A combination theorem for hierarchically quasiconvex subgroups, and application to geometric subgroups of mapping class groups
Giorgio Mangioni


§ ABSTRACT 
We provide sufficient conditions for two subgroups of a hierarchically hyperbolic group to generate an amalgamated free product over their intersection. The result applies in particular to certain geometric subgroups of mapping class groups of finite-type surfaces, that is, those subgroups coming from the embeddings of closed subsurfaces.

In the second half of the paper, we study under which hypotheses our amalgamation procedure preserves several notions of convexity in HHS, such as hierarchical quasiconvexity (as introduced by Behrstock, Hagen, and Sisto) and strong quasiconvexity (every quasigeodesic with endpoints on the subset lies in a uniform neighbourhood). This answers a question of Russell, Spriano, and Tran.


[
    [
    September 9, 2024
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I will achieve in my life - Heaven grant that it be not long - some gigantic amalgamation between the two discrepancies so hideously apparent to me.Virginia Woolf







§ INTRODUCTION



 §.§ A combination theorem for subgroups of HHGs

Given a group G and two subgroups A,B≤ G, it is natural to ask what the subgroup ⟨ A, B⟩_G generated by A and B looks like, and in particular if it is isomorphic to the amalgamated free product A *_C B. In this paper, we prove an amalgamation theorem for when A and B are subgroups of any group G acting “nicely” on a hierarchical space (see Definition <ref> below). This class includes all relatively hyperbolic groups, and all hierarchically hyperbolic groups in the sense of Behrstock, Hagen, and Sisto <cit.> (such as mapping class groups of finite-type surfaces, many 3-manifold groups, many Coxeter and Artin groups, compact special groups…). We give here a special case of the result, postponing the full statement to Section <ref>:


    Let (G,) be a hierarchically hyperbolic group, let A, B≤ G be subgroups and let C=A∩ B. Suppose that there exists a constant M≥ 0 and a domain Y_a∈ for every a∈ (A∪ B)-C, such that the following hold:
    
        
  * max{_Y_a(Cx_0), _Y_a(aCx_0)}≤ M/10;
        
  * _Y_a(Cx_0, aCx_0)≥ M; 
        
  * If a∈ A-C and b∈ B-C, Y_a aY_b;
        
  * In the same setting, _Y_a(Cx_0, bCx_0)≤ M/10.
    
    There exists M_0≥ 0, depending only on (G,), such that, if M≥ M_0, then 
    
    ⟨ A,B⟩_G≅ A *_C B.





 §.§ Amalgamation of geometric subgroups

In understanding the above theorem, one should have in mind the following example. Let G=(S) be the (extended) mapping class groups of a finite-type surface S. Let U, V be two closed, connected, incompressible subsurfaces, such that no connected component of S-U (resp. S-V) is an annulus. These conditions ensure that (U) and (V) naturally embed in G, and we denote the image of such embeddings as geometric embedded subgroups[The terminology "geometric" is due to Paris and Rolfsen <cit.> and denotes the image of the homomorphism (W)→(S) for any closed subsurface W, without further assumptions. Our notation thus denotes the cases when a geometric subgroup is also embedded. If there is a standard name for such subgroups we would be grateful for a reference.]. Let ∂ U∩∂ V=Γ be a common boundary multicurve, and suppose ∂ U-Γ and ∂ V-Γ are “sufficiently entangled” in the complement of Γ, meaning that they are far enough in the curve graph (S-Γ). This condition ensures that the intersection of (U) and (V) is the Dehn Twist flat ^|Γ| supported on Γ. Using the separability of the latter in both (U) and (V), one can find finite-index subgroups A≤(U) and B≤(V), such that every element a∈ A-^|Γ| acts with large translation length on some subsurface Y_a U (Assumption (II)). Furthermore, the entanglement of the boundaries of U and V ensures that, whenever a∈ A-^|Γ| and b∈ B-^|Γ|, the subsurfaces Y_a and Y_b must overlap (Assumption (III)). The above example is analysed more thoroughly in Subsection <ref>, where we prove the following, slightly more general result:



   Let S be a connected finite-type surface, and let (U) and (V) be two geometric embedded subgroups, where each of U and V is either connected or a multicurve. Let Γ=∂ U∩∂ V (where, with a little abuse of notation, the boundary of a multicurve denotes its support). Suppose that ∂ U-Γ, ∂ V-Γ are both non-empty, and that 
    _ (S-Γ)(∂ U-Γ,∂ V-Γ)≥ 4.


   Then there exist finite index subgroups A≤(U), B≤(V), intersecting along the Dehn twist flat ^|Γ|, such that
   
    ⟨ A,B⟩_(S)≅ A *_^|Γ| B.






  §.§.§ Comparison with the literature

Our Theorem <ref> is similar in spirit to Leininger-Reid's result for Veech subgroups along a common multitwist <cit.>. The main difference is that, while the subgroups there are supported on the whole surface S (and indeed every element which does not lie in the intersection is pseudo-Anosov), our result deals with reducible subgroups, and the large translations of the elements are witnessed by pairs of transverse subsurfaces. 

Our theorem also covers the case when U and V are multicurves. In this setting, it should be compared to Loa's result about free product of multitwists supported on “far enough” multicurves <cit.>. While our procedure requires to pass to finite-index subgroups with large translation in the annular domains, theirs applies to the whole Dehn Twist flats supported on the multicurves, but only when the intersection U∩ V is empty. We also stress that Loa's result gives more information about the amalgam, including the fact that it is undistorted in (S) and parabolically geometrically finite, in the sense of <cit.>.

However, we point out that our result about multicurves is just a very special case of a theorem of Koberda, which can be used to produce more general, undistorted RAAGs in (S) (see <cit.> and its quantitative version by Runnels <cit.>). 













    










 §.§ Preserving convexity

If A and B satisfy some property P, it is natural to ask when the subgroup they generate still enjoys P. The feature we focus on in the second half of the paper is hierarchical quasiconvexity (HQC for short), which is the analogue of quasiconvexity in the HHS world. A hierarchically quasiconvex subgroup of a HHG is itself a hierarchically hyperbolic space, and therefore it enjoys numerous properties regarding, for example, a coarse median structure and a quadratic isoperimetric function <cit.>, its asymptotic dimension <cit.>, and the arrangement patterns of top-dimensional quasiflats <cit.>. Therefore, it is relevant to understand when the subgroup generated by two hierarchically quasiconvex subgroups is again hierarchically quasiconvex. We provide sufficient conditions for our amalgamation procedure to preserve hierarchical quasiconvexity (the exact statement is Theorem <ref>):

    Let (G,) be a HHG, let A,B≤ G be two hierarchically quasiconvex subgroups, and let C=A∩ B. Suppose that:
    
        
  * A and B satisfy the hypotheses of Theorem <ref>, for some M≥ 0; 
        
  * A and B fill all squares (Definition <ref>);
        
  * A and B have no drift in the orthogonals (Definition <ref>).
    
    There exists a constant M_0≥ 0, depending on (G,) and the above data, such that, if M≥ M_0, then ⟨ A,B⟩_G≅ A*_C B is hierarchically quasiconvex in (G,).


Roughly, two HQC subgroups A and B fill all squares if, whenever two domains U,V∈ are orthogonal, if A has large projection to U and B to V then the intersection A∩ B also has large projection to one of the domains. This property is equivalent to the fact that A∪ B is hierarchically quasiconvex (see Lemma <ref>). 

Moreover, A and B have no drift in the orthogonals if it does not happen that both A and B have bounded projections to some domain U, which is orthogonal to the domains Y_a and Y_b used to detect the amalgamation. In Subsection <ref> we provide a counterexample where the lack of this property falsifies the conclusion of Theorem <ref>.






