C_w=({w},{w^-1}). The subsets containing w,w'∈ but not w· w',
    C_w,w'=({w,w'},{w· w'}).
     Then,
     
    (⋃_w,w'∈  C_w,w')∪(⋃_w∈ C_w ) ∪ C_={0,1}^∖(),
 
     which proves that () is a closed set.






Recall the definition of (), the invariant random subgroups of the free group  (Definition <ref>).


 
Let μ,ν∈ Prob({0,1}^ℱ). Then,

    
    
  * μ=ν if and only if μ(𝒞(K,L))=ν(𝒞(K,L)) for every finite K,L⊆ℱ. 
    
  * μ is conjugation invariant if and only if for every s∈ S we have s.μ=μ. Hence, μ∈() if and only if for all s∈ S and finite K,L⊆ℱ we have μ(𝒞(K,L))=μ(s.𝒞(K,L)).



A sequence of measures μ_n∈ Prob({0,1}^ℱ) is said to weak* converge to μ if for all continuous f{0,1}^ℱ→ℝ, 

    _H∼μ_n[f(H)] _H∼μ[f(H)].

We denote weak convergence by μ_n μ. 

    In our context, there is a more “down to earth” way of viewing weak* convergence. For every finite subset A⊆, and every  probability measure μ∈({0,1}^), we can define a probability measure μ∩ A∈({0,1}^A)⊆^2^A by 

    ∀ B⊆ A   μ∩ A(B)=μ({H⊆| H∩ A=B}).

Then,

    μ_nμ∀ A⊆, |A|<∞   μ_n∩ Aμ∩ A,

where the convergence on the right is the standard one in the Euclidean space ^2^A.

 
The space ({0,1}^ℱ) is compact in the weak* topology.



Since {0,1}^ℱ is compact, the fact can be deduced from the Banach–Alaoglu Theorem (cf. Theorem 3.15 in <cit.>).





 §.§ Pseudo subgroups

For every B⊆ C⊆, the restriction map R_B⊆ C{0,1}^C→{0,1}^B is 
    ∀ A⊆ C   R_B⊆ C(A)=A∩ B.
 When C= we remove it from the notation and write R_B instead of R_B⊆.
For sets K,L⊆ B⊆ we can define

    _B(K,L)={A⊆ B| K⊆ A, A∩ L=∅}.

Note that _(K,L)=(K,L).
 
    Let K,L⊆ B⊆ C⊆. Then R_B⊆ C(_C(K,L))=_B(K,L) and R_B⊆ C^-1(_B(K,L))=_C(K,L).


    Let A⊆ B and A'⊆ C such that A=A'∩ B. As K,L⊆ B, the intersection of A' with K and L depends only on A'∩ B=A. Hence  
    
    A∈_B(K,L)  A'∈_C(K,L).

  Adding this observation to the fact R_B⊆ C is onto proves the desired claim. 


Now, we can pushforward measures along the inverse system of restrictions. Namely, given  B⊆ C⊆ and a probability measure π on {0,1}^C we can define the probability measure R_B⊆ C*π on {0,1}^B by

    ∀ Y ⊆_ Borel{0,1}^B   R_B⊆ C*π(Y)=π(R_B⊆ C^-1(Y)).


 
   Claim <ref> implies that R_B⊆ C is continuous. Therefore,  R_B⊆ C*({0,1}^C)→({0,1}^B) is continuous.





    


This proof is very standard. As oppose to the other basic claims we attach proofs to, this one doesn't seem to contribute to the understanding of the main theorems. So maybe it should be removed. Waiting for your thoughts on it.
 Let f{0,1}^B→ be a continuous function. Define f{0,1}^C→ be the pullback of f along R_B⊆ C, namely 
    
    ∀ A⊆ C   f(A)=f∘ R_B⊆ C(A)=f(A∩ B).

    The pullback is a composition of continuous functions and hence continuous. For every μ∈({0,1}^C),
    
    _H∼μ[f(H)]=_H∼μ[f∘ R_B⊆ C(H)]=_H∼ R_B⊆ C*μ[f(H)].

       So, if  μ_n μ  in ({0,1}^C), then 
       
    _H∼ R_B⊆ C*μ_n[f(H)]=_H∼μ_n[f(H)]_H∼μ[f(H)]=_H∼ R_B⊆ C*μ[f(H)].

       Since this was true for any continuous f, we have that R_B⊆ C*μ_nR_B⊆ C*μ and we are done.






   Let B⊆. A subset  A⊆ B is said to be a pseudo subgroup of B if there is a subgroup H≤ such that R_B(H)=H∩ B=A. Denote by P(B) the collection of pseudo subgroups of B.[Note that this collection is a closed subset of the subsets of B.]

 
    Let A⊆ B⊆. Then, A is a pseudo subgroup of B if and only if ⟨ A⟩∩ B = A, where ⟨ A⟩ is the subgroup of  generated by A. 




    If there is a subgroup H for which  A=R_B(H)=H∩ B⊆ H, then in particular ⟨ A⟩⊆ H. Because A is always contained in ⟨ A⟩∩ B, we can conclude.



    Let A⊆ B⊆ C⊆. If A is not a pseudo subgroup of B, and A'⊆ C restricts to A (i.e., A'∩ B=A), then A' is not a pseudo subgroup of C.


    By Claim <ref>, if A is not a pseudo subgroup of B, then there is a w∈ B such that w∉ A while w∈⟨ A⟩.  Since w∈ B⊆ C, and A'∩ B=A, the same w satisfies that w∉ A' while w∈⟨ A⟩⊆⟨ A'⟩, proving that A' is not a pseudo subgroup of C.





Let B⊆ be a subset.



  * The set 𝒫_B is the collection of all π∈({0,1}^B)  that are concentrated on pseudo subgroups of B, namely π(P(B))=1. Equivalently, π({0,1}^B∖ P(B))=0.

  * The set 𝒬_B is the subset of _B of distributions that are S-invariant when defined. Namely, assuming  K, L ⊆ B are finite subsets such that 
    ∀ s∈ S∪ S^-1    sKs^-1, sLs^-1⊆ B,
 we have π( _B(K,L)) = π(_B( sKs^-1, sLs^-1)).  We call  probability distributions in Q_B pseudo invariant random subgroups over B, or pseudo IRSs in short.

  * Let _B=R_B*^-1(_B) and _B=R_B*^-1(_B).




    Note that _=_=(()) and _=_=().



Let B⊆ C⊆ be subsets.
    
       
        
  * The restriction to B of a distribution over pseudo subgroups of C, is a distribution over pseudo subgroups of B. Namely,
        
    ∀π∈_C   R_B⊆ C*π∈_B.
 
        Similarly, the restriction to B of a pseudo IRS of C is a pseudo IRS of B. Namely, 
        
    ∀π∈_C   R_B⊆ C*π∈_B.

        
  * The (Borel) probability distributions over subgroups of  are the intersection of pull-backs of all distributions over pseudo subgroups in finite subsets. Similarly, the IRSs of  are the intersection of pull-backs of all pseudo IRSs of finite subsets. Namely,
        
    ⋂_B⊆, |B|<∞_B=(()), ⋂_B⊆, |B|<∞_B=().

    




        
      Let π∈_C.  By definition, 
    π({0,1}^C∖ P(C))=0.
 By Corollary <ref>, 
    R_B⊆ C^-1({0,1}^B∖ P(B))⊆{0,1}^C∖ P(C).
 Hence, by the definition of the pushforward, 
      
    R_B⊆ C*π({0,1}^B∖ P(B))   =π(R_B⊆ C^-1({0,1}^B∖ P(B)))
       ≤π({0,1}^C∖ P(C))
       =0,

    and  so R_B⊆ C*π∈_B.  

        Assume further that π∈_C. By Claim <ref>, for every finite K,L⊆ B, 
    R^-1_B⊆ C(_B(K,L))=_C(K,L).
 
        Thus  
        
    R_B⊆ C*π(_B(K,L))   =π(R_B⊆ C^-1(_B(K,L)))
    =π(_C(K,L)).

        If K,L⊆ B such that sKs^-1,sLs^-1⊆ B for every s∈ S∪ S^-1, then the same applies to C since it contains B, and hence
        
    R_B⊆ C*π(_B(K,L))   =π(_C(K,L))
       =π(_C(sKs^-1,sLs^-1))
       =R_B⊆ C*π(_B(sKs^-1,sLs^-1)).

        Therefore R_B⊆ C*π is in _B and (1) is proved.
 

 
  By clause (1), for every B⊆ C⊆ we have _C⊆_B and _C⊆_B. In particular, when C= we have (())⊆_B and ()⊆_B. Hence  
    (())⊆⋂_B⊆, |B|<∞_B,  ()⊆⋂_B⊆, |B|<∞_B.

     We are left to prove the reverse containments. Recall the open cover of the complement of the subgroups described in the proof of Claim <ref>.
     Given π∈({0,1}^) such that π∉(()), we have 
    π((⋃_w,w'∈  C_w,w')∪(⋃_w∈ C_w ) ∪  C_)=π ({0,1}^∖())>0.

    Hence, by the union bound (Boole's inequality), there is either a pair w,w'∈ such that π(C_w,w')>0, or a single element w∈ such that π(C_w)>0, or π(C_)>0. Thus, if B={_,w,w',w· w',w^-1},  then π∉_B. In particular π∉⋂_B, proving the first reverse containment.
     
     Lastly, let π∈⋂_B.  For every finite K,L⊆, there is a larger enough finite set B such that K,L⊆ B and also sKs^-1,sLs^-1⊆ B for every s∈ S∪ S^-1. Since π∈_B,  by Claim <ref> applied to C=, we have for every s∈ S∪ S^-1 that
    
    π((K,L))   =π(R_B^-1(_B(K,L)))
       =R_B*π(_B(K,L))
       =R_B*π(_B(sKs^-1,sLs^-1))
       =π(R_B^-1(_B(sKs^-1,sLs^-1)))
       =π((sKs^-1,sLs^-1)).

    Hence, by Fact <ref>, (2) is deduced.



    

The following claim is never used in the text. It is used as a reference in the introduction, but it seems we do not need it. Should we remove it?

The Aldous–Lyons Conjecture <ref> is true if and only if for every finite subset B⊆ the set  R_B*(_())  is dense in  R_B*(()).


    Let π∈(), and let (B_t)_t=1^∞ be a sequence of subsets of  such that B_t⊆ B_t+1 and ⋃ B_t=. Since  R_B_t*(_())  is dense in  R_B_t*(()) (as subsets of the finite dimensional vector space ℝ^B_t equipped with some norm ‖·‖), we can find π_t∈_() such that ||R_B_t*π-R_B_t*π_t||≤1/t. By Claim <ref>, ({0,1}^) is compact. 
    Moreover, _B is a closed set when B is finite[This is true for any B⊆, but it is straightforward for finite ones, and we only use it in this case.] and R_B⊆ C* is continuous (Corollary <ref>), so by item (2) of Lemma <ref> () is the intersection of closed sets in the weak* topology, 
 and thus it is closed. 
 Hence, π_t has a subsequence with weak* limit π_∞∈() (without loss of generality, the subsequence is π_t itself). For every s≥ t we have 
    
    ||R_B_t*π-R_B_t*π_∞||≤ ||R_B_t*π-R_B_t*π_s||+||R_B_t*π_s-R_B_t*π_∞||.

    Since R_B_t* is continuous, and since π_sπ_∞, there is a large enough s for which 
    
    ||R_B_t*π_s-R_B_t*π_∞||≤1/s.

    We did not choose a specific norm up until now. By the triangle inequality 
    
    ∀σ,σ'∈({0,1}^B_s)   ||R_B_t⊆ B_s*σ-R_B_t⊆ B_s*σ'||_1≤ ||σ-σ'||_1.

    Hence by choosing ||·||=||·||_1, we have 
    
    ||R_B_t*π-R_B_t*π_s||≤ ||R_B_s*π-R_B_s*π_s||≤1/s.

   Therefore for every s≥ t,
    
    ||R_B_t*π-R_B_t*π_∞||≤2/s.

    But the left hand side is independent of s. Thus R_B_t*π=R_B_t*π_∞ for every natural number t. Let K,L⊆ be finite subsets. Since B_t⊆ B_t+1 and ⋃ B_t=, there is a large enough t for which K,L⊆ B_t. By Claim <ref>, 
    
    R_B_t*π(_B_t(K,L))   =π(R_B_t*^-1(_B_t(K,L)))=π((K,L))
    
         R_B_t*π_∞(_B_t(K,L))   =π_∞(R_B_t*^-1(_B_t(K,L)))=π_∞((K,L))

    which implies π((K,L))=π_∞((K,L)).  By Fact <ref>, we deduce π=π_∞. Therefore π_tπ and _() is weak* dense in (), proving Conjecture <ref>. 

    The reverse implication is immediate from the fact the pushforward along restrictions maps R_B*({0,1}^)→({0,1}^B)  are continuous, as was deduced in Corollary <ref>.


 
    Conjecture <ref> is false if and only if there is a finite set K⊆ and a functional φℝ^{0,1}^K→ℝ, such that 
    
    sup{φ(π)|π∈ R_K(_())}<sup{φ(π)|π∈ R_K(())}.



This formulation of Conjecture <ref> should motivate our approach of studying subgroup tests and their values, which are exactly linear functionals of the above format.




The system of finite subsets of   forms a directed system induced by inclusion. The pushforwards along  restriction maps R_B⊆ C* thus form an inverse system of maps between the compact sets (B) for B⊆, |B|<∞. In this language, Lemma <ref> says that 

    (())=_B_B    and    ()=_B _B.





 §.§ Computational aspects of pseudo invariant random subgroups


 
Given A⊆ B⊆ where |B|<∞, there is an algorithm deciding whether A is a pseudo subgroup of B. 





Given a finite set of elements A={ w_1,...,w_k }⊆, and another element w∈, there is an algorithm to decide whether w∈⟨ A⟩. Indeed, one can use the following:


  * Construct Core(A) the core graph of ⟨ A⟩  via the method of Stallings' foldings  <cit.>.

  * Check whether the path that begins at the basepoint of Core(A) and traverses w is closed.

  * If the path is closed, then w∈⟨ A⟩. Otherwise w∉⟨ A⟩.


By Claim <ref>, A is not a pseudo subgroup of B if and only if there exists a w∈ B such that w∉ A but w∈⟨ A ⟩. Hence, a sufficient and necessary condition for A to be a pseudo subgroup of B is to be accepted by the following algorithm:

    
  * Construct  Core(A).
    
  * For every w∈ B satisfying w∉ A, reject if w is a closed path in Core(A).
    
  * Accept if A passed (2) for every w∈ B.




    Recall that a free group is LERF (locally extended residually finite), i.e., for every finite set A={w_1,...,w_k} and every w∈ such that w∉⟨ A⟩, there exists a finite index subgroup H for which 

    A⊆ H and w∉ H.

This was originally proved by M. Hall <cit.>, and can be proven using the method of Stallings' foldings <cit.>. It follows that for  every A a pseudo subgroup of a finite B, there exists a finite index subgroup H for which R_B(H)=A.



    Let B⊆ be a finite set. Then, the set _B of distributions over pseudo subgroups of B and the set _B of pseudo IRSs over B are computable polytopes in ℝ^{0,1}^B. Namely, they are defined by (computable) integral linear equations and inequalities.


         Recall that _B⊆_B⊆({0,1}^B)⊆^{0,1}^B. Namely, all these distributions  are vectors which are indexed by subsets of B.     
     We describe the defining system of equations and inequalities forcing a vector π∈ℝ^{0,1}^B to belong to _B. First, it needs to be a probability measure, hence  
        
    ∀ A⊆ B   π(A)   ≥ 0,
    ∑_A⊆ Bπ(A)   =1.