In our third and last Theorem, we study when our amalgamation procedure preserves strong quasiconvexity. Recall that, given a metric space X, a subspace Y⊆ X is strongly quasiconvex if every quasigeodesic γ with endpoints on Y lies in a neighbourhood of Y, whose radius only depends on X and on the quasigeodesic constants of γ. Such a subset is also called Morse, as strongly quasiconvex geodesics are exactly the Morse directions. Most of the properties of quasiconvex subsets of hyperbolic spaces hold for strongly quasiconvex subsets of general metric spaces <cit.>. Furthermore, as explored in <cit.>, a subspace of a  hierarchically hyperbolic space is strongly quasiconvex if and only if it is hierarchically quasiconvex and enjoys a further assumption, the orthogonal projection dichotomy. In Theorem <ref> we prove that the latter property is preserved by our amalgamation procedure. This way, we provide a possible answer to <cit.>:

Let (G,) be a HHG, let A,B≤ G be two strongly quasiconvex subgroups of G, and let C=A∩ B. Suppose that A and B satisfy the hypotheses of Theorem <ref>, for some constant M≥ 0.
 
There exists a constant M_0≥ 0, depending on (G,) and the strong quasiconvexity gauge of A and B, such that if M≥ M_0 then ⟨ A,B⟩_G≅ A*_C B is strongly quasiconvex in G.








 §.§ Pindaric excursus: relative hierarchical quasiconvexity?
 
In a relatively hyperbolic group, the "right" notion of convexity of a subgroup is relative quasiconvexity. Indeed, a relatively quasiconvex subgroup inherits a relative hyperbolic structure, and the intersection of two relatively quasiconvex subgroups is again relatively quasiconvex. As both HHGs and relatively hyperbolic groups fall into the category of "relative HHGs", in the sense of Definition <ref> below, one could look for a notion that unifies relative quasiconvexity and hierarchical quasiconvexity.


    Let (G,) be a relative HHG, and let _0 be a collection of domains which is closed under nesting and contains every U∈ such that U is not hyperbolic. Formulate a notion of hierarchical quasiconvexity relative to _0 for subgroups of G, such that:
    
        
  * If (G,) is a HHG and _0=∅, one recovers hierarchical quasiconvexity;
        
  * If (G,ℙ) is relatively hyperbolic and _0=ℙ, one recovers relative quasiconvexity;
        
  * Under suitable conditions on _0, if a subgroup is HQC relative to _0 then it admits a structure of a relative hierarchically hyperbolic space;
        
  * The intersection of two HQC subgroups relative to _0 is HQC relative to _0.
    


We believe a possible approach would be to generalise the notion of transition points on a geodesic in a relatively hyperbolic group, and then try to emulate the characterisation of relative quasiconvexity from <cit.>.

After one finds the right definition, one could attempt to extend Theorem <ref> to relative HQC subgroups, possibly generalising known combination theorems for relatively quasiconvex subgroups (see, among others, <cit.>).






    














 §.§ Outline

Section <ref> provides the background on hierarchically hyperbolic spaces and groups. In Section <ref> we prove the main amalgamation result, Theorem <ref>, which we then apply to certain geometric subgroups of mapping class groups in Section <ref>.

In Section <ref> we strengthen our result to preserve hierarchical quasiconvexity. The proof of Theorem <ref> is in three steps. In Subsection <ref>, we first recall that a subset of a HHS is HQC if and only if it is almost closed under certain quasigeodesics, called hierarchy paths. Next, in Subsection <ref> we determine under which conditions the union A∪ B of two HQC subgroups is again HQC (see Lemma <ref>). Finally, given a hierarchy path connecting two points of A∪ B, in Subsection <ref> we show that it can be decomposed as a union of hierarchy paths with endpoints on cosets of A∪ B, and the conclusion follows from the hierarchical quasiconvexity of the latter cosets.

Finally, Section <ref> is devoted to the proof of the combination result for strongly quasiconvex subgroups, Theorem <ref>.




 §.§ Acknowledgements

Firstly, I would like to thank my supervisor, Alessandro Sisto, for his constant support (even during the summer break) and several hints. Moreover, this paper arose as a side quest in a bigger, slightly unrelated project (talk about serendipity); so I am grateful to Yago Antolin, Matt Cordes, Giovanni Sartori, and Alessandro Sisto for numerous contributions to the first half of the paper, and for letting me publish it in this form. 







§ A CRASH COURSE IN HIERARCHICAL HYPERBOLICITY

We start by recalling some notions from the world of hierarchically hyperbolic spaces and groups, first introduced by Behrstock, Hagen, and Sisto in <cit.>. 

The quasigeodesic space  (X,_X) is a hierarchical space if there exists E≥0, called the hierarchical constant, an index set , whose elements will be referred to as domains, and a set {  U| U∈} of geodesic metric spaces (  U,_U), called coordinate spaces,  such that the following conditions are satisfied:


  * (Projections.)
There is a set {π_U:  X→ 2^  U| U∈} of projections mapping points in X to sets of diameter bounded by E in the various U∈. Moreover, for all U∈, the coarse map π_U is (E,E)–coarsely Lipschitz and π_U( X) is E–quasiconvex in U.


  * (Nesting.) 
 is equipped with a partial order , and either =∅ or  contains a unique –maximal element, denoted by S. When V U, we say V is nested in U. For each U∈, we denote by _U the set of V∈ such that V U. Moreover, for all U,V∈ with V U there is a specified subset ρ^V_U⊂  U with _  U(ρ^V_U)≤ E. There is also a projection ρ^U_V:  U→ 2^ V. (The similarity in notation is justified by viewing ρ^V_U as a coarsely constant map V→ 2^  U.)
 

  * (Orthogonality.) 
 has a symmetric and anti-reflexive relation called orthogonality: we write U V when U,V are orthogonal. Also, whenever V U and U W, we require that V W. We require that for each T∈ and each U∈_T such that {V∈_T| V U}≠∅, there exists a domain W∈_T-{T}, which we call a container for U inside T, such that whenever V U and V T, we have V W. Finally, if U  V, then U,V are not –comparable.


  * (Transversality and consistency.)
If U,V∈ are not orthogonal and neither is nested in the other, then we say U,V are transverse, denoted U V. In this case, there are sets ρ^V_U⊆ U and ρ^U_V⊆  V, each of diameter at most E and satisfying the Behrstock inequality:

    min{_U(π_U(z),ρ^V_U),_V(π_V(z),ρ^U_V)}≤ E

for all z∈  X.

For U,V∈ satisfying V U and for all z∈ X, we have: 

    min{_U(π_U(z),ρ^V_U),_ V(π_V(z)∪ρ^U_V(π_U(z)))}≤ E.
 
 
The preceding two inequalities are the consistency inequalities for points in X.
 
Finally, if U V, then _W(ρ^U_W,ρ^V_W)≤ E whenever W∈ satisfies either V W or V W and WU.
 

  * (Finite complexity.) 
There exists n≥0, the complexity of X (with respect to ), so that any set of pairwise––comparable elements has cardinality at most n.
  

  * (Large links.) 
Let U∈, let z,z'∈ X and let N=__U(π_U(z),π_U(z')). Then there exists {T_i}_i=1,…,⌊ N⌋⊆_U- {U} such that, for any domain T∈𝔖_U-{U}, either T∈_T_i for some i, or _T(π_T(z),π_T(z'))<E.  Also, _U(π_U(z),ρ^T_i_U)≤ N for each i.


  * (Bounded geodesic image.)
For all U∈, all V∈_U- {U}, and all geodesics γ of U, either _  V(ρ^U_V(γ))≤ E or γ∩ N_E(ρ^V_U)≠∅.
 