        Further, for every A⊆ B, we can run the algorithm in Claim <ref> to decide whether A is a pseudo subgroup, and if not to add the constraint π(A)=0. This is the defining system of _B. To get the defining system of _B, we need to add the following:
        For every K,L⊆ B, we can check whether for all s∈ S∪ S^-1, the sets sKs^-1 and sLs^-1 are also contained in B, and if so add the constraints
        
    ∀ s∈ S∪ S^-1   ∑_A∈_B(K,L)π(A)_π(_B(K,L))=∑_A∈_B(sKs^-1,sLs^-1)π(A)_π(_B(sKs^-1,sLs^-1)).

        








 








§ MAIN THEOREM I: APPROXIMATING THE VALUES OF SUBGROUP TESTS
 

Recall the definitions of challenges, tests and values from Section <ref>.
Let  be a test over the generating set S, with challenges (K_i;D_i)_i∈ Q and rational distribution μ over Q. Let K=⋃_i∈ Q K_i, which is a finite subset of (S)=. We can extend the definition of (,·) in the following way: For every  B satisfying K⊆ B⊆, and for every distribution π∈({0,1}^B),  let 

    (,π)=_A∼π_i∼μ[D_i(A∩ K_i)].

The next claim is formalizing the following intuitive idea: Since D_i depends only on the restriction of H to K_i, then replacing π by its push-forward along any restriction R_B will not change its value against .


Under the extended definition of the value, for every B and C such that K⊆ B⊆ C⊆, and for every π∈({0,1}^C), we have

    (,π)=(,R_B⊆ C*π).




    By definition, sampling A∼ R_B⊆ C*π is the same as sampling A'∼π and outputting A=A'∩ B.
    Also, K_i=B∩ K_i since K_i⊆ K⊆ B. Hence,
    
    (,R_B⊆ C*π)   =_A∼ R_B⊆ C*π_i∼μ[D_i(A∩ K_i)]
       =_A'∼π_i∼μ[D_i((A'∩ B)∩ K_i)]
       =_A'∼π_i∼μ[D_i(A'∩ K_i)]
       =(,π).



Recall the first clause of Theorem <ref>: There is a Turing machine M_1 that takes as an input the encoding of a test , and outputs an infinite sequence of non-decreasing numbers (α_t)_t=1^∞⊆ [0,1]  such that lim_t→∞α_t=_().




 
Let σ S→(X) be a finite action. The value of Φ(σ) against  is 
 
    (,Φ(σ))   =_H∼Φ(σ)_i∼μD_i(H∩ K_i)
       =∑_x∈ X∑_i∈ Qμ(i)/|X|· D_i((σ,x)∩ K_i)

 which is a rational combination of  evaluations of D_i. Since the sets K_i, functions D_i and coefficients μ(i) can be calculated in finite time from  the encoding of , and (σ,x)∩ K_i can be calculated from σ for every x∈ X, the value of Φ(σ) against  can be computed in finite time. 

  There is a computable countable enumeration of all (equivalence classes of) finite (S)-actions. To see that, note that a map σ S→(n) is defined by the tuple (n ; {σ(s)}_s∈ S). Since the elements of (n) can be ordered (e.g., lexicographically), as well as the elements of S, the infinite set of tuples (n ; {σ(s)}_s∈ S) inherits a countable enumeration. Denote by σ_t the t^ th action according to this enumeration.
 For t∈ℕ, let 
 
    α_t=max_s≤ t{(,Φ(σ_s))}.

 By our discussion so far, the number α_t can be computed in finite time. Moreover, (α_t)_t=1^∞ is a non-decreasing sequence by definition. 
 Since  Im(Φ)=_(), the sequence (α_t)_t=1^∞ tends to the supremum of (,·) over _(), which is, by (<ref>), _().


We now recall the second clause of Theorem <ref>:  There is a Turing machine M_2 that takes as an input the encoding of a test , and outputs an infinite sequence of non-increasing numbers (β_t)_t=1^∞⊆ [0,1]  such that lim_t→∞β_t=_().







 Let K=⋃ K_i as before, and let B⊆ be a finite subset containing K.  Recall the definition of _B the pseudo invariant random subgroups over B (Definition <ref>), and its pre-image _B. Recall also  Claim <ref> and the discussion before it, where we extended the notion of the value to probability distributions over  subsets of  that contain K. The B-approximation to the ergodic value of  is defined to be

    _B() = sup{(,π)|π∈_B}=sup{(,π)|π∈_B}.


Let {B_t}_t=1^∞ be a sequence of finite subsets of , all containing K, such that B_t⊆ B_t+1 and ⋃_t=1^∞ B_t=. 
We now claim the following:

    
  * _B_t() is computable.
    
  * The sequence _B_t() is non-increasing.
    
  * lim_t→∞_B_t()=_().


By Lemma <ref>, _B is defined by a computable integral system of equations and inequalities. Thus, maximizing a linear functional (,·) over it can be done using linear programming in finite time, proving (1).

By Lemma <ref>, whenever B⊆ C⊆ we have R_B⊆ C*(_C)⊆_B. Hence _B_t+1⊆_B_t, which implies _B_t+1()≤_B_t(), proving (2).

Moreover,  ()=_⊆_B_t  which implies _B_t()≥_().
On the other hand, by Claim <ref> the space ({0,1}^) is compact, and since _B are closed sets they are also compact. Hence there is a distribution π_t∈_B_t satisfying _B_t()=(,π_t). By the compactness of ({0,1}^), there is a weak limit to some subsequence of (π_t)_t=1^∞, which we denote by π_∞. Without loss of generality the subsequence is the original sequence. Since π_s∈_B_t for every s≥ t, and since   _B are closed, we deduce that π_∞∈⋂_B_t=(). By the definition of a weak limit, since (,·) is a continuous functional, we have 

    lim_t→∞_B_t()=lim_t→∞(,π_t)=(,lim_w^*π_t)=(,π_∞)≤_(),

proving (3). Therefore, choosing β_t=_B_t() finishes the proof.




§ EXAMPLES OF TESTS
 




 §.§ Relations verification test
  Given a finite subset R⊆(S), the R-verification test 𝒱_R checks whether R is contained in the sampled subgroup H. It contains a single challenge (R;D) with D= 1_R, namely

    ∀ A⊆ R   D(A)=
        1     A=R,
    
        0     A≠ R.


The sofic value of _R is always 1, e.g., using the Dirac measure concentrated on the group  itself (which associates via Φ with the trivial action of  on a singleton). Moreover, if the conjugation orbit of a finite index subgroup H passes _R with probability 1, then R⊆⋂_w∈ wHw^-1. 




Though the value of the R-verification test does not give us much information about R, we will see  in Section <ref> that the potential robustness of this test is an important property of R.
Moreover, this test has variations where you sample r∈ R according to some distribution and check only that r∈ H and not that all of R is in H. See for example CL_part1,CVY_efficient for more on that.




 §.§ Separation test
  
Given two finite subsets R,L⊆(S), the (R,L)-separation test 𝒱_R||L checks whether R is contained in the sampled subgroup H but L∖ R is not. It contains a single challenge (R∪ L;D) with D= 1_R, namely

    ∀ A⊆ R∪ L   D(A)=
        1     A=R,
    
        0     A≠ R.


 
A group Γ is residually finite if for every finite subset L⊆Γ where _Γ∉ L, there is a finite index normal subgroup N⊴Γ such that L∩ N=∅.




    A finitely presented group Γ=⟨ S|R⟩ is residually finite if and only if for every finite L⊆ such that L∩⟨⟨ R⟩⟩=∅[The subgroup ⟨⟨ R⟩⟩ is the smallest normal subgroup containing R.] there is a finitely described IRS passing _R||L with probability 1. 



    Let Γ=(S)/⟨⟨ R⟩⟩ be residually finite, and L⊆ a finite set disjoint from ⟨⟨ R⟩⟩. By Definition <ref>, there is a finite index normal subgroup N⊴Γ such that L⟨⟨ R⟩⟩∩ N=∅. Using the sequence of epimorphisms
    
    ↠Γ↠Γ/N

    we can define the left action of  on X=Γ/N which passes _R||L with probability 1. 
    
    On the other hand, given a finite subset _Γ∉ L⊆Γ, let  L'⊆ be any set of representatives for L. Since _Γ∉ L, we have L'∩⟨⟨ R⟩⟩=∅. 
    Hence, by our assumption,  there is an  action ρ S→(X) with (_R||L',Φ(ρ))=1. Without loss of generality, ρ is transitive (otherwise, its restriction to any orbit will do).  
    The fact (_R||L',Φ(ρ))=1 implies in particular that R⊆(ρ,x) for any x∈ X. Thus, ρ factors through Γ and defines an action ρ'Γ→(X) by ρ'(w⟨⟨ R⟩⟩)=ρ(w). By the fact (_R||L',Φ(ρ))=1, the permutation ρ(w) has no fixed points for every w∈ L', and thus ρ'(u) has no fixed points for every u∈ L. Hence N=⋂_x∈ X(ρ',x) is a normal subgroup of Γ satisfying N∩ L=∅, as required to deduce Γ is residually finite.



    Let X be a finite set. Given two permutations σ,τ∈(X), we define their normalized Hamming distance to be

    d_H(σ,τ):=|{ x∈ X|σ(x)≠τ(x)}|/|X|=ℙ_x∈ X[σ(x)≠τ(x)].





    A finitely presented group Γ=⟨ S|R⟩ is sofic if for every L⊆(S) such that L∩⟨⟨ R⟩⟩=∅, and for every >0, there is a finite set X and an action ρ S→(X) such that 
    
    ∀ r   ∈ R   d_H(ρ(r),_X)≤,
    ∀ w   ∈ L   d_H(ρ(w),_X)≥ 1-.




It is straightforward to see that soficity is a relaxation of residual finiteness. 



    A finitely presented group Γ=⟨ S|R⟩ is sofic if and only if for every finite L⊆ such that L∩⟨⟨ R⟩⟩=∅ we have _(_R||L)=1.



Assume Γ is sofic, and let  L⊆ be a finite set such that L∩⟨⟨ R⟩⟩=∅. For every >0, there is an action ρ_ S→(X_), such that 
  
    ∀ r   ∈ R   d_H(ρ_(r),_X_)≤,
    ∀ w   ∈ L   d_H(ρ_(w),_X_)≥ 1-.

    Hence,
    
    (_R||L,Φ(ρ_))   =_x∈ X_[∀ r∈ R,∀ w∈ L  ρ_(r).x=x, ρ_(w).x≠ x]
       ≥ 1-∑_r∈ R_x∈ X_[ρ_(r).x≠ x]-∑_w∈ L_x∈ X_[ρ_(w).x=x]
       ≥ 1-∑_r∈ Rd_H(ρ_(r),_X_)-∑_w∈ L(1-d_H(ρ_(w),_X_))
       ≥ 1-|R|-|L|.

    Since |R| and |L| are fixed, and >0 is arbitrarily small, we have _(_R||L)=1.

    On the other hand, if for every L we have _(_R||L)=1, then for every >0 there is an action ρ S→(X) with (_R||L,Φ(ρ))≥ 1-. Now, since 
    
    (_R||L,Φ(ρ))   =_x∈ X[∀ r∈ R,∀ w∈ L  ρ(r)x=x, ρ(w)x≠ x]
       ≤min(min_r∈ R(_x∈ X[ρ(r)x=x]), min_w∈ L(_x∈ X[ρ(w)x≠ x]))
       =min(min_r∈ R(1-d_H(ρ(r),_X)), min_w∈ L(d_H(ρ(w),_X))),

    we deduce that
 
    ∀ r   ∈ R   d_H(ρ(r),_X)≤,
    ∀ w   ∈ L   d_H(ρ(w),_X)≥ 1-.

    and Γ is sofic.



 §.§ Formula satisfaction test

Let φ(y_1,...,y_n) be a boolean formula. The test _φ is defined over the formal set of generators S={x_1,...,x_n}. It has a single challenge (K;D), where K=S. Note that the characteristic function of every subset A⊆ S is already in the form 1_A{x_1,...,x_n}→{0,1}, and thus induces an assignment for the formal variables. The decision predicate D accepts A⊆ S if 1_A induces a satisfying assignment for φ, i.e.,

    D(A)=φ( 1_A(x_1),..., 1_A(x_n)).

It is straightforward that (_φ)=1 if and only if φ is satisfiable. 

There is a more interesting version of this test, when the formula φ is a 3CNF. Recall that a 3CNF is an and of clauses of the form φ_t=y_i^_1 y_j^_2 y_k^_3, where y_i,y_j,y_k are taken from a fixed set of formal generators (as before), _1,_2,_3∈{0,1}, and we interpret y^0=y and y^1=¬ y. Now, checking that φ is satisfied by 1_A is the same as going over all the clauses and checking that they are all satisfied. Thus, we can define a randomized version of this check by associating a challenge (K_t;D_t) with each clause φ_t, by letting K_t={x_i,x_j,x_k} and D_t(A)=φ_t( 1_A(x_i), 1_A(x_j), 1_A(x_k)). Then, by choosing a uniform distribution over challenges,  the associated test _φ checks whether a uniformly chosen clause is satisfied by 1_A. The famous PCP theorem <cit.> says that any 3CNF φ can be (efficiently) transformed into a not much larger 3CNF φ', such that if _(_φ)=1, then _(_φ')=1, and if _(_φ)=0, then  _(_φ')≤1/2.
The scaled up version of this argument shows that any  problem (see <cit.>) can be reduced to approximating the sofic value of a test.[This last remark is meaningless as long as we do not specify the exact way we encode tests. The point  is that there is a natural way to encode tests such that  can be reduced to approximating the sofic value immediately from <cit.>. In any case,  this paper is devoted to proving a much stronger conclusion, which is that approximating the sofic value is as hard as the Halting Problem.]




    





§ ROBUSTNESS
 
 In this section we define  a rigidity property of tests called robustness (see Definition <ref>). Loosely, a test is robust if every almost optimal strategy against it is close to an optimal one. Though this notion is natural, one needs to clarify what is the measure of distance between strategies. 



 §.§ Edit distance

We first define a generalized version of the normalized Hamming distance (<ref>). Let X and Y be two finite sets, and assume X⊆ Y. Given two permutations σ∈(X) and τ∈(Y), we define their normalized Hamming distance with errors to be

    d_H(σ,τ)=1-|{ x∈ X|σ(x)=τ(x)}|/|Y|.

If for every y∉ X we let σ(y)=error ∉ Y,
then we can define equivalently 

    d_H(σ,τ)=ℙ_y∈ Y[σ(y)≠τ(y)],

which is the way we defined the normalized Hamming distance in (<ref>). Since the generalized version is the same as the usual Hamming distance when Y=X, we use the same notation for both and just call them the Hamming distance from now on.

Let S→ℝ_>0 be a function, which we call the significance function. 
Given two actions ρ S→(X) and φ S→(Y), we  define their -weighted distance as follows:

    d^
    (ρ,φ)=∑_s∈ S(s)· d_H(ρ(s),φ(s)).[Though the Hamming distance is always bounded by 1, the weighted distance may be larger than 1.]

For every θ∈(Y), let φ^θ S→(Y) be φ^θ(s)=θφ(s) θ^-1.
Now, the -weighted edit distance[
There is a more graph theoretical way of viewing the edit distance, in the spirit of the  section <ref>. It essentially measures how much one Schreier graph needs to be changed to get to the other graph. For more on that, see Section 4 of <cit.> and the Γ-graph notion in <cit.>.] of ρ and φ is  the following minima: 

    d^_ edit(ρ,φ)=min_θ∈(Y)(d^(ρ,φ^θ)).

Note that the edit distance is independent of the specific embedding of X in Y.
The edit distance induces a metric on _(). For every π,π' ∈_(), let

    d^_ edit(π,π')=inf{ d^_ edit(ρ,φ) |ρ∈Φ^-1(π),φ∈Φ^-1(π')} ,

where recall the definition of Φ in (<ref>).






  §.§.§ A graph theoretic perspective
 


 Let Γ be a  group with S a finite  generating set. Let X be a set and σΓ→(X) an action of Γ on X. The generalized Schreier graph  (σ,S) is a directed graph with vertex set X and edges labeled by S. The edges of (σ,S) are defined as follows: For every x∈ X  and s∈ S, there is a directed edge from  σ(s).x to x labeled by s, namely

    σ(s).x x.

We may say that an edge is labeled by some inverse of s∈ S, such as xy. By that we mean the edge labeled by s and oriented in the other direction, namely yx. This is natural as σ(s^-1)=σ(s)^-1. Moreover, when S generates Γ, it includes words with inverses, and so when one associates words in the generators to paths in a labeled graph, this interpretation is natural (cf. <cit.>). 


    This  generalizes the usual notion of a Schreier coset graph (Γ,H,S) by choosing σΓ→_H\^Γ to be the natural action of Γ on right H-cosets. 
 
The Schreier graph does not provide any new data about σ, but it  provides a graph theoretic perspective for approaching IRSs.
For example, a fact that we repeatedly use is that orbits of the action are in correspondence with the connected components of the Schreier graph. Moreover, the edit distance is better understood as a distance between graphs: Given two labeled graphs, one wants to transform one to the other. Whenever the endpoint of an edge labeled by s is changed, there is an associated cost (s). The same cost is incurred when deleting completely or adding such an s-labeled edge. The goal is to minimize the cost of moving between the graphs (up  to isomorphism). The edit distance is exactly the minimal cost of such a transformation.







  §.§.§ Topology induced by the edit distance

 It turns out that regardless of the significance function , the topology induced by the edit distance is strictly stronger than the weak^* topology:

  We have the following:


  * Given two sequences π_n,π'_n∈_() such that π_n converges  to π_∞∈() in the weak^* topology and lim_n→∞d^_ edit(π_n,π'_n)=0, then π'_n converge to the same π_∞ in the weak^* topology.

  * There are sequences π_n∈_() that converge in the weak^* topology but are not Cauchy sequences in the edit distance topology.




    The above proposition should be contrasted with the amenable case, studied in <cit.>, where the authors showed that the converse of (1) is true when the free group is replaced by an amenable group. See <cit.> for a different proof of the above clause (2).





Let min() be the minimum value of . Since S is finite, min()>0. So d^_ edit≥min()d_ edit where d_ edit is the edit distance with significance function equal to the constant 1. So lim_n→∞d_ edit(π_n,π'_n)=0. This reduces the problem to the ordinary edit distance.

Let K,L be finite subsets of . By Remark <ref>, we have

    lim_n→∞π_n((K,L)) = π_∞((K,L)).

So it suffices to prove

    lim_n→∞ |π_n((K,L))-π'_n((K,L))| = 0.

Let r>0 be a large enough radius so that K∪ L is contained in the ball B(r) ⊂ of radius r with respect to the word metric induced by S.

Because π_n and π'_n are finitely described, there are actions ρ_n S→(X_n) and ρ'_n S→(Y_n) which induce π_n and π'_n respectively. We may assume that for each n, either X_n ⊂ Y_n or Y_n ⊂ X_n. After conjugating if necessary, we may further assume

    d_ edit(ρ_n,ρ'_n) = d(ρ_n,ρ'_n) = ∑_s∈ S d_H(ρ_n(s),ρ'_n(s)).


Given an integer k ≥ 1, let (k) be the set of all v∈ X_n ∪ Y_n such that there exists g in the ball B(k)⊂ with ρ_n(g)^-1v ρ'_n(g)^-1v. The definition of edit distance gives

    |(1)| ≤ d_ edit(ρ_n,ρ'_n) |X_n ∪ Y_n|.

Suppose v ∈(k) ∖(k-1) for some k≥ 2. Then there is a g ∈ B(k) such that ρ_n(g)^-1v ρ'_n(g)^-1v. Let g=s_1⋯ s_k with s_i ∈ S ∪ S^-1. Since v ∉(k-1), if h=s_1⋯ s_k-1, then w:=ρ_n(h)^-1v = ρ'_n(h)^-1v and ρ_n(s_k)^-1w ρ'_n(s_k)^-1w. Hence w ∈(1) and so v ∈ B(k-1)(1). Therefore,

    |(k)| ≤ |B(k-1)| |(1)| ≤ |B(k-1)|d_ edit(ρ_n,ρ'_n) |X_n ∪ Y_n|.

Since |B(k-1)| is independent of n, and d_ edit(ρ_n,ρ'_n) 0, we can deduce that

    lim_n→∞|(k)|/|X_n ∪ Y_n| = 0

for all k. 

Because K ∪ L ⊂ B(r), if v ∉(r), then (ρ_n, v) ∈(K,L) if and only if (ρ'_n, v) ∈(K,L). Hence

    |π_n((K,L))-π'_n((K,L))| ≤|(k)|/|X_n ∪ Y_n|→ 0

as n→∞. This finishes the proof of (1).

To prove item (2), recall that if G=(V,E) is a graph then a subset I ⊂ V is independent if no edge in E has both endpoints in I. If G is finite then the independence ratio of G is α(G)=|I|/|V| where I is an independent set of maximum cardinality. If ρ:S →(n) is an action then we let α(ρ)=α(G) where G is the associated action graph.

For any actions ρ and ρ',  we have d_edit(ρ,ρ') ≥ |α(ρ)-α(ρ')|. So it suffices to show there exist action sequences {ρ_n}_n, {ρ'_n}_n whose corresponding IRSs weak^* converge to the same IRS and  |α(ρ)-α(ρ')|>c for some constant c>0.

By using a first moment argument, it can be shown that if ρ_n:S →(n) is chosen uniformly at random and if G_n is the associated action graph then with high probability α(G_n)<1/2 - c where c>0 is a constant depending only on |S| (which we assume is at least 2). Moreover,  there are sets Ω_n ⊂(n)^S with |Ω_n|/|(n)^S| → 1 as n→∞ such that if ρ_n ∈Ω_n for all n and π_n are the corresponding IRSs then π_n weak^* converges to the trivial subgroup {}. This can be derived by estimating the expected number of short cycles in the graph G_n <cit.>. A more precise estimate on Ω_n is obtained in <cit.>.

Therefore, there exists a sequence {ρ_n} of actions such that if π_n are the corresponding IRSs then π_n weak^* converges to the trivial subgroup {} and α(G_n) < 1/2 - c.

On the other hand, there exist ρ'_n:S →(n) whose action graphs are bi-partite such that if π'_n is the corresponding IRS then π'_n also weak^* converges to the trivial subgroup {}. Since the independence ratio in this case is 1/2, it follows that the sequence π_1,π'_1,π_2,π'_2,… cannot be Cauchy in the edit distance.

















 §.§ Robustness

Given s∈ S, we can define a map v_s→ℕ that given a reduced word w∈ counts the number of appearances of s or s^-1 in w. Namely, if w=∏_j=1^ℓ s_j^_j,  where s_j∈ S and _j∈{± 1}, is a reduced word, then v_s(w)=∑_j=1^ℓ 1_s(s_j).

Let  be 
a test with its usual associated data S, (K_i;D_i)_i∈ Q and μ. We define _ S→ℝ_>0, the significance function associated with , as follows: 

    _(s)=_i∼μ[∑_w∈ K_i v_s(w)].


 
    Let >0, and  be 
a test. Let X and Y be finite sets, and ρ S→(X) and φ S→(Y) be actions. Assume d^__ edit(ρ,φ)≤. Then |(,Φ(ρ))-(,Φ(φ))|≤.



    Assume without loss of generality that X⊆ Y and that d^__ edit(ρ,φ)=d^_(ρ,φ). Let i∈ Q. For every w=∏_j=1^ℓ s_j^_j∈, we have
    
    _y∈ Y[ 1_(ρ,y)(w)≠ 1_(φ,y)(w)]≤_y∈ Y[ρ(w).y≠φ(w).y].

Inequality (<ref>) is a consequence of the following implication: If  ρ(w) acts on y the same way as φ(w), then in particular  w is either in both (ρ,y) and (φ,y) or in none of them. For 1≤ k≤ℓ, let w_k be the suffix of w from position k, namely w_k=∏_j=k^ℓ s_j^_j. Moreover, let w_ℓ+1 be the empty word _. Then 

    _y∈ Y[ρ(w).y≠φ(w).y]≤_y∈ Y[⋁_1≤ k≤ℓρ(w_k). y ≠φ(w_k). y].

Inequality (<ref>) is a consequence of  the following implication: If the endpoint of the path beginning at y and labeled by w^-1 in (ρ,S) is different than the endpoint of  the same path in (φ,S), then the two paths diverged at some point. Now,
   
    _y∈ Y[⋁_1≤ k≤ℓρ(w_k). y ≠φ(w_k) .y]≤∑_k=1^ℓ_y∈ Y[ρ(w_k). y ≠φ(w_k). y ρ(w_k+1). y = φ(w_k+1). y].

Inequality (<ref>) is a consequence of the union bound and the logic tautology ⋁_j=1^m φ_j=⋁_j=1^m(φ_j¬φ_j-1), where φ_0= False. But, since 
   
    ρ(w_k) .y ≠φ(w_k) .y ρ(w_k+1) .y = φ(w_k+1) .y ρ(s_k^_k)∘ρ(w_k+1).y≠φ(s_k^_k) ∘ρ(w_k+1).y,

   and since ρ(w_k+1).y is uniformly distributed given that y is uniformly distributed, we can deduce that 
    
    _y∈ Y[ρ(w_k) .y ≠φ(w_k) .y ρ(w_k+1). y = φ(w_k+1) .y]
        ≤_y∈ Y[ρ(s_k) .y≠φ(s_k) .y ]_=d_H(ρ(s_k),φ(s_k)).

   All in all, 

    _y∈ Y[ 1_(ρ,y)(w)≠ 1_(φ,y)(w)]≤∑_j=1^ℓ d_H(ρ(s_j),φ(s_j))=∑_s∈ Sv_s(w)d_H(ρ(s),φ(s))

Hence, by the union bound and inequality (<ref>), we have

    _i∼μ_y∈ Y[(ρ,y)∩K_i≠(φ,y)∩K_i]   ≤_i∼μ[∑_w∈ K_i∑_s∈ S v_s(w)d_H(ρ(s),φ(s))]
       =∑_s∈ S_i∼μ[∑_w∈ K_iv_s(w)]_(<ref>)d_H(ρ(s),φ(s))
       =∑_s∈ S_(s)d_H(ρ(s),φ(s))
       =d^_(ρ,φ)≤.

Lastly, by combining  

    |(,ρ)-(,φ)|   ≤_i∼μ_y∈ Y[D_i((ρ,y)∩K_i)≠ D_i((φ,y)∩K_i)]
       ≤_i∼μ_y∈ Y[(ρ,y)∩K_i≠(φ,y)∩K_i]

with 
 (<ref>), we deduce the claim.



    For those familiar with  robustness of non-local games, d^__ edit plays a somewhat analogous role to the state-dependent distance (cf. Section 4 of <cit.>, for example).


Robustness is a reverse implication  to the one in Claim <ref>.


Let   be a test and δℝ_≥0→ℝ_≥0 a function satisfying δ()0. We say that  is δ-robust if: 


  * There exists an optimal strategy, namely π∈_() such that (,π)=_().

  * Almost optimal strategies are close to optimal ones, namely, if  π'∈_() satisfies (,π')≥_()-, then there is an optimal strategy π such that d^__ edit(π,π')≤δ().




Let Γ=⟨ S|R⟩ be a finitely presented group. Then, Γ is said to be δ-homomorphism stable, where δ()0, if for every ρ S→(X) where  _r∈ R[d_H(ρ(r),_X)]≤ there is a Γ-action φ S→(Y), where X⊆ Y and such that _s∈ S[d_H(ρ(s),φ(s))]≤δ().


    This notion is more commonly known as flexible pointwise group stability in permutations. See GlebskyRivera,ArzhantsevaPaunescu,BLT,BeckerLubotzky,CL_part1,CVY_efficient for various results in this theory.


    The group Γ=⟨ S|R⟩ is δ-homomorphism stable if and only if the  R-verification test _R is δ'-robust. The exact relation between δ and δ' can be calculated explicitly — they are constant multiples of one another, where the constants depend only on S and R.





§ TAILORED NON-LOCAL GAMES


We commonly use both {0,1} and _2 for the set with 2-modular arithmetic.



 §.§ Non-local games preliminaries

We repeat the definitions which appeared in Section <ref> of the introduction. Since this paper is not focused on non-local games, we treat them somewhat technically. For formal definitions associated with the complexity class ^*, we refer to <cit.>. For the connection between ^* and nonlocal games, a good starting point is <cit.>.


   A (synchronous) non-local game  consists of a finite graph G=(V,E), a length function ℓ V→ℕ, formal sets of generators S_x={^x,i| 1≤ i≤ℓ(x)} for every vertex x∈ V, a distribution μ over the edges E, and decision functions D_xy{0,1}^S_xy→{0,1} for every edge xy∈ E, where S_xy=S_x∪ S_y. We denote by S the set ⋃_x∈ V S_x.


   Anyone familiar with the definition of a non-local game will immediately notice that the standard formalism is different than the one we chose here. Usually, a (synchronous) non-local game is defined as a pair of finite sets X,A, commonly referred to as the question and answer sets respectively, together with a   distribution μ over pairs of questions X× X and a decision predicate D X× X× A× A→{0,1} satisfying D(x,x,a,b)=0 for any x∈ X and a≠ b∈ A. 

   It is quite straightforward to move between the definitions.   A way one can extract the data of Definition <ref> from the above is as follows: Let ℓ be the constant function Λ=⌈log |A|⌉, and fix an embedding of A into {0,1}^Λ. The vertices V of the underlying graph G will be X, and the support of μ will be the edge set E⊆ X× X. The formal generator ^x,i corresponds to the i^ th bit of the answer a when x∈ X is asked as a question. 
   Then, given that x,y were asked, a pair of answers a,b can be encoded as a map γ S_x∪ S_y→{0,1}, where 
   
    γ|_S_x=a   and  γ|_S_y=b .
 
  The  notation (<ref>) will be used repeatedly in the text. 
   Lastly, D_xy(γ)=D(x,y,a,b), where γ is the aforementioned encoding. 

   In the other direction, we can define A to be all bit strings shorter than Λ=max_x∈ Vℓ(x), X to be the vertex set V, and μ stays the same. Finally, D(x,y,a,b)=D_xy(γ) when a,b are of the correct lengths, i.e. a=ℓ(x),b=ℓ(y), and is 0 when either of them is of the incorrect length. Since we consider only synchronous strategies, the condition D(x,x,a,b) is never checked in practice for a≠ b and we can assume it was satisfied beforehand.
Because of this correspondence, we may refer to ℓ as the answer length function.



Non-local games are often dramatized as two-prover interactive proofs with one round <cit.>. This perspective  may help some readers to absorb the upcoming technicalities better. 
Two players, that can share some resources beforehand (random bits in the classical case and entangled qubits in the quantum case), are separated spatially — e.g., they are seated in far away rooms. 
A referee  samples a pair of questions and sends one to each player. 
The players then use their shared resources to come up with answers, and send them back to the referee. 
The referee then decides, using the decision predicate, whether the players won or lost. 
The decision predicate as well as the distribution over possible questions are assumed to be known to the players beforehand.  
We demonstrate this dramatization in Example <ref>.
   


The following is the quantum mechanics' analogue of a probability distribution. Similar to the way probability distributions (over finite sets) are collections of non-negative real numbers that add up to 1, the quantum analogue would be “non-negative” matrices that add up to the identity. 