  * (Partial realisation.) 
Let {V_j} be a family of pairwise orthogonal elements of , and let p_j∈π_V_j( X)⊆  V_j. Then there exists z∈  X, which we call a partial realisation point for the family, so that:


  * _V_j(z,p_j)≤ E for all j,

  * for each j and 
each V∈ with V_j V, we have 
_V(z,ρ^V_j_V)≤ E, and

  * for each j and 
each V∈ with V_j V, we have _V(z,ρ^V_j_V)≤ E.



  * (Uniqueness.) For each κ≥ 0, there exists
θ_u=θ_u(κ) such that if x,y∈ X and
_ X(x,y)≥θ_u, then there exists V∈ such
that _V(x,y)≥κ.

We often refer to , together with the nesting and orthogonality relations, and the projections as a hierarchical structure for the space X. 



    Notice that, if E is a hierarchical constant for (X, ), then so is any E'≥ E. Hence, throughout the paper we will always implicitly assume that every hierarchical constant is strictly positive.



Where it will not cause confusion, given U∈, we will often suppress the projection map π_U when writing distances in U, i.e., given A,B⊆ X and P⊆ U we shall write d_U(A,B) for d_ U(π_U(A),π_U(B)) and  d_U(A,P) for d_ U(π_U(A),P). Furthermore, when V_1, V_2∈ are such that V_i U or V_i U for i=1,2, we will write d_U(V_1,V_2) for d_ U(ρ^V_1_U,ρ^V_2_U).



A hierarchical space is 


  * hierarchically hyperbolic if every coordinate space is E-hyperbolic;

  * relatively hierarchically hyperbolic if every coordinate space is either E-hyperbolic or -minimal. Notice that this includes relatively hyperbolic spaces and groups, as explained in <cit.>.



All properties of hierarchically hyperbolic spaces whose proofs do not involve the hyperbolicity of coordinate spaces also hold for hierarchical spaces in general. In particular, the following is proved as in <cit.>, which only uses the partial realisation axiom:

 Let (X, ) be a hierarchical space. For every U,V,W∈ such that U V and both ρ^U_W, ρ^V_W are defined, then _W(ρ^U_W,ρ^V_W)≤ 2E.



Combining Lemma <ref> and Axiom <ref> we get:

    Let U,V,W∈. Suppose that _V(U,W) is well-defined and strictly greater than 2E. Then U W.


Moreover, possibly after enlarging the HHS constant E, we get the following variant of the bounded geodesic image axiom, which is proved by combining the consistency inequalities with the original bounded geodesic image axiom, as in <cit.>:


Let (X, ) be a hierarchical space. Let x,y∈ X and let U,V∈ be such that U V. Then either _U(x,y)≤ E or every geodesic [π_V(x), π_V(y)]⊆ V must pass E-close to ρ^U_V.















Let R≥0 and let (b_U)_U∈∈∏_U∈2^   U be a tuple such that for each U∈, the U–coordinate  b_U has diameter ≤ R.  Then (b_U)_U∈ is R–consistent if for all V,W∈, we have 
    min{_V(b_V,ρ^W_V),_W(b_W,ρ^V_W)}≤ R

whenever V W and 

    min{_W(b_W,ρ^V_W),_V(b_V∪ρ^W_V(b_W))}≤ R

whenever V W.


Later we will need the following, which is <cit.> (notice that it holds for hierarchically hyperbolic spaces):


Let (X,) be a hierarchically hyperbolic space. Then for each R≥1, there exists θ=θ(R) so that,  for any R–consistent tuple (b_U)_U∈, there exists x∈ X such that _V(x,b_V)≤θ for all V∈.


Observe that the uniqueness axiom (Definition (<ref>)) implies that the realisation point x for (b_U)_U∈, provided by Theorem <ref>, is coarsely unique.  




Let (X,) be a hierarchical space. An automorphism consists of a map g: X→X, a bijection g^♯: → preserving nesting and orthogonality, and, for each U∈, an isometry g^♢(U):   U→ (g^♯(U)) for which the following two diagrams commute for all U,V∈ such that U V or U V:

    Xrgdπ_U   Xdπ_g^♯ (U)
      Urg^♢ (U)     (g^♯ (U))

and

    Urg^♢ (U)dρ^U_V     (g^♯ (U))dρ^g^♯ (U)_g^♯ (V)
      Vrg^♢ (V)     (g^♯ (V))

Whenever it will not cause ambiguity, we will abuse notation by dropping the superscripts and just calling all maps g.


We say that two automorphisms g,g' are equivalent, and we write g∼ g', if g^♯=(g')^♯ and g^♢(U)=(g')^♢(U) for each U∈.  Given an automorphism g, a quasi-inverse g for g is an automorphism with g^♯=(g^♯)^-1 and such that, for every U∈, g^♢(U)=g^♢(U)^-1. Since the composition of two automorphisms is an automorphism, the set of equivalence classes of automorphisms forms a group, denoted Aut().


    A finitely generated group G acts on a hierarchical space (X,) by automorphisms if there is a group homomorphism G→Aut(). Notice that this induces a G-action on X by uniform quasi-isometries.




If a group G acts on a (relative) HHS (X,), in such a way that the action on X is metrically proper and cobounded and the action on  is cofinite, then G is called a (relative) hierarchically hyperbolic group, and any quasi-isometry between G and X given by the Milnor-Švarc Lemma endows G with the (relative) HHS structure of X (possibly with a larger HHS constant). 



















§ DETECTING AN AMALGAMATED FREE PRODUCT IN A HHG

We are now ready to prove Theorem <ref> from the introduction, in the following extended form.

    Let G be a group acting on a hierarchical space (X,), and fix a basepoint x_0∈ X.  Let A_1,…,A_n≤ G be subgroups, and let C be a subset of the intersection ⋂_l=1^n A_l be a subset. Suppose that there exists a constant M≥ 100E, where E is a HHS constant of X, and a domain Y_a∈ for every a∈ (⋃_i A_i)-C, such that the following hold:
    
        
  *  max{_Y_a(Cx_0), _Y_a(aCx_0)}≤ M/10;
        
  *  _Y_a(Cx_0, aCx_0)≥ M; 
        
  *  If a∈ A_i-C and b∈ A_j-C where i≠ j, Y_a aY_b;
        
  *  In the same setting, _Y_a(Cx_0, bCx_0)≤ M/10.
    
    Then C coincides with the pairwise intersections A_i∩ A_j for every i≠ j, and in particular C = ⋂_l=1^n A_l. Moreover, the natural map 
    _C A_∙:= A_1 *_C … *_C A_n→⟨ A_1,…,A_n⟩_G

    is an isomorphism. 


 Given a word w=g_1… g_k c∈_C A_∙-{1}, we shall show that one can find a collection of pairwise transverse domains W_1,…, W_k, one for every subword g_1… g_j of w, such that the projection of g_1… g_j Cx_0 on W_i only changes when one passes from g_1… g_i-1 Cx_0 to g_1… g_i Cx_0. In a way, the domain W_i detects “the i-th step of w”, and as a consequence of assumption <ref> all subwords g_1… g_j with j≥ i+1 cannot undo the translation on W_i. In particular, one gets that _W_k(Cx_0, wCx_0) is greater than some positive constant, depending on M and E, and therefore w is non-trivial in G as it acts non-trivially on (X,). 


We now move to the proof of Theorem <ref>. First notice that, whenever i≠ j, C=A_i∩ A_j. Indeed C⊆ A_i∩ A_j, and if a∈ (A_i∩ A_j)-C then by assumption <ref> applied to a=b we would get that _Y_a(Cx_0, aCx_0)≤ M/10, in contrast with assumption <ref>.


Now, we want to prove that the natural epimorphism _C A_∙→⟨ A_1,…,A_n⟩_G is injective. In other words, given any non-trivial word w= g_1… g_k c∈_C A_∙-{1}, where  c∈ C and every two consecutive g_i and g_i+1 belong to different factors, we have to show that w≠_G 1. This is clearly true if k≤ 2, so we focus on the case k≥3. 