    A positive operator valued measure (POVM) of dimension n with outcomes in a set A  is a mapping A→ M_n× n(ℂ) such that _a is a positive (semi-definite) matrix for every a∈ A and ∑_a∈ A_a=_n. It is called projective, or a projection valued measure (PVM), if in addition _a is an orthogonal projection for every a∈ A, namely (_a)^2=_a=(_a)^*, where * is the conjugate transpose operation. 
Given that  is a PVM and A={0,1}^Λ,  we define for 1≤ i≤Λ the i^ th marginal of  to be ^i{0,1}→ M_n× n(ℂ)  by 

    ^i_0=∑_aΛ→{0,1}
     a_i=0_a   and  ^i_1=∑_aΛ→{0,1}
     a_i=1_a.

Thus, _a=∏_i=1^Λ^i_a_i. Furthermore, the matrix ^i=^i_0-^i_1 is an order 2  unitary, which we call the i^ th binary observable. All in all, a PVM can be given either as a map {0,1}^Λ→ M_n× n(ℕ), or as a map [Λ]→ U(n) such that the images of  are commuting involutions — i.e., they square to the identity.

As its name suggests, every POVM  defines a probability distribution over its answer set A as follows

    [a  is sampled]=τ(_a),

where τ is the dimension normalized trace on n× n matrices. Such an answer is said to be sampled according to  and we denote it by a∼.


    In the case of a PVM with outcome set A={0,1}^Λ, the observables {^i}_i=1^Λ generate a unitary representation of _2^Λ. The _a's in this case can be read from the unitary representation using the Fourier transform of the representation. Hence, for the case of A={0,1}^Λ, a PVM can be given as a collection of mutually perpendicular ortohgonal projections that sum up to the identity, or as a unitary representation of _2^Λ, and one can move from one perspective to the other without losing information.


     A (synchronous quantum) n-dimensional strategy for  is a map ρ S→ U(n), where the images are involutions, and ρ(S_x) commutes for every fixed x∈ V — by saying that a set of matrices (or later permutations) commutes, we mean that every pair of elements in this set commute.
     Another way of viewing strategies is by saying that they associate with every vertex x∈ V an n-dimensional PVM (in observable form).
     The strategy ρ is said to be commuting along edges if ρ(S_xy) commutes for every edge xy∈ E. 

     
     Such a strategy defines for every edge xy∈ E a probability distribution over functions γ S_xy→{0,1} as follows:
     
    [γ S_xy→{0,1}  is sampled]=τ(∏_∈ S_x^_γ()∏_∈ S_y^_γ()),

where τ is again the dimension normalized trace on n× n matrices, ^_0ℂ^n→ℂ^n is the projection on the (+1)-eigenspace of ρ(), and ^_1ℂ^n→ℂ^n is the projection on its (-1)-eigenspace.[Note that ρ()=^_0-^_1, which exactly shows how to move from this observable form of the PVMs induced by ρ back to its projection based form.] We say that γ was sampled according to ρ if it has this distribution, and we denote it by γ∼ρ.

For γ S_xy→{0,1}, let ^x_γ=∏_∈ S_x^_γ(). Then, the map ^xy{0,1}^S_xy→ M_n× n(ℂ) defined by ^xy_γ=^x_γ^y_γ^x_γ  is  a POVM. Thus, γ∼ρ is a special case of sampling according to a POVM (<ref>), which in this case is ^xy. In the spirit of the notation in (<ref>), it may be appropriate to write ^x_γ=^x_a and ^y_γ=^y_b.


    The images of  according to ρ are usually called binary observables, and the value γ() is usually called the measurement outcome.


 
    There is a procedural (i.e., algorithmic) way of  sampling  γ S_xy→{0,1} according to ρ: 
Since ρ(S_x)⊆ U(n) commutes, these matrices have a mutual orthonormal basis of eigenvectors B_x. Similarly, ρ(S_y) have a mutual orthonormal basis of eigenvectors B_y.
 First, sample v⃗∈ B_x uniformly at random. 
 Then, sample u⃗∈ B_y according to the  squared length of its projection on v, namely 
    [u⃗∈ B_y  is sampled|v⃗∈ B_x  was sampled]=|⟨v⃗|u⃗⟩ |^2,

        where ⟨·| ·⟩ is the standard inner product on ℂ^n, i.e., ⟨v⃗|u⃗⟩=∑_i=1^n v_i· u_i.
 Finally,  γ S_xy→{0,1} is determined as follows — for ∈ S_x, γ()=
        0    ρ()v⃗=v⃗,

        1    ρ ()v⃗ =-v⃗,, and for ∈ S_y, γ()=
        0    ρ()u⃗=u⃗,

        1    ρ ()u⃗ =-u⃗..
        All in all, 
        
    [ (v⃗,u⃗)  is sampled]=|⟨v⃗|u⃗⟩|^2/n,

        which is independent of the ordering between x and y. Note also that the probability a specific γ is measured is  independent of the  chosen bases B_x and B_y.



The function γ S_xy→_2 is sampled in the above procedure if for the resulting pair (v⃗,u⃗) we have v⃗∈( ∏_∈ S_x^_γ()) and u⃗∈(∏_∈ S_y^_γ()). Denote by ^x_γ the product ∏_∈ S_x^_γ(), and similarly ^y_γ. So, 

    [γ  is sampled]=∑_v⃗∈(^x_γ)
    u⃗∈(^y_γ)|⟨v⃗|u⃗⟩ |^2/n.

By the fact that B_x is a mutual orthonormal basis of eigenvectors for ρ(S_X), every v⃗∈ B_x is either in the image of ^x_γ or in its kernel. Therefore, the above sum can be written as 

    ∑_v⃗∈(^x_γ)
    u⃗∈(^y_γ)|⟨v⃗|u⃗⟩ |^2/n=∑_v⃗∈ B_x
    u⃗∈ B_y|⟨^x_γv⃗|^y_γu⃗⟩ |^2/n=∑_v⃗∈ B_x
    u⃗∈ B_y|⟨^y_γ^x_γv⃗|u⃗⟩ |^2/n.

Since B_y is an orthonormal basis and ^x_γ,^y_γ are orthogonal projections,  for every v⃗∈ B_x we have 

    ∑_u⃗∈ B_y|⟨^y_γ^x_γv⃗|u⃗⟩ |^2=^y_γ^x_γv⃗^2=⟨^y_γ^x_γv⃗|^y_γ^x_γv⃗⟩ =⟨^x_γ^y_γ^x_γv⃗|v⃗⟩.

Thus,

    ∑_v⃗∈ B_x
    u⃗∈ B_y|⟨^y_γ^x_γv⃗|u⃗⟩ |^2/n=∑_v⃗∈ B_x⟨^x_γ^y_γ^x_γv⃗|v⃗⟩/n=τ(^x_γ^y_γ^x_γ)=τ(^x_γ^y_γ),

where the last equality uses the cyclicity of the trace and the fact ^x_γ is a projection. All in all, 

    [γ  is sampled]=τ(^x_γ^y_γ),

as needed.


 
The formula (<ref>) for the probability a specific γ is sampled is motivated by the Born measurement rule from quantum mechanics. In quantum mechanics, a (projective) measurement is specified by choosing a PVM  A→ B(ℋ) acting on a separable Hilbert space ℋ (Definition <ref> provided the finite dimensional case). 
Given a quantum state (unit vector) ψ∈ℋ, a measurement of ψ returns the outcome a with probability 
    [a   is the  measurement outcome]=ψ^* _a ψ.
 
If there are two systems (e.g. two “players”), then a Hilbert space ℋ_A, ℋ_B is associated to each of them
respectively, such that the joint Hilbert space is ℋ=ℋ_A ⊗ℋ_B. 
The joint distribution on outcomes is then given by probabilities 

    [(a,b)  are the  measurement outcomes]=ψ^* (_a^x ⊗ (^y_b)^†) ψ,

where † is transposition with no complex conjugation, as opposed to *.[It is common to drop the transposition † on the ℋ_B POVM in (<ref>). But, in this case, a transposition will appear in the calculation using traces (<ref>). So, we decided on this equivalent formulation.]
The case where ℋ_A and ℋ_B have the same (finite) dimension n, and ψ= n^-1/2∑_i e_x,i⊗ e_y,i for orthonormal bases {e_x,i} and {e_y,i} of ℋ_A and ℋ_B respectively, termed “maximally entangled,” yields 

    [(a,b)  are the  measurement outcomes]=τ(^x_a ^y_b).
 
 


    The  value of the strategy ρ against the non-local game  is

    (,ρ)   =_xy∼μ_γ∼ρ[D_xy(γ)]
       =∑_xy∈ E∑_γ S_xy→_2μ(xy)D_xy(γ)τ(∏_∈ S_x^_γ()∏_∈ S_y^_γ()).

 The (synchronous) quantum value of , ^*(), is the supremum over all possible strategies against .
 




The main result in <cit.> is a reduction from the Halting Problem to approximating  the quantum value of a game:

There exists a polynomial time algorithm that takes as input (the encoding of) a Turing machine M and outputs (the encoding of) a non-local game _M such that:

    
  * Sampling xy∈ E according to  μ and evaluating  D_xy(·) from the encoding of the  game _M  can be done in time poly(|M|), where |M| is the bit-length of the encoding of M.
    
  * If M halts, then there exists a quantum strategy ρ that commutes along edges, for which  (_M,ρ)=1. In particular, ^*(_M)=1.
    
  * If M never halts, then ^*(_M)≤1/2.


The  goal of the next section is to show that a somewhat stronger variation of Theorem <ref>, which is proved in <cit.>,  provides a negative solution to the Aldous–Lyons Conjecture <ref>.



 §.§ Permutation strategies

Our goal in  Section <ref> is to relate values of subgroup tests to values of non-local games and vice versa. The strategies for subgroup tests are finitely described IRSs, which are (induced by) maps of the form σ S→(n), while quantum strategies for non-local games are maps of the form ρ S→ U(n) adhering to certain conditions. These already seem quite related, as embedding (n) into U(n) as permutation matrices is natural, and indeed transforms every (nice enough) finitely described IRS into a quantum strategy. But, there is a problem with doing this naively: 
Say that we want to construct a quantum strategy, such that  for a certain ∈ S, regardless of anything else,  γ() will always be 1. Then, we can achieve it by choosing ρ()=-. But, - is not a permutation matrix. Therefore, any σ that will be naively transformed into a quantum strategy by embedding (n) into U(n) as permutation matrices cannot ensure a specific  is always evaluated to 1. 
To amend that, we use a tool developed in the study of solution groups of linear constraint system games (cf.  <cit.>): 
We add a special generator  that plays the role of -. 
Since - is a central involution without fixed points, the image of  under σ needs to behave the same. 
This forces σ to act on an even sized set. 
Now, if we want σ()=σ() to translate into γ() being always 1, we will need to restrict ourselves to the (-1)-eigenspace of σ(). This leads us to the following two definitions. 


    A permutation strategy for a non-local game  is a map 
σ S∪{}→(2n)
with the following properties:

    
  * The image of  is a central involution with no fixed points. Namely, 
    
    σ()^2=,  d_H(σ(),)=1,
       ∀∈ S   σ()σ()=σ()σ().

    
  * The rest of the images are involutions as well. Namely,
    
    ∀∈ S   σ()^2=.

    
  * The image of S_x commutes. Namely, for every fixed vertex x∈ V, 
    
    ∀,'∈ S_x   σ()σ(')=σ(')σ().





    Let σ S∪{}→(2n) be a permutation strategy for . Embed (2n) in U(2n) as permutation matrices. Note that in this viewpoint, U(2n) is acting naturally on functions f [2n]→ℂ, and the matrices are represented with respect to the standard basis, which consists of the indicators 
    
    ∀⋆∈ [2n]    1_⋆(♢)=
            1    ♢=⋆,
    
            0    ♢≠⋆.

    Moreover, for a permutation ζ∈(2n) and function f [2n]→ℂ, we have
    
    ∀⋆∈ [2n]   (ζ.f)(⋆)=f(ζ^-1.⋆).

    By (1) of Definition <ref>, σ() is a fixed point free involution, and thus its (+1)-eigenspace W^+ and (-1)-eigenspace W^- are of the same dimension n. Note that W^+ consists of all functions f [2n]→ℂ that are constant along -labeled edges in the Schreier graph[Generalized Schreier graphs were defined in  Section <ref>.] (σ,S∪{}), while W^- consists of all functions that flip their signs along -labeled edges. Let =^_1ℂ^2n→ W^- be the orthogonal  projection on W^-. Note that ^*=^_1 and ^*=_W^-.
    
    We can choose bases for W^+ and W^- as follows. 
    If we denote 
    
    ⋆'=σ().⋆

    for every ⋆∈ [2n], then ⋆⋆' is the perfect matching induced by -labeled edges in  (σ,S∪{}). 
    So, an orthogonal basis of W^+ may be taken to be  B^+={ 1_⋆+ 1_⋆'}_⋆∈ [2n], while for W^- one takes  B^-={ 1_⋆- 1_⋆'}_⋆∈ [2n] —  B^- contains for each vector also its opposite, so this is a basis when choosing a representative for each such pair. The union of these bases is an orthogonal basis for ℂ^2n. Furthermore, we later refer to B^- as the standard basis of W^-≅ℂ^n.



    Given that ρ S∪{}→(2n) is a permutation strategy, there are n orbits of σ(), and σ acts on them in a well-defined manner. If we arbitrarily denote each orbit as {+⋆,-⋆}, instead of {⋆,⋆'} a la (<ref>), then σ acts naturally on the set Ω_±={±}×Ω in a way that commutes with the sign flip. Hence, a permutation strategy can be defined as a map into signed permutations, where  is mapped to the sign flip, and such that the variables at each vertex are mapped to commuting involutions. This point of view may be helpful for understanding some of the next claims and remarks. We decided not to use the language of signed permutations in this paper, though in the companion paper <cit.> it is more pronounced. 



 Let σ S∪{}→(2n) be a permutation strategy for . Recall from Remark <ref> that W^- is the (-1)-eigenspace of σ(), and that ℂ^2n→ W^- is the orthogonal projection on W^-. Then,
the quantum strategy ρ S→End(W^-) induced by σ is  the mapping to the W^--corners
    
    ρ()=σ()^*.





    Let  σ be a permutation strategy. Then, the quantum strategy ρ induced by σ, as in Definition <ref>, is indeed a quantum strategy for , as in Definition <ref>.


    The criterion for being a quantum strategy as in Definition <ref> is to be a map into unitaries such that the image of the generators S_x associated with any specific vertex commute, and all the images are involutions.
First,
 
    ρ()^*=(σ()^*)^*=σ()^* ^*=σ()^-1 ^*.

    Now,  the fact that σ() commutes with  σ() for any ∈ S implies that ^*σ()=σ()^*. Therefore, 
    
    ρ()^*ρ()   =σ()^-1^*σ()^*
       =σ()^-1σ()__2n^*^*__W^-
       =^*=_W^-,

   and the image of ρ is contained in U(W^-). Item (2) of Definition <ref> states that  σ()^-1=σ() for all ∈ S,  and combined with (<ref>) gives
   
    ρ()^*=σ()^-1^*=σ()^*=ρ(),

   i.e., the images of ρ are involutions. Finally, since σ(S_x) commutes for every fixed x∈ V, we have
   
    ∀,∈ S_x   ρ()ρ()   =σ()^*σ()^*
       =^*σ()σ()^*
       =^*σ()σ()^*
       =σ()^*σ()^*
       =ρ()ρ(),

   which finishes the proof.



[Classical permutation strategies]
Let (2)={,ζ} be the group of permutations acting on two elements. For every fixed f S→{0,1}, we can define an action σ S∪{}→(2) as follows

    σ()=ζ  ;  σ()=    f()=0,
    ζ    f()=1.