    For every i=1,…, k let Y_i=Y_g_i, and set 
    
    C_i=g_1… g_i-1Cx_0,   W_i=g_1… g_i-1Y_i,
 so that
    
    _W_i(C_i,C_i+1)= _Y_i(Cx_0,g_iCx_0)≥ M.


    We break the rest of the proof of Theorem <ref> into a series of Claims. 

    
        W_i W_i+1 for every i=1,…, k-1.
    
    
    Notice that Y_i and g_iY_i+1 are transverse by assumption <ref>, since g_i and g_i+1 lie in different factors of the amalgamation. Therefore, the domains W_i=(g_1… g_i-1)Y_i and W_i+1=(g_1… g_i-1)g_i Y_i+1 must be transverse as well, since the G-action preserves transversality. 
    

    
        For every i=2, …, k, _W_i(W_i-1, C_i)≤ M/10+E. 
        
        Symmetrically, for every i=1, …, k-1, _W_i(W_i+1, C_i+1)≤ M/10+E.
    

    
        We only prove the first statement, as the second follows analogously. As _W_i-1(C_i-1, C_i)≥ M≥ 4E, by the Behrstock inequality one of the following happens:
        
            
  * Every point of π_W_i(C_i) is E-close to ρ^W_i-1_W_i. Then the conclusion clearly follows.
            
  * Every point of π_W_i(C_i-1) is E-close to ρ^W_i-1_W_i. Then we have that
            
    _W_i(C_i, W_i-1)≤_W_i(C_i, C_i-1)+E≤ M/10+E,

            where we used Assumption <ref>. 
        
    

    
        For every i=2, …, k-1, _W_i(W_i-1, W_i+1)≥ 4/5M-4E>6E.  
    
    
    We simply notice that
    
    _W_i(W_i-1, W_i+1)≥_W_i(C_i, C_i+1)-_W_i(C_i, W_i-1)-_W_i(C_i+1, W_i+1)-

    
    -_W_i(ρ^W_i-1_W_i)-_W_i(ρ^W_i+1_W_i)≥ 4/5M-4E.

    






    
        
        
        
        
        
        
        
        
        
        

        
        
        
        


    
    
    We continue with a general statement about families of pairwise transverse domains in a hierarchical space:
    
        Let (X,) be a hierarchical space, and let {W_1, …, W_k}⊂ be a collection of domains such that  W_i W_i+1 for every i=1,…, k-1. If _W_j(W_j-1,W_j+1)> 6E for every j=2,…, k-1, then {W_1, …, W_k} are pairwise transverse, and _W_j(W_i,W_r)> 2E for every i<j<r.
    
    Notice that the hypothesis of Lemma <ref> are satisfied by the W_is we are considering, in view of Claims <ref> and <ref> and of the fact that M≥ 100E.

    
        We proceed by induction on k. If k=3 we just need to show that W_1 W_3. This is true since _W_2(W_1,W_3)> 6E, and we can invoke Corollary <ref>.

        Now assume that the theorem is true for every collection of at most k-1 elements. In particular, by applying the inductive hypothesis to the collections {W_1, …, W_k-1} and to {W_2, …, W_k}, we get that _W_j(W_i,W_r)> 2E whenever i<j<r and (i,r)≠(1,k). Thus, we only need to show that _W_j(W_1,W_k)> 2E for every j=2,…, k-1, and again Corollary <ref> will imply that W_1 W_k. Now
        
    _W_j(W_1,W_k)≥_W_j(W_j-1,W_j+1)-_W_j(W_j-1,W_1)-

        
    -_W_j(W_j+1,W_k)-_W_j(ρ^W_j-1_W_j)-_W_j(ρ^W_j+1_W_j).

        Since by Behrstock inequality we have that _W_j(W_j-1,W_1)≤ E, and similarly _W_j(W_j+1,W_k)≤ E, we get that 
        
    _W_j(W_1,W_k)> 6E-4E= 2E,

        as required.
    

    
    For every i≠ j and every x∈ C_j, _W_i(x, W_j)≤ E.
    

    
        
        
        
        
        
        We assume that j<i, as the case j>i is dealt with analogously. If by contradiction _W_i(x, W_j)> E, then the Behrstock inequality implies that _W_j(x, W_i)≤ E. Moreover _W_j(W_i,W_j+1)≤ E, either by combining the Behrstock inequality with Lemma <ref> (if j≤ i-2) or because W_i=W_j+1 (if j=i-1). Thus we get
         
    _W_j(x,C_j+1)≤_W_j(x,W_i)+_W_j(ρ^W_i_W_j)+_W_j(W_i,W_j+1)+

         
    +_W_j(ρ^W_i+1_W_j)+_W_j(W_j+1,C_j+1)≤ M/10+5E,

        where we used that _W_j(W_j+1,C_j+1)≤ M/10+5E by Claim <ref>. But then
        
    M≤_W_j(C_j,C_j+1) ≤_W_j(x,C_j+1) M/10+5E,

        giving a contradiction as M≥ 100E.
    

    

    
        _W_k(C x_0, w Cx_0)=_W_k(C_1, C_k+1)≥ 9/10M-5E>0.
    
    The Claim concludes the proof of Theorem <ref>, because it proves that w acts non-trivially on X, and therefore is non-trivial in G.
    
    
    This is just a matter of putting all the above Claims together. Indeed, one has that
     
    _W_k(C_1, C_k+1)≥_W_k(C_k,C_k+1)-_W_k(C_k, W_k-1)-_W_k(ρ^W_k-1_W_k∪ρ^W_1_W_k∪ C_1).

    The first term of the right-hand side is at least M by assumption <ref>. The second one is at most M/10+E by Claim <ref>. Regarding the third one, we have that 
    
    _W_k(ρ^W_k-1_W_k∪ρ^W_1_W_k∪ C_1)≤

    
    ≤_W_k(ρ^W_k-1_W_k)+_W_k(W_k-1,W_1)+_W_k(ρ^W_1_W_k∪ C_1)≤ E+E+(2E)=4E,

    where we used Claim <ref> to bound the last term. Hence 
    _W_k(C_1, C_k+1)≤ 9/10 M-5E,
 as required.
    
    
    
    
    
    

Before moving forward, we point out the following lemma, which combines some of the above Claims and will be useful later.















    
 For every 1≤ j<i≤ k and every x∈ C_j,

    _W_i(x, C_i)≤ M/10+5E.

Symmetrically, for every 1≤ i+1<j≤ k and every x∈ C_j,

    _W_i(x, C_i+1)≤ M/10+5E.



 
This is just a combination of some inequalities from the above proof. Indeed

    _W_i(x, C_i)≤_W_i(x,W_j)+_W_i(W_j)+_W_i(W_j, W_i-1)+_W_i(W_i-1)+_W_i(W_i-1, C_i).

The first term is at most E by Claim <ref>; the third term is at most E by the Behrstock inequality, combined with Lemma <ref>; the last term is at most M/10+E by Claim <ref>. Thus

    _W_i(x, C_i)≤ 4E+M/10+E=M/10+5E.

The second inequality follows analogously. 




§ AMALGAMATION OF GEOMETRIC SUBGROUPS ALONG COMMON BOUNDARIES

We not describe an application of Theorem <ref> to mapping class groups of finite-type surfaces. The subgroups we shall amalgam are mapping class groups of embedded subsurfaces, which overlap “sufficiently” away from a collection of common boundary components. The precise result is Theorem <ref> below, which in turn is Theorem <ref> from the introduction. 



 §.§ Notation and setting

We gather here the (fairly standard, but sometimes subtle) notation we shall need to prove Theorem <ref>.