It is straightforward to see that the quantum strategy associated with σ satisfies ρ()=(-1)^f(), and thus γ S_xy→{0,1} that is sampled according to ρ is deterministically f|_S_xy.
Strategies  of this form are usually called deterministic. By taking direct sums of such actions (for potentially different f's), we can get any (rational) distribution over deterministic strategies. These strategies are usually called classical. So, every (rational) classical strategy can be obtained as a permutation strategy. 





    Let σ S∪{}→(2n) be a permutation strategy, and ρ the associated quantum strategy. Then, if for a word w∈(S∪{}) we have ρ(w)=, then σ(w)=.


    Recall from Remark <ref> that the images of ρ act naturally, and in the same way as σ,  on W^-, the (-1)-eigenspace of σ(), which consist of functions f [2n]→ℂ that flip along -edges. Recall also the notation ⋆'=σ().⋆ from (<ref>).
    So, the fact that ρ(w)= means that for any  function f∈ W^-, σ(w).f=f as well.  
    This is true in particular for B^-={ 1_⋆- 1_⋆'}_⋆∈ [2n], the basis of W^- described in the same remark. Note that for every ζ∈(2n) we have ζ. 1_⋆= 1_ζ.⋆.
    Hence, for every ⋆∈[2n], we have

    1_⋆- 1_⋆'=σ(w).( 1_⋆- 1_⋆')= 1_σ(w).⋆- 1_σ(w).⋆'.

By evaluating the functions on both sides of the above equation on the point ⋆∈ [2n], we get

    1= 1_σ(w).⋆(⋆)- 1_σ(w).⋆'(⋆),

which means σ(w).⋆=⋆. Since ⋆ was any point in [2n], σ(w)= and we are done.





     A non-local game  is said to have a perfect permutation strategy if there exists a permutation strategy σ such that the  quantum strategy ρ induced by it, as in Definition <ref>, satisfies (,ρ)=1.
 


In the rest of this subsection we further analyze permutation strategies, and specifically discuss a procedural sampling — similar to  Claim <ref>  — of a γ S_xy→{0,1} according to ρ which is induced by a permutation strategy σ. 



 §.§ Interlude — Fourier bases of _2^m actions



    Let σ_2^k→(_2^k) be the standard action of _2^k on itself, namely σ(w).v=v+w for every v,w∈_2^k. The Fourier basis of σ is the following collection of functions, denoted by _2^k: For every u∈_2^k, let u_2^k→ℂ be 

    ∀ w∈_2^k   u(w)=(-1)^⟨ u,w⟩/√(2^k),

where ⟨ u,w⟩=∑_i=1^k u_iw_i is the standard bilinear form on _2^k.[We try to keep ⟨·,·⟩ for the bilinear form over a finite vector space, and ⟨· |·⟩ for the complex inner product.]



    The collection _2^k is an orthonormal basis of eigenvectors for the action  σ. 


    The action σ acts as permutations on the standard basis of ℂ^_2^k, and thus extends to act on the whole vector space of functions from _2^k to ℂ. Using the standard inner product ⟨·|·⟩ on  ℂ^_2^k as before, we can check that 
    
    ∀ u∈_2^k   ⟨u|u⟩ =∑_w∈_2^ku(w)·u(w)=∑_w∈_2^k1/2^k=1.

    Furthermore, 
    
    ∀ u≠ v∈_2^k   ⟨v|u⟩   =∑_w∈_2^kv(w)·u(w) 
       =1/2^k∑_w∈_2^k(-1)^⟨ v+u,w⟩
       =0,

    where the last equality is derived from the fact v+u≠0⃗. Lastly, 
    
    ∀ u,v,w∈_2^k   (σ(w).v)(u)   =v(u+w)
       =(-1)^⟨ u+w,v⟩/√(2^k)
       =(-1)^⟨ w,v⟩·(-1)^⟨ u,v⟩/√(2^k)
       =(-1)^⟨ w,v⟩v(u),

    and _2^k is indeed a collection of mutual eigenvectors for σ.


Recall basic group actions definitions. 
Let G be a group, and σ G→(X),τ G→(Y) be two homomorphisms, which induce actions of G on X and Y respectively. A map 
f X→ Y  is a factor (of G-actions)  — also known as a morphism of G actions or a  (σ,τ)-equivariance — if
for all w∈ G and x∈ X, we have f(σ(w).x)=τ(w).f(x).
Note that for every set S⊆ G, such a map induces a morphism of oriented edge-labeled graphs between (σ,S) and (τ,S).
The actions σ and τ are said to be equivalent if there exists a bijective factor between them, which in the graph case would be a (edge-labeled oriented) graph isomorphism between (σ,S) and (τ,S).
Given two actions, we can define their sum  σ⊕τ G→(X⊔ Y), where  ⊔  is the  disjoint union, to be

    ∀ w∈ G, z∈ X⊔ Y   
     (σ⊕τ(w)).z=σ(w). z     z∈ X,
    τ(w). z     z∈ Y.


and their  product σ⊗τ S→(X× Y) to be

    ∀ s∈ S, x∈ X,y∈  Y   (σ⊗τ(s)).(x,y)=(σ(s).x ,τ(s).y).


If σ is a transitive action and  x∈ X, then it is equivalent to the  action of G on G/(σ,x), where (σ,x) is defined (as in Section <ref>) to be
(σ,x)={w∈ G|σ(w).x=x}.
The bijective factor in this case is associating with σ(w).x the coset w·(σ,x), and recalling that by transitivity every y∈ X is of the form σ(w).x for some w∈ G. So,  every action σ G→(X) is equivalent to the sum of actions of G on quotients of it G/H, for some collection of subgroups H≤ G. 

Assume now that G is _2^m for some positive integer m. Let σ G→(X) be a transitive action. Since every subgroup H of  G is normal, this action factors through  some _2^k for k≤ m. Namely, we can find a bijective factor between X and the action of _2^k on itself: 
 Fix some ⋆∈ X; (σ,⋆) is a subspace of _2^m, and we implicitly assumed it is isomorphic to _2^m-k; for every ♢∈ X, there is an element g of _2^m for which σ(g).⋆=♢, and thus σ(g').⋆=♢ for every g'∈ g+(σ,⋆); so we can associate g+(σ,⋆) with ♢ and _2^m/(σ,⋆) with _2^k.  
For every g∈_2^m, let g̃∈_2^k be its image in the bijection between the quotient space _2^m/(σ,⋆) and _2^k. Then,  

    ∀ v∈_2^k   σ(g).v=(-1)^⟨ v,g̃⟩v.

In particular, for s∈ S⊆_2^m, the action of s on v can be read from what happens along edges labeled by s in (σ,S) — if v(⋆)=v(σ(s).⋆), then v is in the (+1)-eigenspace of σ (s), and otherwise it is in its (-1)-eigenspace.

All in all, given a not necessarily transitive action σ of _2^m on X, its orbits have Fourier bases as in Definition <ref> (extended to being zero on all vertices outside of the orbit), and  the disjoint union of these will be called  the Fourier basis of σ. 
Note that since the bijections between the orbits and _2^k can be chosen in many ways, the Fourier basis of σ is determined only up to the sign of each basis vector. 





 §.§ Procedural sampling according to permutation strategies

Let σ S∪{}→(2n) be a permutation strategy for , and recall Remark <ref> and the notation therein. 
By Definition <ref>, σ(S_x∪{}) consists of commuting involutions. 
Therefore, the group generated by σ(S_x∪{}) is a quotient of _2^S_x∪{} — this can be acquired by linearly extending the map 1_↦σ() for every ∈ S_x∪{}, where 1_S_x∪{}→_2 is the indicator of . 
This means that 

    σ_x=σ|_S_x∪{}
 
induces an action of  _2^S_x∪{} on [2n]. 
Choose B_x ⊂ℂ^2n to be the Fourier basis of σ_x, as was defined in the Interlude Section <ref>.
Let B_x^-= B_x∩ W^-⊂ B_x be the collection of n eigenvectors in B_x on which σ() acts as (-1), namely all v⃗∈ B_x such that  σ().v⃗=-v⃗. These are the Fourier basis elements that flip along edges labeled by  in (σ_x,S_x∪{}). Then, B_x^- is an orthonormal basis of eigenvectors for ρ(S_x).


    Let  σ S∪{}→(2n) be a permutation strategy for , and ρ S→ U(W^-) the quantum strategy associated to σ. Then, the sampling procedure of γ S_xy→{0,1} according to ρ can be described  as follows:
    
        
  * Sample ⋆∈ [2n] uniformly at random. Then, it is part of an orbit O_x⊆ [2n] of σ_x (defined in (<ref>)) and O_y⊆ [2n] of σ_y.
        
  * Choose a pair v⃗∈ B^-_x and u⃗∈ B^-_y such that v⃗ is supported on O_x and u⃗ is supported on O_y, with probability 
        
    2⟨v⃗|u⃗⟩ ^2/|O_x∩ O_y|.

        
  * For ∈ S_x, let γ()=0 if ρ().v⃗=v⃗ and 1 otherwise. Similarly, for ∈ S_y, let γ()=0 if ρ().u⃗=u⃗ and 1 otherwise.
    
    We may use the notation γ∼σ instead of γ∼ρ to refer to sampling according to a permutation strategy.



The probability of sampling ⋆∈ [2n] which is in O_x∩ O_y is |O_x∩ O_y|/2n. Thus, the probability of sampling a specific pair v⃗∈ B_x^-,u⃗∈ B_y^- whose intersection of supports is O_x∩ O_y is
    
    |O_x∩ O_y|/2n·2⟨v⃗|u⃗⟩^2/|O_x∩ O_y|=⟨v⃗|u⃗⟩ ^2/n.

    This shows that the above sampling procedure provides the same distribution as the one in Claim <ref>.

    IS THIS PROOF REDUNDANT.
    We mainly need to show that the above is genuinely a sampling scheme, which boils down to the second step defining a probability distribution, i.e.,
    
    ∑_v⃗∈ B_x^-Supp(v⃗)=O_x 
    u⃗∈ B_y^-Supp(u⃗)=O_y⟨v⃗|u⃗⟩ ^2=|O_x∩ O_y|/2.

    This is because, once this is true, 
    the probability of getting a specific pair is still ⟨v⃗|u⃗⟩ ^2/n. To see that, 
    note that  the probability of sampling ⋆∈ [2n] which is in O_x∩ O_y is |O_x∩ O_y|/2n, and thus the probability of sampling a specific pair v⃗,u⃗ whose intersection of  supports is O_x∩ O_y is
    
    |O_x∩ O_y|/2n·2⟨v⃗|u⃗⟩^2/|O_x∩ O_y|=⟨v⃗|u⃗⟩ ^2/n.

    Note that if v⃗ and u⃗ do not have a common support, then their inner product is zero and the calculation remains valid.

Let 1_O_x be the indicator of O_x (it is actually a scalar multiple of a basis element in B_x, though not in B_x^-). Let u⃗· 1_O_x be the restriction of u⃗ to O_x, namely the function which agrees with u⃗ on O_x, and outputs 0 everywhere else. 
Since u⃗ is a Fourier basis vector supported on O_y, its non-zero entries are all of absolute value 1/√(|O_y|). Thus, ⟨u⃗· 1_O_x|u⃗· 1_O_x⟩=|O_x∩ O_y|/|O_y|. 
On the other hand, u⃗· 1_O_x is supported on O_x. 
Moreover, since  preserves O_x and O_y (by their definition), whenever ⋆∈ O_x∩ O_y, also σ().⋆∈ O_x∩ O_y. 
Thus, u⃗· 1_O_x is a function that flips along  labeled edges in (σ,S∪{}). All in all, we have 

    u⃗· 1_O_x=∑ _v⃗∈ B_x^-Supp(v⃗)=O_x ⟨v⃗|u⃗· 1_O_x⟩_⟨v⃗|u⃗⟩v⃗,

    and in particular 
    
    u⃗· 1_O_x^2=∑ _v⃗∈ B_x^-Supp(v⃗)=O_x ⟨v⃗|u⃗⟩^2.

    Summing up over the |O_y|/2 many u⃗∈ B^-_y which are supported on O_y, we deduce the claim.











 §.§ Tailored games

 As permutation strategies allow us to translate finitely described strategies of subgroup tests to quantum strategies of non-local games, the soon to be defined Z-aligned permutation strategies of tailored non-local games will allow us to move in the other direction. It is harder to motivate our choices beforehand in this case, but we hope that Section <ref>  clarifies what may seem cryptic in this section. We begin by restricting the family of non-local games that we focus on — for the reader familiar with linear constraint system games, we note that this family is a generalization of them.
 

Colloquially,  a tailored non-local game is one where D_xy reads part of γ, and decides according to this partial view which parity checks to apply on the whole of γ.[We considered calling such games controlled linear, since it is more informative. But, since conditionally linear is a term used in MIPRE,Tailored_MIPRE, and terms containing linear are generally overused, we decided to use a less informative notion.]

Formally,
   a tailored non-local game  is equipped with an extra structure, described shortly, and its decision functions D_xy behave canonically with respect to this extra data.
    Instead of a single length function ℓ,  has two length functions ℓ^ V→ℕ and ℓ^ V→ℕ, and ℓ=ℓ^+ℓ^. Before, the length function described the size of the formal generating set at each vertex. 
    Now, the formal set of generators S_x at x∈ V will be a disjoint union of the sets S_x^ and S_x^, where S_x^={^x,,i| 1≤ i≤ℓ^(x)} and S_x^={^x,,i| 1≤ i≤ℓ^(x)}. The elements of  S_x^ are  called readable variables at x∈ V and the elements of S_x^ the linear or unreadable variables at x. 
    Let S→{0,1} be the readability function, namely, the indicator of whether ∈ S is  readable or not.
      In addition,  is equipped with  a collection of controlled linear constraints functions L_xy that take as input a function γ^ S_x^∪ S_y^→_2, and outputs a collection of subsets of S_xy∪{}. Namely,
    
    L_xy_2^S_x^∪ S_y^→_2^_2^S_xy∪{}.

    The image of L_xy is interpreted as a collection of linear constraints that will be verified by the decision function.   
    Finally, the decision function D_xy(γ) behaves as follows: It restricts γ to the readable variables, namely looks at γ^=γ|_S_x^∪ S_y^ S_x^∪ S_y^→_2, and calculates L_xy(γ)=L_xy(γ^). Then, it extends γ such that γ()=1. Finally, for every  α∈ L_xy(γ), we have α S_xy∪{}→_2,  and D_xy verifies that 
    
    ⟨α,γ⟩=∑_∈ S_xy∪{}α() ·γ()=0.
 
    Namely, L_xy(γ) consists of linear constraints that γ needs to satisfy. 



    Though being tailored  seems to be quite a restrictive form for a non-local game, every game can be tailored in a trivial manner. First, all variables are declared readable, namely ℓ^=ℓ and ℓ^=0. Then, if the decision function D_xy decided to accept γ according to the original game, then it lets L_xy(γ) be empty (and thus all linear conditions will be satisfied regardless of what γ is). And, if D_xy decided to reject γ according to the original game, 
    then it chooses L_xy(γ) to contain the singleton {} as the single subset appearing in  L_xy. Note that {} represents the linear equation 1·γ()=0, which is 1=0, and thus cannot be satisfied. 


The last remark raises the question: What have we gained by defining tailored non-local games, if any game can be tailored in a straightforward manner? 


   A permutation strategy σ for a tailored non-local game  is said to be Z-aligned if the assignments  σ() for readable variables act on each point in [2n] either like the identity  or like σ(). This is equivalent to ρ() being a diagonal matrix when presented according to  the standard basis B^- of W^-.[ The standard basis is usually called the Z-basis in quantum information theory, and thus the name Z-aligned strategies.]  



    The classical permutation strategies described in Example <ref> are Z-aligned. But, one can construct permutation strategies that induce a classical strategy in the usual sense (i.e., whose images are all commuting) without being Z-aligned. 

It is clearer now why the way one tailors a non-local game matters: The existence of a perfect Z-aligned permutation strategy for the game  depends on it. Let us demonstrate this with binary linear constraint system (LCS) games, and specifically with the magic square game. For  a full description of  the magic square game we refer to <cit.> or <cit.>, and for a general  introduction to LCS games see <cit.>. 

[Linear constraint system games]
   Let A be an m× n matrix with _2 coefficients, and let b⃗ be a column vector in _2^m. Classically, such a pair defines a system of linear equations Ax⃗=b⃗ over _2.  It also defines a certain non-local game (A,b⃗) which is the quantum counterpart of this classical system of equations. Let us provide a dramatization of this game, in the spirit of Remark <ref>. In  (A,b⃗), the referee chooses a random linear constraint in Ax⃗=b⃗ (i.e., a row) and sends it to one of the players, and chooses  a random variable that appears  in the chosen constraint (i.e., a column whose intersection with the chosen row is non-zero) to the other player. It then expects the row player to respond with an assignment to the variables appearing in this constraint, and from the column player to respond with an assignment to its single variable. The referee declares this to be a winning condition if the row player's assignment satisfied the constraint, and both players agree on the value of the variable given to the column player. 
   The column player is thought to hold a global assignment and answer according to it, while the row player is assumed to have a satisfying assignment to each constraint — intuitively, by  checking consistency between the players, the referee verifies that the global assignment of the column player indeed satisfies all the constraints. 
   
   Let us define this game rigorously. 
   The vertices in the underlying graph of (A,b⃗) will be indexed by the rows and columns of the matrix A, namely {r_i| i∈ [m]} and {c_j| j∈ [n]}. There is an edge between a row r_i and a column c_j if and only if A_ij=1. The length of every column vertex is 1, and we denote by _j the variable associated with the j^ th column c_j. The length of each row vertex is the number of 1's in the row, and we associate variables S_r_i={_ij'| A_ij'=1} to every row r_i. The decision function D_r_ic_j gets as input an assignment γ to _j and {_ij'| A_ij'=1}, and accepts if and only if 

    ∑_j' A_ij'=1γ(_ij')=b_i   and  γ(_j)=γ(_ij).

Though for our discussion the distribution μ over edges in this game is not important, one can consider the following standard sampling scheme: 1) Choose a row uniformly at random. 2) Choose a uniform column out of the support of the chosen row.

Let us describe a non-trivial tailoring of  (A,b⃗). First, all variables are unreadable, namely ℓ^=0 and ℓ^=ℓ. Given that the edge r_ic_j was sampled, the system L_r_ic_j will consist of two checks, which are derived from 
(<ref>):[note that since there are no readable variables, L_r_ic_J is constant.] 

    α_ consistency()   =
        0    ≠_j,_ij
    
        1    otherwise
    α_ linear()   =
        0    =_j
    
        1    =_ij'∈ S_r_i
    
        b_i    =

Then, α_ consistency will force D_r_ic_j to check that γ(_j)=γ(_ij), and α_ linear will force it to check that ∑_j' A_ij'=1γ(_ij')=b_i, and thus it acts the same way as before. 


   The difference between the above tailored form of (A,b⃗) and the one suggested in Remark <ref> may seem technical. But, here all the variables are unreadable, and in the version of Remark <ref> version all variables are readable. 
   If all variables of a tailored game are readable, for it to have a perfect  Z-aligned permutation strategy is the same as having a perfect classical strategy — which in turn is the same as for the linear system Ax⃗=b⃗ to have a solution. 
   But, when all the variables are unreadable, there could be a perfect Z-aligned permutation strategy without Ax⃗=b⃗ having a solution. 
   For example, the perfect strategy for the Mermin–Peres magic square game (see for example <cit.>) can be derived from a permutation strategy (see our companion paper <cit.>), and since there are no readable variables in this case,  it is automatically Z-aligned.
   Furthermore, the magic square game has no perfect classical strategy. This shows that if we care about perfect Z-aligned permutation strategies, the way we tailor a game matters.







§ MAIN THEOREM II: TESTS ASSOCIATED WITH TAILORED GAMES


The goal of this section is to define a mapping from tailored non-local games to subgroup tests such that: 

    
  * Completeness: Perfect Z-aligned permutation strategies that commute along edges for the tailored game are translated to perfect finitely described strategies of the subgroup test.
    
  * Soundness: Almost perfect finitely described strategies for the subgroup test are close to almost perfect Z-aligned permutation strategies for the tailored non-local game. 
 

Throughout this section,  is a tailored non-local game with underlying graph G=(V,E),  length functions ℓ^,ℓ^ V→ with ℓ=ℓ^+ℓ^ and Λ =max_x∈ V(ℓ(x)), formal sets of generators S_x=S_x^∪ S_x^ for every vertex x∈ V, where S_x^={^x,,i| 1≤ i≤ℓ^(x)} and S_x^={^x,,i| 1≤ i≤ℓ^(x)}, a readability function S→{0,1} where S=⋃_x∈ V S_x, a distribution μ over  edges E, and decision functions D_xy_2^S_xy→_2 for every edge xy∈ E with a controlled linear constraints map L_xy_2^S_x^∪ S_y^→_2^_2^S_xy∪{}. 




    The synchronous subgroup test =() associated with the tailored non-local game  is defined as follows: The set of generators S̃ in the test  is S∪{}. The set  Q̃ which parametrizes the challenges of  is the edge set E of , and the distribution μ̃ over challenges is μ of . The decision function D̃_xy works as follows (and the subsets K̃_xy can be derived from this description): Given a subgroup H of (S̃), 
    
        
  *  It verifies that
    
    ∉ H  ;  ^2∈ H  ;  ∀∈ S_xy   [,]∈ H.

    
  *  It verifies that
    
    ∀∈ S_xy      ^2∈ H,
    ∀,'∈ S_x      [,']∈ H,
    ∀,'∈ S_y      [,']∈ H.

        
  *  
    It verifies that 
    
    ∀∈ S_x^∪ S_y^   ∈ H  or  ∈ H.

        
  *  If H passed all the previous checks, then the following function γ̃^ S_x^∪ S_y^→_2 can be calculated: γ̃^()=0 if ∈ H and γ̃^()=1 if ∈ H. After recovering this γ̃^, the subset 
        
    L̃_xy(H)=L_xy(γ̃^)

        can  be calculated.
        So, the decision function verifies that 
        
    ∀α∈L̃_xy(H)   ^α()·∏_∈ S_x^α()·∏_∈ S_y^α()∈ H,

          where the product is according to some pre-fixed ordering of S_x and S_y.
    
  
    If any of the above checks did not go through, D̃_xy rejects H, and otherwise it accepts it.



    Let us motivate the checks that the decision function D̃_xy is applying on H. Let σS̃→(X) be a finitely described strategy. If σ always passes clause:Check_1_in_associated_test, it means that  acts as a fixed point free central involution on X, which is part of the requirement of a permutation strategy  for  (see Definition <ref>). If σ always passes clause:Check_2_in_associated_test, then the images of all other variables in S̃ are involutions, and for any fixed x∈ V, σ(S_x) commutes, which are again requirements for σ to be a permutation strategy of . If σ always passes clause:Check_3_in_associated_test, then the readable variables always act as either  or , which implies σ is a Z-aligned permutation strategy for  (see Definition <ref>). So, the goal of the first three Checks is to force any finitely described strategy of  to be a permutation Z-aligned strategy for . Lastly, clause:Check_4_in_associated_test verifies “the same”  linear relations as , which means that the winning probability of σ  against   and  are similar.


We are ready to formulate our main theorem:

    Let  be a tailored non-local game with max_x∈ V (ℓ(x))=Λ>0, and let =() be its associated synchronous subgroup test, as in Definition <ref>. 
    
        
  *  Completeness: If  has a perfect Z-aligned permutation strategy that commutes along edges, then  has a perfect finitely described strategy.
        
  *  Soundness: If  has a finitely described strategy σ with (,σ)≥ 1-, then there exists a quantum strategy ρ such that (,ρ)≥ 1-CΛ ^4· 2^6Λ, where C>0 is a universal constant.
    

The next section is devoted to the proof of Theorem <ref>.  


    There exists a polynomial time algorithm that takes as input (the encoding of) a Turing machine M and outputs (the encoding of) a tailored non-local game _M such that:

    
  *  Sampling xy∈ E according to  μ and evaluating  D_xy(·) from the encoding of the  game _M  can be done in time poly(|M|), where |M| is the bit-length of the encoding of M.
    
  *  If M halts, then _M has a perfect  Z-aligned permutation strategy that commutes along edges.
    
  *  If M never halts, then ^*(_M)≤1/2.


Theorem <ref> is proved in <cit.>.

   The Aldous–Lyons Conjecture <ref> has a negative solution.


The idea is as follows. Given a Turing machine M, we can define  the tailored non-local game  _M calculated in Theorem <ref>, and then calculate the associated subgroup test (_M)  from Definition  <ref>. In the next paragraph, the following two facts are proved:

    
  * If M halts, then (_M) has sofic value 1.
    
  * On the other hand, if M does not halt, then the sofic value of (_M) is smaller than 1-λ(M), where λ(M)=2^-c|M|^c for some universal constant c>0 independent of M. 
 
 Recall that by Theorem <ref>, there is a computable sequence approaching _((_M)) from above and another computable sequence approaching _((_M)) from below. If the Aldous–Lyons Conjecture <ref> has a positive solution, then these values are the same, and as Corollary <ref> states, one can approximate _((_M)) up to any predetermined additive error θ. As λ(M) can be computed directly from |M|, one can choose θ=λ(M)/2, and thus deduce whether _((_M))<1 or _((_M))>1-λ(M). This in turn allows one to decide whether M halts or not. Since the Halting Problem is undecidable, this is a contradiction, which implies that Conjecture  <ref> must have a negative solution.



Let us prove the two bullets above. Since the running time of the verification procedure of _M bounds Λ=max_x∈ V(ℓ(x)), we can use clause:1_tailored_MIP* of Theorem <ref> to deduce that Λ=poly(|M|). 
    If M halts, then by clause:2_tailored_MIP* of Theorem <ref> and clause:1_in_values_associated_test of Theorem <ref>, we have _((_M))=1.
    If M does not halt, then by clause:3_tailored_MIP* of Theorem <ref>, we have ^*(_M)≤1/2. Hence, by clause:2_in_values_associated_test of Theorem <ref>,
    we have
    
    1/2≥ 1-CΛ^4· 2^6Λ(1-_((_M))).

    Rearranging this inequality, we get 
    
    _((_M))≤ 1-1/2CΛ^4· 2^6Λ.

    The quantity 1/2CΛ^4· 2^6Λ can be bounded from below by λ(M)=2^-c|M|^c, where c>0 is a universal constant that depends on the constants implicit in the notation  Λ=poly(|M|) (guaranteed by Theorem <ref>), as well as the constant C>0 appearing in Theorem <ref>. This finishes the proof.







§ PROVING THEOREM <REF>






 §.§ Proving the completeness in Theorem <ref>


This is the easy direction, we have done most of the work towards it, and here we use the notation that we have established before. Recall that S̃=S∪{}. Let σ S∪{}→(2n) be a perfect Z-aligned permutation strategy that commutes along edges for the tailored non-local game . Further recall that the finitely described IRS Φ(σ) samples a subgroup H of  as follows — it takes a uniform ⋆∈ [2n] and outputs H=(σ,⋆). By Definition <ref>,

    d_H(σ(),)=1   ;   σ()^2=  ;  ∀∈ S   σ()σ()=σ()σ().

Therefore, for every ⋆∈ [2n], 

    ∉(σ,⋆)   ;  ^2∈(σ,⋆)  ;  ∀∈ S   [,]∈(σ,⋆),

and when Φ(σ) runs against  it  always passes clause:Check_1_in_associated_test. Also by  Definition <ref>,

    ∀∈ S σ()^2=  ;  ∀ x∈ V, ∀,'∈ S_x   σ()σ(')=σ(')σ().

Therefore, for every ⋆∈ [2n],

    ∀∈ S ^2∈(σ,⋆)  ;  ∀ x∈ V, ∀,'∈ S_x   [,']∈(σ,⋆),

and when Φ(σ) runs against  it always passes clause:Check_2_in_associated_test.
By Definition <ref>, for every ⋆∈ [2n] we have 

    ∀∈^-1(1)   σ().⋆=⋆  or  σ().⋆=σ().⋆.

Namely, for every ⋆∈ [2n],

    ∀∈^-1(1)   ∈(σ,⋆)  or  ∈(σ,⋆),

and when Φ(σ) runs against  it always passes clause:Check_3_in_associated_test.

We are left to show that Φ(σ) always passes clause:Check_4_in_associated_test.
To that end, we need a couple of claims.
    The first is a well known and commonly used observation in quantum information theory. For the convenience of the reader, we provide a proof.

    Let S be a finite set, ρ S→ U(n) such that ρ(S) commutes[This claim can be generalized so that a weaker assumption is used. See Section 3 of <cit.>.] and all the images are involutions.  Then, given a fixed α S→_2, we have ∑_∈ Sα()γ()=0  with probability 1 when  γ∼ρ if and only if ∏_∈ Sρ()^α()=. Namely,

    _γ∼ρ[⟨α,γ⟩=0]=1 ∏_∈ Sρ()^α()=.



Assume without loss of generality that α is fully supported. Otherwise, we can focus on ρ|_Supp(α) and proceed in the same manner. 
    As before, we can write ρ()=_0^-_1^, where ^_0 is the projection on the (+1)-eigenspace of ρ(), and _1^ is the projection on its (-1)-eigenspace.  By Definition <ref>, 

    _γ∼ρ[∑_∈ Sγ()=0]   =∑_γ S→_2
    ∑γ()=0τ(∏_∈ S_γ()^).

Note that ^_0+^_1=, and thus 
∑_γ S→_2τ(∏_∈ S_γ()^) is always 1. 
Therefore, 

    τ(∏_∈ Sρ())   =τ(∏_∈ S(^_0-^_1))
       =∑_γ S→_2
    ∑γ()=0τ(∏_γ()^) -∑_γ S→_2
    ∑γ()=1τ(∏_γ()^)
       =2·∑_γ S→_2
    ∑γ()=0τ(∏_γ()^)-1
       =2·_γ∼ρ[∑_∈ Sγ()=0]-1.

Thus,  ∏ρ()= if and only if τ(∏ρ())=1 if and only if  _γ∼ρ[∑_∈ Sγ()=0]=1, which finishes the proof.



    Let σ S∪{}→(2n) be a Z-aligned permutation strategy for a tailored game , and let ρ be the quantum strategy associated with σ (as in Definition <ref>). 
    Then, the restriction of  γ∼ρ to the readable variables in S_xy (which we denoted by γ^) depends only on the vertex ⋆∈ [2n] sampled in the beginning of the procedure outlined in Claim <ref>.
    Furthermore, if we denote the sampled L_xy(γ) by L_xy(⋆), emphasizing its dependence on the sampled vertex, then L̃_xy((σ,⋆))=L_xy(⋆), where L̃_xy was defined in (<ref>).


    Let ⋆∈ [2n] be some fixed vertex, O_x be the σ_x orbit of ⋆, and ∈ S_x be a readable variable. Since σ is Z-aligned, either σ().⋆=⋆ or σ().⋆=σ().⋆. Now, as discussed in Section <ref>, σ acting on  O_x is equivalent to an action of _2^k on itself. Furthermore, checking whether  v⃗∈ B^-_x that is supported on O_x is in the (+1) or (-1) eigenspace of σ() depends only on whether v⃗ flips along an -labeled edge in (σ,S) for some vertex in O_x. Since v⃗∈ B^-_x, it must flip along -labeled edges. So, if σ().⋆=⋆ then σ()v⃗=v⃗ and γ()=0, and if σ().⋆=σ().⋆ then σ()v⃗=-v⃗ and γ()=1. 
    Since this was independent of the specific v⃗ that we considered, the first part of the claim is deduced.  
   This proves that γ̃^()=γ^(), where γ̃^() was defined in clause:Check_4_in_associated_test of Definition <ref>, and therefore L̃_xy((σ,⋆))=L_xy(γ̃^)=L_xy(γ^)=L_xy(⋆) which is the second part of the claim.