  §.§.§ Curves, surfaces, and mapping classes

In what follows, let S be a possibly disconnected surface of finite-type (that is, an oriented, compact surface from which a finite number of points is removed). If S is connected, let (S) be the (extended) mapping class group of S, that is, the group of isotopy class of self-homeomorphisms of S fixing the boundary pointwise. If instead S is disconnected, its mapping class group is defined as the direct product of the mapping class groups of its connected components.

By curve we denote the isotopy class of an embedding 𝕊^1↪ S which is essential, meaning that it does not bound a disk with at most one puncture. A multicurve is a collection of pairwise disjoint, non-isotopic curves. We often see a multicurve as a subsurface of S, by replacing each curve with a closed annulus whose core is the curve. Given a curve γ, let T_γ be the Dehn Twist around γ (see e.g. <cit.>).

The curve graph S is the simplicial graph whose vertices are all curves on S, and where adjacency corresponds to disjointness. This definition does not apply to some surfaces of small complexity:

    
  * If S is either a sphere with four punctures, or a torus with at most two punctures, then two curves are adjacent in S if and only if their intersection number is minimal among all pairs.
    
  * If S is an annulus then its annular curve graph, which we still denote by S, is a quasiline. We won't need the actual definition, and we refer to <cit.> for further explanations.
  

The following is an easy consequence of known facts, but we provide a proof for completeness. Recall that a subgroup H of a group G is separable if, for every g∈ G-H, there exists a finite quotient ψ G→G such that ψ(g)∉ψ(H). 

    Let S be a finite-type surface, and let Γ⊆∂ S be a collection of boundary components. Then the Dehn twist flat ^|Γ| supported on Γ is a separable subgroup of (S). 



Let S be the surface obtained from S by gluing a once-punctured disk  onto each boundary component belonging to Γ, and let {p_1,…, p_r} be the punctures added this way. By e.g. <cit.>, the quotient (S)/^|Γ| is isomorphic to the group (S, {p_1,…,p_k}), which is the subgroup of (S) of all mapping classes fixing the punctures {p_1,…,p_k} pointwise. 

Now let g∈(S)-^|Γ|, and let h be its image in (S, {p_1,…,p_k}), which is therefore non-trivial. Now, (S, {p_1,…,p_k})≤(S), and the latter is residually finite (see e.g. <cit.> and the discussion below it for punctured surfaces); hence we can find a finite quotient G of (S, {p_1,…,p_k}) where h projects non-trivially. Then, if we take the composition (S)→(S, {p_1,…,p_k})→ G, we get a finite quotient such that the image of ^|Γ| is trivial while the image of g is not.







  §.§.§ The marking graph

Let (S) be the marking graph of S, as defined in <cit.>. A vertex x of (S) consists of a multicurve of maximal cardinality, called the support of x and denoted by supp(x), and for every α∈supp(x) a choice of a set p∈α of diameter at most 1, called the transversal associated to α. By e.g. <cit.>, which in turn builds on observations from <cit.>, (S) has the following HHS structure:
    
        
  * The domain set  is the collection of subsurfaces of S;
        
  *  is containment and  is disjointness of subsurfaces (up to isotopy);
        
  * For every U∈, the associated coordinate space U is the curve graph of U, and the projection π_U (S)→ U is the subsurface projection.
    
    Furthermore, (S) acts geometrically on (S), and therefore inherits the HHG structure described above.



 §.§ The result

By subsurface we mean the isotopy class of an embedding U↪ S, where U is a finite-type surface whose connected components cannot be pairs of pants. The boundary ∂ U of a connected, non-annular subsurface U is defined as the closure of U minus its interior: if U is an annulus, with a little abuse of notation we define its boundary as the core of the annulus. The boundary of a disconnected subsurface is the union of the boundaries of its connected components.

If U is a closed subsurface, we can extend every mapping class on U to the identity on S-U, and we get a homomorphism (U)→(S). This map is injective if and only if every curve in ∂ U is essential in S, and no two boundary components are isotopic (see e.g. <cit.>). In this case, we call (U) a geometric embedded subgroup of (S). Our main theorem describes when two such subgroups span a free product, amalgamated along a common boundary Dehn twist flat:


   Let S be a connected finite-type surface, and let (U) and (V) be two geometric embedded subgroups, where each of U and V is either connected or a multicurve.
   
   Let Γ=∂ U∩∂ V, and let Γ_U=∂ U-Γ (resp. Γ_V=∂ V-Γ). Suppose that Γ_U, Γ_V are both non-empty, and that _ (S-Γ)(Γ_U,Γ_V)≥ 4.

   Then there exist finite index subgroups A≤(U), B≤(V), intersecting along the Dehn twist flat ^|Γ|, such that
   
    ⟨ A,B⟩_(S)≅ A *_^|Γ| B.




The proof will highlight some useful techniques to verify the requirements of Theorem <ref>. Such tools often use only the existence of a HHG structure for the ambient group, and can therefore be exported to more general settings. 


We start with some considerations on the two subsurfaces and their mapping class groups:
    
        Given two subsurfaces W U and W' V, if none of the two subsurfaces is a sub-multicurve of Γ, then W W'.
     
    
If by contradiction W and W' were either disjoint, or one contained in the other, then there would be two curves δ⊂ W and δ'⊂ W, both disjoint from Γ, such that _ (S-Γ)(δ, δ')≤ 1. However, notice that δ would either belong to, or be disjoint from, ∂ U, since W U, and the same is true for δ' and ∂ V. Therefore
        _ (S-Γ)(δ, δ')≥_ (S-Γ)(Γ_U, Γ_V) -2 ≥ 2, contradicting our assumption.
    

    
        (U)∩(V)=^|Γ|.
    
    
        Clearly ^|Γ|≤(U)∩(V). Conversely, pick an element g∈(U)∩(V). By e.g. <cit.>, there exists a power k such that g^k=∏_i=1^l g_i, where each g_i is either a partial pseudo-Anosov or a power of a Dehn Twist, and the supports {R_i} of the g_i are all pairwise disjoint, closed subsurfaces. 
        
        Now, as g is supported on both U and V, each R_i must be nested in both U and V, and therefore must be nested in Γ by Claim <ref>. In other words, g^k∈^|Γ|.
        
        Now, if one between U and V is a multicurve we immediately get that g∈^|Γ| as well, and we are done. Otherwise, the above argument shows that g projects to a torsion element of (U), where U is the surface obtained by capping every curve in Γ with a once-punctured disk. However, as ∂ U-Γ is non-empty, U still has non-empty boundary, so <cit.> yields that (U) is torsion-free (more precisely, the cited result is stated for surfaces with negative Euler characteristic; however we can always embed (U) in the mapping class group of such a surface, for example by gluing a sphere with five disks removed along one of the boundaries of U). Then again g∈^|Γ|, as required.
    


Next, we produce the subgroups A, B. Extend Γ to a multicurve Γ' of maximal cardinality, and let x_0∈(S) be any marking supported on Γ'. Let
    
    C_0sup{_W(x_0,V) | W U,WΓ}+sup{_W'(x_0,U) | W' V,W'Γ}.

The above quantity is well-defined, as Claim <ref> tells us that, whenever W U is not a sub-multicurve of Γ, we have that W V, so that the projection of V to W is well-defined. Furthermore, C_0 is finite. Indeed, let y_V∈ (S) be a marking whose support contains ∂ V. Then, whenever W is nested in U but not in Γ, the projection ρ^V_W coincides with the subsurface projection of y_V to W. In other words, 
    
    _W(x_0,V)=_W(x_0,y_V)≤ E_ (S)(x_0,y_V)+E,

    where E is a HHS constant for ( (S), ), and we used that subsurface projections are E-coarsely Lipschitz. We can similarly define a marking y_U whose support contains ∂ U, and use it to bound the second term of Equation (<ref>). Notice that, in view of the above argument, we can rewrite Equation (<ref>) in the following form:
    
    C_0= sup{_W(x_0,y_V) | W U,WΓ}+sup{_W'(x_0,y_U) | W' V,W'Γ}.