We can now come back to our proof. Let us, in the spirit of previous notations, denote  σ_xy=σ|_S_xy∪{}. Let ⋆∈ [2n], and let O_xy be the orbit of ⋆ under σ_xy. Since σ commutes along edges (see Definition <ref>), σ_xy induces an action of _2^S_xy∪{}. Since (σ_xy) is commuting, the action of readable variables in O_xy is constant, namely if a readable variable acts as  on some ⋆∈ O_xy, it will act that way on all ⋆∈ O_xy, and similarly for acting as . Furthermore, by Claim <ref>, the linear constraints sampled in the associated game L̃_xy at a specific vertex ⋆∈ O_xy are the same as the ones sampled by the tailored non-local game.
Joining these observations, there is a set of constraints L such that for every ⋆∈ O_xy we have  L̃_xy((σ,⋆))=L_xy(⋆)=L.  So, we can focus on the orbit O_xy, and study the corner of ρ, which was the quantum strategy induced by σ, with respect to the projection ^O_xyℂ^2n→ℂ^O_xy. 
Since ρ is a perfect strategy, for every γ that was sampled (according to Claim <ref>) by first choosing ⋆∈ O_xy, and for every α∈ L, we have ∑α()γ()=0. 
By Claim <ref>,  this means that ∏ρ()^α() is the identity when restricted to functions supported on O_xy that flip along -labeled edges. By Claim <ref>, we can deduce that ∏σ()^α() acts as the identity on O_xy. Since this was true for every α in L, and L=L̃_xy((σ,⋆)) for every ⋆∈ O_xy, we can conclude that σ always passes clause:Check_4_in_associated_test when run against . All in all, σ is a perfect strategy for , proving the completeness clause of Theorem <ref>.



 §.§ Proving the soundness in Theorem <ref>


The plan is as follows:

    
  * Recall that S̃= S∪{}.  Given a finitely described strategy σS̃→(n) for , with (,σ)≥ 1-, we  perturb it bit by bit until it passes with probability 1 clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test from the definition of  (while potentially worsening the probability of passing  clause:Check_4_in_associated_test). 
    
  * Then, we can bound from below the value of this perturbed strategy using the mechanisms of Section <ref>. 
    
  * Since the resulting perturbed strategy satisfies the first three checks of , it is a Z-aligned permutation strategy for . For such strategies, we prove that their value against  induces a lower-bound on their value against . This will conclude the proof.
   

The following three propositions are the manifestation of the above plan. We abuse notations and denote by μ the marginal distribution on vertices of 𝒢, namely

    ∀ x∈ V   μ(x)=1/2∑_xy∈ Eμ(xy).

Also, recall the notion of the significance function _  associated with a test , defined in (<ref>), and the notion of the edit distance d^_ edit (<ref>) between finitely described IRSs. Finally, recall that  Λ=max_x∈ V(ℓ(x)), and assume Λ>0.

   Let σS̃→(n) be a finitely described strategy  for , with (,σ)≥ 1-. Denote by _x the losing probability in  given that an edge containing x∈ V was sampled as the challenge — note that _x∼μ[_x]=. 
   Then, there is a strategy σ'S̃→(2m), where m= ⌈n/2⌉, that passes with probability 1 clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test in the definition of the associated test , and which is close to σ in the following sense:
   
    d_H(σ(),σ'())≤  C_0
    ∀ x∈ V, ∈ S_x      d_H(σ(),σ'())≤  C_0· 2^2Λ·Λ^3(+_x).

    for some universal constant C_0>0.



     Every  finitely described strategy 
 σS̃→(2n)   for  that passes with probability 1 clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test in the definition of the associated test  is a Z-aligned permutation strategy for  and vice versa. Moreover, there is a universal constant C_1>0 such that  if  (,σ)≥ 1-, then (,ρ)≥ 1-C_1· 2^2Λ where ρ is the quantum strategy associated with σ (Definition <ref>). 


    Let  be the test associated with the tailored non-local game .  Then, there is a universal constant C_2>0, such that the significance _τ̃ associated with the test   satisfies
    
    _()   ≤ C_2· 2^2Λ,
    ∀ x∈ V, ∈ S_x   _()   ≤μ(x)· C_2· 2^2Λ.



    We did not try to optimise the parameters in the three preceding propositions. Actually, we decided to choose a simpler version of the associated synchronous test (Definition <ref>) that provides an exp(Λ) deterioration in the value, as seen in Proposition <ref>. We have variations of tailored games and the associated synchronous test which allow for only a poly(Λ) deterioration in the value, but we could not find any reduction that is independent of Λ.


Now, we can deduce the soundness clause of Theorem <ref> as a corollary of the preceding propositions.


    Let σS̃→(n) be the given finitely described strategy with value ≥ 1-. By Proposition <ref> and Proposition  <ref>, there is a strategy σ'S̃→(2⌈n/2⌉) passing clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test with probability 1, and such that the significance weighted distance between σ and σ' is
    
    d^_(σ,σ')   =∑_s∈S̃_(s)· d_H(σ(s),σ'(s))
       =_()d_H(σ(),σ'())+∑_x∈ V∑_∈ S_x_()d_H(σ(),σ'())
       ≤ C_2· 2^2Λd_H(σ(),σ'())+∑_x∈ V∑_∈ S_xμ(x)· C_2· 2^2Λ d_H(σ(),σ'())
       ≤ C_0C_2·2^2Λ+∑_x∈ V|S_x|·μ(x)· C_2· 2^2Λ· C_0· 2^2Λ·Λ^3(+_x)
       ≤ C_0C_2·2^2Λ+ C_0C_2Λ ^4 · 2^4Λ(∑_x∈ Vμ(x)(+_x))__x∼μ[+_x]=2
       ≤ 3C_0C_2Λ ^4· 2^4Λ.

    Therefore, by Claim <ref>, 
    (,σ')≥ 1--3C_0C_2Λ ^4· 2^4Λ≥ 1-4C_0C_2Λ ^4· 2^4Λ.
 By applying Proposition <ref> on σ',  its associated quantum strategy ρ' satisfies  
    
    (,ρ')≥ 1-4C_0C_1C_2Λ ^4· 2^6Λ.

By    choosing C=4C_0C_1C_2, we deduce the claim.


We are left to prove Propositions <ref>, <ref> and 
 <ref>. This is done in a reverse manner.




   Note that  participates in most of  the checks in  regardless of the specific challenge. 
   In clause:Check_1_in_associated_test, it appears 3+2|S_xy| times, and since |S_xy|≤ 2Λ, this is bounded by 3+4Λ. 
   In clause:Check_2_in_associated_test it is not used. 
   In clause:Check_3_in_associated_test it is used at most |S_xy| times, which is bounded from above  by 2Λ. 
   Lastly, the number of times   appears in L_xy(γ) is bounded by the number of subsets of S_xy∪{} that contain it, which is 2^|S_xy|≤ 2^2Λ.   All in all, the significance of  is at most (3+4Λ)+2Λ+2^2Λ≤ 4· 2^2Λ. 
    
    Now, a variable  in S_x appears only if we sampled an edge in E with x as one of its endpoints. If this happened, it will appear twice in clause:Check_1_in_associated_test, 2+2|S_x| times in clause:Check_2_in_associated_test, at most twice in clause:Check_3_in_associated_test, and at most 2^|S_xy∪{}|-1 times in clause:Check_4_in_associated_test. Again, |S_x|≤Λ and |S_xy|≤ 2Λ, and thus  appears at most 2+2+2Λ+2+2^2Λ≤ 3· 2^2Λ times when an edge containing x is sampled. This means that the significance of  is at most μ(x)· 3· 2^2Λ. Choosing C_2=4 completes the proof.




Let σS̃→(2n) be a finitely described strategy that passes clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test of  with probability 1. Then, 
from perfectly passing clause:Check_1_in_associated_test and  clause:Check_2_in_associated_test, σ is a permutation strategy, and by perfectly passing clause:Check_3_in_associated_test  it is Z-aligned. The other direction was proved in the completeness analysis in Section <ref>. We are going to prove that 

    (,ρ)≥ 1- 2· 2^2Λ(1-(,σ)),
 and thus C_1=2 satisfies the claim.

We start by giving expressions for the success probability of σ in , and of the associated ρ in 𝒢 respectively. 
Fix an edge xy∈ E, and denote by σ_x=σ|_S_x∪{} and σ_y=σ|_S_y∪{} the appropriate restrictions of the permutation strategy σ.  The intersection of orbits O_x∩ O_y, where O_x is an orbit of σ_x and O_y is an orbit of σ_y, partitions [2n] into disjoint sets. Fixing a pair of orbits O_x,O_y such that O_x∩ O_y≠∅,  by Claim <ref>, there is a set L_xy^O_x,O_y  of subsets of  S_xy∪{} such that 

    ∀⋆∈ O_x∩ O_y   L^O_x,O_y_xy=L_xy(⋆)=L̃_xy((σ,⋆)).
 
For α∈𝔽_2^S_xy∪{}, define

    ^α=∏_∈ S_x∪{}^α()   and    ^α=∏_∈ S_y^α().

Let Υ̃^O_x,O_y(α) be the set of all vertices ⋆∈ O_x∩ O_y such that σ(^α).⋆≠σ(^α).⋆, namely all vertices in the intersection of orbits that do not satisfy the subgroup test constraint induced by α.  Let 

    Υ̃^O_x,O_y_xy = ⋃_α∈ L^O_x,O_y_xyΥ̃^O_x,O_y(α),

and let 

    Υ̃_xy=⋃_O_x,O_yΥ̃^O_x,O_y_xy,

where the union is over all orbits of σ_x and σ_y that have a non-empty intersection.
Then Υ̃_xy is the set of all  ⋆∈ [2n] such that D̃_xy( (σ,⋆))=0, hence

    (,σ)=1- _xy∼μ|Υ̃_xy|/2n =1-_xy∼μ_⋆∈ [2n]_xy(⋆)   ,

 where
 
    _xy(⋆)=|Υ̃_xy^O_x,O_y|/|O_x∩ O_y|

 and O_x and O_y are the unique σ_x and σ_y orbits for which ⋆∈ O_x∩ O_y.
 
Recall that according to the procedural sampling described in  Claim <ref>, any pair v⃗∈ B_x^-, u⃗∈ B^-_y of Fourier basis elements supported on σ_x-orbit O_x and σ_y-orbit O_y respectively, is sampled with probability 2⟨v⃗|u⃗⟩^2/|O_x∩ O_y| (conditional on the sampled ⋆ being in O_x∩ O_y), and it induces an “assignment” γ S_xy∪{}→_2 for which γ()=1.
 Let Υ^O_x,O_y(α) be the set of all such (v⃗,u⃗) for which 
  γ does not satisfy a specific linear constraint α∈_2^S_xy∪{}, namely ∑_∈ S_xy∪{}γ()α()≠ 0. 
  Let  
 
    Υ^O_x,O_y_xy = ⋃_α∈ L^O_x,O_y_xyΥ^O_x,O_y(α)

 be the collection of Fourier basis pairs (v⃗,u⃗)∈ B_x^-× B_y^- supported on O_x,O_y, such that the assignment γ induced by the pair is rejected by D_xy. 
 Finally let 
 
    Υ_xy=⋃_O_x,O_yΥ^O_x,O_y_xy⊆ B^-_x× B^-_y,

 which is exactly the collection of possible sampled pairs by the quantum strategy ρ induced by σ, given that the edge xy was asked,  that produce an assignment γ satisfying D_xy(γ)=0. 
 Hence,

    (,ρ)=1-_xy∼μ∑_(v⃗,u⃗)∈Υ_xy⟨v⃗|u⃗⟩^2/n= 1-_xy∼μ_⋆∈ [2n]δ_xy(⋆) ,

where 

    δ_xy(⋆) =∑_(v⃗,u⃗)∈Υ^O_x,O_y_xy2⟨v⃗|u⃗⟩ ^2/|O_x∩ O_y|

and O_x,O_y are the unique orbits for which ⋆∈ O_x∩ O_y.
Observe the following claim.

    For every α∈𝔽_2^S_xy∪{},  σ_x-orbit O_x and σ_y-orbit O_y such that O_x∩ O_y≠∅, we have

    ∑_(v⃗,u⃗)∈Υ^O_x,O_y(α)⟨v⃗|u⃗⟩ ^2 ≤ 1/2|Υ̃^O_x,O_y(α)|  .


Let us show how to deduce our conclusion assuming the  validity of this claim, and then we will address the claim itself.
Given ⋆∈ O_x∩ O_y, we have

    δ_xy(⋆)     (<ref>)=∑_(v⃗,u⃗)∈Υ^O_x,O_y_xy2⟨v⃗|u⃗⟩ ^2/|O_x∩ O_y|
          ≤ ∑_α∈ L_xy^O_x,O_y ∑_(v⃗,u⃗)∈Υ^O_x,O_y(α)2⟨v⃗|u⃗⟩ ^2/|O_x∩ O_y|
        Claim <ref>≤∑_α∈ L_xy^O_x,O_y |Υ̃^O_x,O_y(α)|/|O_x∩ O_y| .

Now, for every α∈ L_xy^O_x,O_y,  we have by (<ref>) that Υ̃^O_x,O_y(α)⊆Υ̃^O_x,O_y_xy, which using the above analysis means 

    δ_xy(⋆)≤∑_α∈ L_xy^O_x,O_y |Υ̃^O_x,O_y_xy|/|O_x∩ O_y|≤ 2^2Λ+1·_xy(⋆),

where the last inequality uses (<ref>) and  the fact |L^O_x,O_y_xy|≤|_2^S_xy∪{}|≤  2^2Λ+1. 
Therefore,

    1-(𝒢,ρ)(<ref>)=_xy∼μ_⋆∈ [2n][δ_xy(⋆)]≤ 2· 2^2Λ_xy∼μ_⋆∈ [2n][_xy(⋆)](<ref>)=2· 2^2Λ(1-( 𝒯̃,σ)),

proving (<ref>) and thus the proposition. 




Let Γ̃^O_x,O_y(α) be the complement of Υ^O_x,O_y(α) in O_x∩ O_y, namely 

    Γ̃^O_x,O_y(α)= (O_x∩ O_y) ∖Υ̃^O_x,O_y(α).

Recall the notation ^α and ^α from (<ref>).
For ⋆∈ O_x∩ O_y, we have σ(^α).⋆∈ O_x and σ(^α).⋆∈ O_y by the fact O_x is a σ_x-orbit and O_y is a σ_y-orbit. If ⋆∈Γ̃^O_x,O_y(α), then by its definition we have ♢:=σ(^α).⋆=σ(^α).⋆, and ♢∈ O_x∩ O_y.  As σ(^α) and σ(^α) are involutions, we have σ(^α).♢=⋆=σ(^α).♢, which implies ♢∈Γ̃^O_x,O_y(α).  All in all, Γ̃^O_x,O_y(α) is invariant under the actions of ^α and ^α — graphically, if we add ^α and ^α labels to the Schrier graph of the action σ on [2n], then they agree on every ⋆∈Γ̃^O_x,O_y(α) and are contained in it.

For (v⃗,u⃗) ∈Υ^O_x,O_y(α)  we know that  σ(^α)v⃗ =±v⃗ while σ(^α)u⃗=∓u⃗. 
This means that for every ^α-labeled edge between vertices in Γ̃^O_x,O_y(α), v⃗ and u⃗ agree on one of its endpoints and evaluate to opposites on the other endpoint. 
This implies

    ⟨v⃗|u⃗⟩   =∑_⋆∈ O_x∩ O_yv⃗(⋆)u⃗(⋆)
       =∑_⋆∈Υ̃^O_x,O_y (α)v⃗(⋆)u⃗(⋆)+∑_⋆∈Γ̃^O_x,O_y(α)v⃗(⋆)u⃗(⋆)
       =∑_⋆∈Υ̃^O_x,O_y (α)v⃗(⋆)u⃗(⋆)+1/2∑_⋆∈Γ̃^O_x,O_y(α)v⃗(⋆)u⃗(⋆)+v⃗(σ(^α).⋆)u⃗(σ(^α).⋆)_=0
       =⟨v⃗· 1_Υ̃^O_x,O_y (α)|u⃗⟩,

where  1_Υ̃^O_x,O_y is the indicator function of the set Υ̃^O_x,O_y, and · is the pointwise product of functions.
Thus,

    ∑_(v⃗,u⃗)∈Υ^O_x,O_y(α)⟨v⃗|u⃗⟩^2   =∑_v⃗Supp(v⃗)=O_x(∑_u⃗
      (v⃗,u⃗)∈Υ^O_x,O_y(α)⟨v⃗· 1_Υ̃^O_x,O_y (α) |u⃗⟩^2)
       ≤∑_v⃗Supp(v⃗)=O_x(∑_u⃗Supp(u⃗)=O_y⟨v⃗· 1_Υ̃^O_x,O_y (α) |u⃗⟩^2)
       =(*),

and since 1_Υ̃^O_x,O_y(α) is supported on O_y, we have 

    ∑_u⃗Supp(u⃗)=O_y⟨v⃗· 1_Υ̃^O_x,O_y(α) |u⃗⟩^2=v⃗· 1_Υ̃^O_x,O_y (α)^2=|Υ̃^O_x,O_y (α)|/|O_x| .

Since there are |O_x|/2 many v⃗∈ B^-_x which are supported on O_x, we deduce that
(*)≤|Υ̃^O_x,O_y(α)|/2,
as desired.









































































































A short detour into (quite elementary) group stability results is needed for proving  Proposition <ref>.


 §.§ Interlude — Group stability


 [Almost involutions are close to involutions]
 Let ζ∈(X). Then,
there is τ∈(X) such that τ^2= and
d_H(ζ,τ)= d_H(ζ^2,). 


Let W={ x∈ X |ζ^2 .x=x}.
Note that the restriction ζ|_W is an involution W→ W.
Define 

    τ(x)=ζ(x)     x∈ W,
    
    x     x∉ W.

Then τ^2= and d_H(ζ,τ)=1-|W|/|X|=d_H(ζ^2,).








[Almost fixed point free are close to fixed point free]
        Let ζ∈(X) be an involution. Assume d_H(ζ,)≥ 1-. Then, there is an involution with no fixed points τ∈(Y), where |Y|=2·⌈|X|/2⌉, such that d_H(ζ,τ)≤ 2.


    Let W={x∈ X|ζ.x=x}. Since d_H(ζ,)=1-|W|/|X|, we can deduce that |W|≤|X|. If W is odd, add a vertex to it and make it even. Now that W is even sized, we can choose any perfect matching on it and define τ to be the involution induced by this perfect matching. Extend τ to the rest of X by letting it act as ζ on the vertices out of W. Then the resulting τ is an involution with no fixed points, and 
    
    d_H(ζ,τ)≤|W|+1/|X|≤ 2.



[Almost commuting involutions are close to commuting involutions]
    Given two involutions ζ,τ∈(X), there is an involution τ'∈(X) such that
    
    [τ',ζ]=   and   d_H(τ,τ')≤ d_H([τ,ζ],).

    
    
   Let δ=d_H([τ,ζ],), and let W={x∈ X |ζτ .x=τζ .x}. Then |W|= (1-δ)|X|. Moreover, for every x
    ∈ W, 
    
    ζτ.(ζ. x)   =ζ .(τζ .x)=ζ^2 τ .x=τ .x=τζ .(ζ .x)ζ .x∈ W;
    τζ. (τ .x)   =τ .(ζτ .x)= τ^2 ζ .x=ζ.x=ζτ .(τ .x)τ .x∈ W.

    Hence, W is invariant under the actions of ζ and τ. Thus, the following is well defined
    
    τ'(x)=τ(x)     x∈ W,
    
                x     x∉ W.

    We therefore can conclude that d_H(τ,τ')≤ 1-|W|/|X|=δ, and 
    
    ∀ x∈ W   τ'ζ.x_∈ W   =τζ. x=ζτ .x=ζτ'. x;
    ∀ x∉ W   τ'ζ. x_∉ W   = ζ .x=ζτ'.x,

    which implies [τ',ζ]= as required.
 
 
     Claim <ref> can be proved even without assuming that the permutations are involutions (while allowing both permutations to change and not only τ). This was originally proved in <cit.> and was later reproved with effective bounds in <cit.>.
 


The following is a special case of item 1 of Theorem 2 in <cit.>.
[Almost actions of finite groups are close to actual actions]
    Let G be a finite group, and let f G→(X) be a function. Assume that for every g,h∈ G, d_H(f(gh),f(g)f(h))≤. Then, there is a homomorphism φ G→(X) such that 
    
    ∀ g∈ G   d_H(f(g),φ(g))≤ |G|^2.



    Let x∈ X be a vertex such that 
    
    ∀ g,h∈ G   f(gh).x=f(g)f(h).x.

    Then, if O⊆ X is the orbit of x with respect to the action of (the group generated by) (f), then for every y∈ O we have f(gh).y=f(g)f(h).y. To prove that, it is enough to prove it for a single step, namely for  y=f(k).x for some k∈ G (the rest is by induction). So, for every g,h∈ G, we have
    
    f(gh).y=f(gh)f(k).x=f(ghk).x=f(g)f(hk).x=f(g)f(h)f(k).x=f(g)f(h).y,

    where all equalities are either f(k).x=y or (<ref>). This means that the restriction of f to its action on O induces a homomorphism from G to (O).
    
    We assumed for every g,h∈ G, that d_H(f(gh),f(g)f(h))≤. Therefore, there are at most |G|^2 many vertices z in X for which there is some g,h∈ G such that f(gh).z≠ f(g)f(h).z. So, the union of orbits as O consists of at least (1-|G|^2)|X| of the vertices in X. So, if we define φ(g).x=f(g).x whenever x satisfies (<ref>), and φ(g).x=x otherwise, then φ is an action of G it satisfies for every g∈ G, d_H(f(g),φ(g))≤ |G|^2.

 






 §.§ Proof of Proposition <ref>

Throughout this proof, we use the term triangle inequality for iterative applications of the Hamming distance's triangle inequality,  as was used  in the proof of Claim <ref>.


   Let σS̃→(n) be a finitely described strategy  for , with (,σ)≥ 1-. By clause:Check_1_in_associated_test, d_H(σ()^2,)≤. By Claim <ref>, there is a function σ_1S̃→(n) that agrees with σ on all ∈ S, but σ_1() is an involution and d_H(σ(),σ_1())≤. Again by clause:Check_1_in_associated_test, d_H(σ(),)≥ 1-, and by the triangle inequality d_H(σ_1(),)≥ 1-2. Let m=⌈n/2⌉. Then, by Claim <ref>, there is a function σ_2S̃→(2m) such that for every ⋆∈ [n]⊆ [2m], we have σ_2().⋆=σ_1().⋆=σ().⋆, and σ_2() is a fixed point free involution with d_H(σ_1(),σ_2())≤ 4. All in all, for every ∈ S∪{} we have d_H(σ_2(),σ())≤ 5.

   Now, let _x be the losing probability when an edge containing x is sampled in . Then, _x∼μ[_x]=. Furthermore, by clause:Check_2_in_associated_test, for every ∈ S_x we have d_H(σ()^2,)≤_x. Therefore, by the triangle inequality, d_H(σ_2()^2,)≤ d_H(σ_2()^2,σ()^2)+d_H(σ()^2,)≤ 10+_x. By Claim <ref>, there is a function σ_3S̃→(2m)  such that  σ_3()=σ_2(), and for every x∈ V and ∈ S_x both σ_3()^2= and d_H(σ_3(),σ_2())≤ 10+_x. Now, by clause:Check_1_in_associated_test, for every ∈ S_x we have d_H(σ()σ(),σ()σ())≤_x. Thus, by the triangle inequality, d_H(σ_3()σ_3(),σ_3()σ_3())≤ 30+2_x. Since σ_3() and σ_3() are involutions, we may apply Claim <ref> and find a function σ_4S̃→(2m) such that σ_4()=σ_3(), σ_4()σ_4()=σ_4()σ_4() and d_H(σ_4(),σ_3())≤ 30+2_x. Again by the triangle inequality, we deduce that d_H(σ(),σ_4())≤ 5, and that for ∈ S_x we have d_H(σ(),σ_4())≤ 40 +3_x.

   Recall that S_x={^x,i| i≤ℓ(x)} (ignoring readability). We extend σ_4 to _2^S_x in the following way: For ξ S_x→_2, let
   
    σ_4(ξ)=∏_i=1^ℓ(x)σ_4()^ξ(),

   where the product is ordered according to the index i. By clause:Check_2_in_associated_test, for every ,'∈ S_x we have 
    d_H(σ()σ('),σ(')σ())≤_x.

   Thus, by the triangle inequality, d_H(σ_4()σ_4('),σ_4(')σ_4())≤ 160+ 13_x. Note also that the images of σ_4 are involutions. Therefore, for every ξ_1,ξ_2 S_x→_2 we have 
   
    d_H(σ_4(ξ_1+ξ_2),σ_4(ξ_1)σ_4(ξ_2))≤ℓ(x)^2(160+ 13_x)≤Λ^2(160+ 13_x).

   By Claim <ref> applied to each _2^S_x individually, we have a function σ_5S̃→(2m) such that σ_5(S_x) commutes and consists only of involutions, and 
    d_H(σ_4(),σ_5())≤ |_2^S_x|^2 ·Λ^2(160+13_x)≤ 2^2Λ·Λ^2(160+13_x).
 Furthermore, because of the way σ_5 is constructed in the proof of Claim <ref>, by letting σ_5()=σ_4(), the image of  is still a central involution with no fixed points. So, σ_5 always passes clause:Check_1_in_associated_test and clause:Check_2_in_associated_test. 

   Since σ passes clause:Check_3_in_associated_test with probability _x when an edge with x∈ V  is sampled, we deduce that every readable variable ∈ S_x^ satisfies

    _⋆∈ [n][σ().⋆=⋆ or σ().⋆=σ().⋆]≥ 1-_x.

Hence, by the triangle inequality, 
   
    _⋆∈ [2m][σ_5().⋆=⋆ or σ_5().⋆=σ_5().⋆]   ≥ 1-_x-d_H(σ_5(),σ())-d_H(σ_5(),σ())
       ≥ 1-_x-5-2^2Λ·Λ^2(160+13_x)
       ≥ 1-2^2Λ·Λ^2(165+14_x).

   By the fact that σ_5(S_x) commutes for every x∈ V,  the property σ_5().⋆=⋆ implies σ_5().♢=♢ for every ♢∈ O_x, where O_x is the orbit of ⋆ induced by σ_5(S_x) (which we usually denoted as (σ_5)_x beforehand). Since σ_5() commutes with all images, the same is true for the property σ_5().⋆=σ_5().⋆.  
   Hence, in each orbit of σ_5(S_x), either all vertices ⋆ satisfy σ_5().⋆=⋆ or σ_5().⋆=σ_5().⋆ or none of them. 
   We can thus define a function σ_6 that agrees with σ_5 on the orbits in which all readable variables ∈ S_x^ act appropriately, and on the other orbits we just define for every ∈ S_x that σ_6().⋆=⋆ (note that we keep  acting the same no matter what). So, we have 
   
    ∀ x∈ V, ∈ S_x    d_H(σ_6(),σ_5())   ≤ |S_x^|· 2^2Λ·Λ^2(165+14_x)
       ≤ 2^2Λ·Λ^3(165+14_x),

and now σ_6 always passes clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test. By the triangle inequality, d_H(σ_6(),σ())≤ 5 and for every ∈ S_x we have

    d_H(σ_6(),σ())   ≤ 2^2Λ·Λ^3(165+14_x)+2^2Λ·Λ^2(160+13_x)
       +30+2_x+10+_x+5
       ≤ 2^2ΛΛ ^3(370+30_x).

Choosing σ'=σ_6 and C_0=370, we deduce the claim.



    How better can the parameters in Propositions <ref>, <ref> and <ref> be? We know how to  make the parameters in Proposition <ref>  polynomial in Λ by implementing a Hadmamard code in every vertex x∈ V and using <cit.> or <cit.> in the analysis instead of <cit.> — since we did not see any theoretical gain by doing that, and it was a lengthier proof, we decided not to implement this parameter improvement. We are also not sure whether the dependence in Proposition <ref> is needed or not. Finally, the significance analysis performed in Section <ref> is very crude. Can a better analysis allow to remove the dependence on Λ altogether, at least for some rich enough family of games?




    



§ SUBGROUP STRATEGIES

We need to decide if and where to keep this

    A subgroup strategy for a non-local game  consists of an IRS κ∈((  S∪{})) with the following properties for κ-a.e. subgroup H.

    
  * The image of  is a central involution with no fixed points. Namely, we have 
    
    ^2 ∈ H, ∉ H,
       ∀∈ S   [,]∈ H.

    
  * The rest of the images are involutions as well. Namely,
    
    ∀∈ S   ^2 ∈ H.

    
  * The image of S_x commutes. Namely, for every fixed vertex x∈ V, and 
    
    ∀,'∈ S_x   [,'] ∈ H.





 Let κ∈((  S∪{})) be a subgroup strategy for .
The quantum strategy ρ induced by κ is defined as follows. Let =(S∪{}) and sub() be the space of pairs (H,Hg) with H ∈sub() and Hg ∈ H \. Let κ̃ be the measure on this space which disintegrates as 

    dκ̃(H,Hg) = dκ(H).

In other words, we are putting counting measure on the fibers of the map sub() →sub(). So if F is a non-negative Borel function on sub() then

    ∫ F(H,Hg) dκ̃(H,Hg) = ∫∑_Hg F(H,Hg)  dκ(H).

Let L^2(κ̃) denote the associated L^2-space. 

For f ∈, define an operator σ̃(f) on L^2(κ̃) by

    (σ̃(f)·ϕ)(H,Hg) = ϕ(f^-1Hf, f^-1Hg) =ϕ (f^-1Hf, f^-1Hf · f^-1g).

It is an exercise to verify that σ̃ defines a homomorphism from  into the unitary group of L^2(κ̃). 

Let L^2_1(,κ̃) ⊂ L^2(κ̃) be the subspace of functions ϕ such that σ̃()·ϕ = -ϕ. Let 

    ^_1: L^2(κ̃) → L^2_1(,κ̃)

be the orthogonal projection map onto L^2_1(,κ̃). Let

    ^_1: L^2(κ̃) → L^2(κ̃)

be the composition of ^_1 with the inclusion map L^2_1(,κ̃) → L^2(κ̃).

Note that (^_1)^*^_1=^_1 and ^_1(^_1)^*=_n.

Given a Hilbert space ℋ, let B(ℋ) denote the algebra of bounded operators on ℋ. Define ρ S→ B( L^2_1(,κ̃)) by
      
    ρ()=^_1 σ̃()(^_1)^*.








    Let  κ be a subgroup strategy. Then, the quantum strategy ρ induced by κ, as in Definition <ref>, is indeed a quantum strategy for , as in Definition <ref>.


The proof is the exact same as the proof of Claim <ref>. 








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