    Now, let F⊂(U) be the subset of all elements a such that, for every Y∈, 
    _Y(y_U, a(y_U))≤ 12C_0+100E+100

    Notice that F is finite, by the fact that (S) acts metrically properly on (S), combined with the uniqueness axiom (<ref>) for the HHS ((S),). By Lemma <ref> there exists a finite-index subgroup A≤(U) containing ^|Γ|, such that if a∈ F∩ A then a∈^|Γ|. One can define B analogously. Notice that 
    ^|Γ|≤ A∩ B≤(U)∩(V)=^|Γ|,

    where the last equality is Claim <ref>.
    

We are finally ready to show that A and B satisfy the hypotheses of Theorem <ref>, and therefore

    ⟨ A,B⟩_(S)≅ A *_^|Γ| B.

We first describe the subsurfaces Y_g we shall use to prove the Theorem. For every a∈ A-^|Γ| there exists a domain Y_a such that _Y_a(y_U, a(y_U))≥ 12C_0+100E+100, by how we chose A.

We first notice that Y_a must be nested in U. Indeed, the action of (U) fixes ∂ U and every curve α∈supp(y_U) which is disjoint from U, and the transversal for each such α is moved within distance at most 4 in α (see e.g. <cit.>). Furthermore, if a surface Y is either disjoint from U, or properly contains U, then the subsurface projection of both y_U and a(y_U) only depends on the above data. Therefore _Y(y_U, a(y_U))≤ 4 whenever YU. 

Furthermore, suppose that every Y_a as above was nested in Γ. Then we could multiply a by a suitable multitwist in ^|Γ| to find an element of (A∩ F)-^|Γ|, contradicting our choice of A. Thus pick any Y_a as above which is nested in U but not in Γ. 

Now let M=10C_0+100E+40, and choose x_0 as above. We now verify the four requirements from Theorem <ref>, for every a∈ A-^|Γ| and every b∈ B-^|Γ|:


  * <ref> The action of ^|Γ| fixes every curve α∈Γ'-Γ, and the associated transversal (again, up to distance 4 in α). Therefore, as Y_a U but is not nested in Γ, the subsurface projection of ^|Γ| x_0 to Y_a is coarsely the same as that of x_0, and the same is true for a^|Γ| x_0. In particular max{_Y_a(^|Γ| x_0), _Y_a(a^|Γ| x_0)}≤ 4≤ M/10.

  * <ref> By the above discussion we get _Y_a(^|Γ| x_0,a^|Γ| x_0)≥_Y_a(x_0,ax_0)-8. Furthermore  Equation (<ref>) yields that

    _Y_a(x_0,ax_0)-8≥_Y_a(y_U,ay_U)-2C_0-8≥ M.


  * <ref> As Y_a U, Y_b V and none is nested in Γ, Claim <ref> tells us that Y_a Y_b.

  * <ref> Firstly _Y_b(^|Γ| x_0,a^|Γ| x_0)≤_Y_b(x_0,ax_0). Moreover

    _Y_a(ax_0, Y_b)≥_Y_a(ax_0, x_0)-_Y_a(x_0, Y_b)-_Y_a(ρ^Y_b_Y_a)≥ M-C_0-E≥ 2E.

Hence by the Behrstock inequality _Y_b(ax_0, Y_a)≤ E. In turn, this means that

    _Y_b(x_0, ax_0)≤_Y_b(x_0, Y_a)+ _Y_b(ρ^Y_a_Y_b)+_Y_b(Y_a, ax_0)≤ C_0+2E≤ M/10.


The proof of Theorem <ref> is now complete.






































§ A COMBINATION THEOREM FOR HIERARCHICALLY QUASICONVEX SUBGROUPS

This Section is devoted to the proof of Theorem <ref> from the introduction, regarding the hierarchical quasiconvexity of the amalgamated free product of two HQC subgroups A,B of a HHG (G,). 




 §.§ Quasiconvexity and friends


A subspace Z of a geodesic metric space is R-quasiconvex, for some constant R≥ 0, if every geodesic segment with endpoints on Z is contained in the R-neighbourhood of Z.








For the rest of the section, let (X,) be a HHS. 


A subspace Y⊆ X is κ-hierarchically quasiconvex, for some κ [0,+∞)→ [0,+∞), if:
    
        
  * For every U∈, π_U(Y) is κ(0)-quasiconvex in U;
        
  * Realisation: for every x∈ X and every R∈[0,+∞), if _U(x,Y)≤ R for every U∈ then _X(x,Y)≤κ(R).
    



    It follows from the definition, together with the fact that coordinate projections are coarsely Lipschitz, that if Y and Z are two subspaces of X within Hausdorff distance d, and if Y is κ-hierarchically quasiconvex, then Z is κ'-hierarchically quasiconvex, for some function κ' only depending on κ, d, and (X,). This observation will be used repeatedly throughout the section.


An equivalent definition of hierarchical quasiconvexity, which more closely resembles Definition <ref>, involves being closed under certain quasigeodesic paths, called “hierarchy paths”:

    For λ≥ 1, a (not necessarily continuous) path γ [a,b]⊂ℝ→ X is a λ–hierarchy path if
    
        
  * γ is a (λ, λ)-quasigeodesic,
        
  * for each W∈, the path π_w(γ) is an unparameterised (λ,λ)-quasigeodesic, meaning that it becomes a (λ,λ)-quasigeodesic after precomposing it with an increasing function g[0,l]→[a,b] mapping 0 to a and l to b.
    



    <cit.> states that any two points of X are connected by a λ_0-hierarchy path, for some constant λ_0 only depending on (X, ). From now on, we will assume that the HHS constant E has been chosen greater than this λ_0.





    A subset Y⊆ X is κ–hierarchically quasiconvex if and only if there exists a function Λ [1,+∞)→ [0,+∞) such that every λ-hierarchy path with endpoints on Y is contained in the Λ(λ)-neighbourhood of Y. Moreover, κ and Λ each determine the other.


The following combines <cit.>:
 Let Y⊆ X be κ-hierarchically quasiconvex. There exists a coarsely Lipschitz, coarse retraction _Y X→ Y, called the gate on Y, such that, for every x∈ X and every W∈, π_W(_Y(x)) uniformly coarsely coincides with the coarse closest point projection of π_W(x) to the κ(0)-quasiconvex subset π_W(Y)⊆ W.


Given two HQC subspaces A,B, the gate of one onto the other can be characterised in terms of the coarse intersection of A and B:


    Let A and B be κ-hierarchically quasiconvex subsets of X, and let _X(A,B)=r. There exist non-negative constants R_0 and D_1, both depending only on κ and r, such that _Haus(N_R_0(A)∩ N_R_0(B), _B(A))≤ D_1.


Furthermore, if G is a group and A, B are subgroups of G, then by <cit.> there exist D_2, depending on R_0 and on a choice of a word metric for G, such that _Haus(N_R_0(A)∩ N_R_0(B), A∩ B)≤ D_2. Thus, for HQC subgroups of a HHG, the gate of one onto the other is within finite Hausdorff distance from the actual intersection:


    Let A and B be κ-hierarchically quasiconvex subgroups of a HHG (G,). There exists a constant L, which only depends on (G,) and κ, such that _Haus(A∩ B, _B(A))≤ L.





 §.§ Hierarchical quasiconvexity of a union

In hyperbolic spaces, a union of quasiconvex subspaces is again quasiconvex. We provide a proof of this fact for completeness:

    Let X be a δ-hyperbolic space, and let Y,Y' be two R-quasiconvex subspaces. Then Y∪ Y' is R-quasiconvex, where 
    R=R+2δ+_X(Y,Y')+1.
 



    Let γ be a geodesic segment with endpoints a,b∈ Y∪ Y'. If a and b are both in Y, or both in Y', then γ is in the R-neighbourhood of Y∪ Y', by quasiconvexity of each subset. Thus, suppose without loss of generality that a∈ Y and b∈ Y'. Let p∈ Y and p'∈ Y' be such that _X(p,p')≤_X(Y,Y')+1, and choose geodesics [a,p], [p, p'], and [p', b]. Then γ lies in the 2δ-neighbourhood of [a,p]∪ [p, p']∪ [p', b], as geodesic quadrangles are 2δ-slim in hyperbolic spaces. In turn, [a,p]∪ [p, p']∪ [p', b] is contained in the (R+_X(Y,Y')+1)-neighbourhood of Y∪ Y', and the conclusion follows.



Now we want to establish an analogue of Lemma <ref> for HQC subspaces of a HHS. More precisely, we shall prove that the necessary and sufficient condition for the union to be HQC is the following. Recall that, given two subspaces C⊆ A of a metric space X, we say that C is R-dense in A if A⊆ N_R(C).


    Let (X,) be a HHS, and let A,B⊆ X. We say that A and B fill all squares if there exists a constant T such that, for every U,V∈ such that U V, either π_U(_A(B)) is T-dense in π_U(A), or π_V(_B(A)) is T-dense in π_V(B). 

Notice that, if A and B fill all squares for some constant T, then they also fill all squares for any bigger constant T'≥ T.


Definition <ref> roughly forbids the following situation, which gives the name to the property. Let X=ℝ^2 with the HHS structure coming from the usual Cartesian coordinates, let A be the x-axis and B the y-axis. Both A and B are hierarchically quasiconvex, as they correspond to factors of the product structure. However, A∪ B is not hierarchically quasiconvex, as it does not satisfy the realisation property: any point p∈ℝ^2 has the same x-coordinate as some point in A and the same y-coordinate as some point in B, but can be arbitrarily far from A∪ B. In other words, A and B leave some gaps in the square. 












    Let (X,) be a HHS, and let A,B⊆ X be κ-hierarchically quasiconvex subspaces. Then A and B fill all squares (Definition <ref>), for some constant T≥ 0, if and only if A∪ B is κ-hierarchically quasiconvex, where T and κ each determine the other (together with κ and _X(A,B)).


We split the two implications of Theorem <ref> into Lemmas <ref> and <ref> below.



    Let (X,) be a HHS, and let A,B⊆ X be κ-hierarchically quasiconvex subspaces. If A∪ B is κ-hierarchically quasiconvex then A and B fill all squares (Definition <ref>), for some constant T which only depends on (X,), κ, and κ.
    
    
 Fix a HHS constant E for (X,). Assume by contradiction that A and B do not fill all squares. This means that, for every T∈[0,+∞), one can find two orthogonal domains U,V∈ such that min{_U(A), _V(B)}≥ T, but neither π_U(_A(B)) is T-dense in π_U(A) nor π_V(_B(A)) is T-dense in π_V(B). 

From now on, we will say that a quantity is uniform if it does not depend on T, but only on (X,), κ, and κ. We shall eventually choose T greater than every uniform constant we will find along the way, and this will yield the desired contradiction.

Since π_U(_A(B)) is not T-dense in π_U(A), there exists a point a∈ A such that _U(a,C)> T. Similarly, choose b∈ B such that _V(b,C)> T. Now consider the following tuple of coordinates:
    
    x_W=π_W(a) W U;
    π_W(b) W V;
    π_W(a) W U  W V;
    ρ^U_W∪ρ^V_W

    where, in order to make the definition more compact, we slightly abused the notation by setting ρ^U_W=∅ if U W, and similarly for V. Notice that the tuple {x_W}_W U is E-consistent, as a is a point of X and therefore satisfies the consistency inequalities by Axiom (<ref>), and for the same reason {x_W}_W V and {x_W}_W U, W V are E-consistent as well. Arguing as in <cit.>, one can show that the whole tuple {x_W}_W∈ is K_1-consistent for some uniform constant K_1. Then by the Realisation Theorem <ref> we can find p∈ X and a uniform constant K_2 such that _W(p,x_W)≤ K_2 for every W∈.

    
        There exist uniform constants D≥ 0 and K_3≥ K_2 such that, if T≥ D, then _W(p,A∪ B)≤ K_3 for every W∈.
    
    
        We refer to how we defined x_W in Equation (<ref>). If either W is nested into U or V, or W is orthogonal to both, then π_W(p) is K_2-close to either π_W(a) or π_W(b), and we have nothing to prove. Thus suppose, without loss of generality, that either U W or U W. Then π_W(p) is K_2-close to ρ^U_W∪ρ^V_W, and in particular 
        
    _W(p,U)≤ K_2+_W(ρ^V_W)+_W(U,V)≤ K_2+3E,

        where we used that _W(U,V)≤ 2E by Lemma <ref>. Now let P_U be the product region associated to U, as defined in <cit.>. For our purposes, P_U can be thought of as the subspace of all z∈ X such that, for every Y∈ such that U Y or U Y, the projection π_Y(z) coincides with ρ^U_Y, up to some error which is bounded in terms of E. By <cit.>, there exists uniform constants D,λ,ν≥ 1 such that, if T≥ D, then there exists a λ-hierarchy path γ connecting a to _A(B)⊆ A which passes ν-close to the product region P_U. Since A is κ-hierarchically quasiconvex, by Lemma <ref> we have that γ is contained in some uniform neighbourhood of A. Thus the distance between A and P_U is uniformly bounded. In turn, since the projection map π_W X→ W is E-coarsely Lipschitz, π_W(A) is uniformly close to π_W(P_U), which in turn uniformly coarsely coincides with ρ^U_W⊆π_W(p). 
    
    Since A∪ B is κ-quasiconvex, the realisation property for A∪ B tells us that p is κ(K_3)-close to A∪ B, and without loss of generality we can assume that _X(p,A)≤κ(K_3). Then _B(p) is uniformly close to _B(A), as gate maps are coarsely Lipschitz. However, by how gate maps are constructed (see Definition <ref>), we have that π_V(_B(p)) uniformly coarsely coincides with the projection of π_V(p) to the quasiconvex subset π_V(B), and π_V(p)=π_V(b) already lies in π_V(B).
    Summarising, there is some constant K_4, only depending on (X,), κ, and κ, such that _V(b, _B(A))≤ K_4, and this is against our choice of b if we choose T>K_4.




       Let (X,) be a HHS, and let A,B⊆ X be κ-hierarchically quasiconvex subspaces. If A and B fill all squares (Definition <ref>), for some constant T≥ 0, then A∪ B is κ-hierarchically quasiconvex, where κ only depends on (X,), κ, _X(A,B), and T.



Firstly, by Lemma <ref> the projection π_U(A∪ B) is κ(0)-quasiconvex, where κ(0) is a constant depending on κ(0), D_X(A,B), and a HHS constant E for (X,). 

Thus we are left to prove the realisation property for A∪ B. 
To this purpose, fix R≥ 0, let x∈ X be such that _W(x,A∪ B)≤ R for every W∈, and set 
    Ω=κ(0)+R+T+E(D+KD+K+4),
 
where K is the constant from <cit.>, depending only on κ and (X, ), such that for every b∈ B

    _X(b, _A(b))≤ KD+K.


We claim that there exists F∈{A,B} such that _W(x,F)≤Ω for every W∈. This will imply that _X(x,A∪ B)≤κ(Ω), so we can set κ(R)=κ(Ω). If this is not the case, let U,V∈ be (necessarily distinct) domains such that _U(x,B)> Ω and _V(x,A)> Ω. Notice that, as Ω≥ R, we must have that _U(x,A)≤ R and _V(x,B)≤ R. There are a few configurations to analyse, depending on the relation between U and V.
    
        
  * U V: By the Behrstock inequality, Definition <ref>.(<ref>), we can assume without loss of generality that _U(x,ρ^V_U)≤ E. Moreover, since Ω≥ 4E and (ρ^V_U)≤ E, we have that _U(B,ρ^V_U)> 2E, and again the Behrstock inequality tells us that π_V(B) has diameter at most 3E. Thus 
        
    _V(x,A)≤_V(x,B)+_V(B)+_V(A,B)≤ R+3E+E(D+1),

        where we used that _V(A,B)≤ E(D+1) as the projection π_V X→ V is (E,E)-coarsely Lipschitz. But then _V(x,A)≤Ω, against our assumption.

        
  * U V:  Since _U(x,B)>Ω≥ E, the bounded geodesic image Lemma <ref> yields that every geodesic connecting π_V(x) to π_V(B) must pass E-close to ρ^U_V. Since _V(x,B)≤ R we get that _V(x,U)≤ R+E, and in turn 
        
    _V(U,A)≥_V(x,A)-_V(x,U)-_V(ρ^U_V)≥ 2E+κ(0).
 
        Now, every geodesic γ⊆ V connecting two points of π_V(A) is contained in the κ(0)-neighbourhood of π_V(A), and therefore is at least 2E-far from ρ^U_V. Then again the bounded geodesic image Lemma <ref> tells us that π_U(A) has diameter at most E, and as in the previous case we get that _U(x,B)≤Ω, yielding a contradiction. Notice that the same argument covers the case when V U.
        
        
  * U V: Since A and B fill all squares, we can assume without loss of generality that π_U(_A(B)) is T-dense in π_U(A), so there exists b∈ B such that _U(x,_A(b))≤ R+T. In turn,
        
    _U(x,B)≤_U(x,b)≤_U(x,_A(b))+_U(_A(b), b)≤ R+T+E(KD+K+1)≤Ω,

        where again we used that π_U is (E,E)-coarsely Lipschitz. This again yields a contradiction.
    










 §.§ Hierarchical quasiconvexity of an amalgam
 
We devote the rest of the Section to the proof of Theorem <ref>, regarding the amalgamation of HQC subgroups. First, a definition.


Let Z be a subset of the hierarchically hyperbolic
space (X,). Define P^1_λ(Z) to be the union of all λ–hierarchy paths between points in Z. One can then inductively set P^n_λ(Z)=P^1_λ(P^n-1_λ(Z)) for all n≥ 2. 

Recall that every two points of X are connected by a λ_0-hierarchy path, where, λ_0 is the constant from Remark <ref>; thus, for every λ≥λ_0 and every n≥ 1, the hull P^n_λ(Z) is non-empty and contains Z. 


The following restates the core result of <cit.>, which was crucial in establishing the equivalence between hierarchical quasiconvexity and being closed under hierarchy paths:

    There exist N∈ℕ, λ≥λ_0, and κ, all depending only on (X,),  such that, for every Z⊆ X, the N-th hierarchy path hull P^N_λ(Z) is κ-hierarchically quasiconvex.



    For every θ≥ 0, let H_θ(Z) be the θ-quasiconvex hull of Z, as defined in e.g. <cit.>. By <cit.>, there exists θ_0 such that, for every Z⊂ X and every θ≥θ_0, the hull H_θ(Z) is κ_θ-hierarchically quasiconvex, where κ_θ only depends on θ and the HHS structure. Moreover, <cit.> states that there exist N and λ as above, and a constant θ≥θ_0, such that _Haus(P^N_λ(Z), H_θ(Z))≤ D, where D only depends on λ, θ, and (X, ). Thus, as P^N_λ(Z) is within Hausdorff distance at most D from a κ_θ-hierarchically quasiconvex set, it is itself κ-hierarchically quasiconvex by Remark <ref>, for some function κ depending on κ_θ and D (and therefore, ultimately, only on (X,)).



We now introduce the technical requirement for Theorem <ref>, whose necessity will be discussed in Subsection <ref> below.
 Let (G,) be a HHG, let A,B≤ G be two subgroups satisfying the hypothesis of Theorem <ref>, and let C=A∩ B. We say that A,B have no drift in the orthogonals if there exists R≥0 such that the following hold. For every a,b∈ (A∪ B)-C belonging to different subgroups, and for every domain U∈ which is orthogonal to both a^-1Y_a and Y_b, we have that either π_U(A) or π_U(B) is R-dense in U.




    Let (G,) be a HHG, let A,B≤ G be two κ-hierarchically quasiconvex subgroups, and let C=A∩ B. Suppose that:
    
        
  * There exists M≥100E such that A and B satisfy the hypotheses of Theorem <ref>;
        
  * A and B fill all squares (Definition <ref>), for some constant T≥ 0;
        
  * A and B have no drift in the orthogonals (Definition <ref>), for some constant R≥ 0.
    
    There exist a positive constant 𝔐 and a function 𝔨[0,+∞)→ [0,+∞), both depending only on κ, T, R, and (G,), such that, if M≥𝔐, then ⟨ A,B⟩_G≅ A*_C B is 𝔨-hierarchically quasiconvex in G.




Theorem <ref> is a consequence of the following technical statement:

    There exist functions K'[0,+∞)→ [0,+∞) and M[0,+∞)→ [100E,+∞), depending only on κ, T, R, and (G,), such that the following holds. Let λ be as in Lemma <ref>. For every K≥ 0, if M≥ M(K) then 
    
    P^1_λ(N_K(A*_C B))⊆ N_K'(K)(A*_C B).




 Set K_0=0, and iteratively define M_n=M(K_n-1) and K_n=K'(K_n-1) for every n=1,…, N, where N is the integer from Lemma <ref>. Now let 𝔐=max_i=1,…, N M_n. If M≥𝔐, then 

    P^N_λ(A*_C B)⊆ P^N-1_λ(N_K_1(A*_C B))⊆…⊆ N_K_N(A*_C B).

Now, P^N_λ(A*_C B) is κ-HQC, where κ is the function from Lemma <ref> which only depends on (G,). Furthermore, as P^N_λ(A*_C B) and A*_C B are within Hausdorff distance at most K_N, Remark <ref> implies that A*_C B is itself 𝔨-hierarchically quasiconvex, for some function 𝔨 only depending on K_N and κ. As by hypothesis K_N only depends on κ, T, R, and (G,), this concludes the proof of Theorem <ref>.



Let γ be a λ-hierarchy path between points of N_K(A*_C B). If we connect each endpoint of γ with any point of A*_C B within distance K, we get a λ'-hierarchy path γ' which extends γ, for some constant λ'≥λ depending only on K, λ, and the HHS constant of G. Up to the action of A*_C B on itself, we can assume that the endpoints of γ' are the identity element 1 and some w=g_1… g_k c, where c∈ C, g_i∉C for all i=1,…, k, and every two consecutive g_i and g_i+1 belong to different factors of the amalgamation. 


A and B satisfy the hypothesis of Theorem <ref>, whose data include a basepoint x_0∈ G and a domain Y_a∈ for every a∈ (A∪ B)-C. As in Notation <ref>, for every i=1,…, k let Y_i=Y_g_i, and set 

    C_i=g_1… g_i-1Cx_0,   W_i=g_1… g_i-1Y_i.

Moreover, for every i=1,…, k let 

    A_i=
        g_1… g_i-1 Ag_i∈ A;
    
    g_1… g_i-1 Bg_i∈ B.
 
This way, A_i contains both C_i and C_i+1.




We shall prove Lemma <ref> assuming that M≥ M(K), where

    M(K)=.
 
In the expression above, L=L(κ, ) is the constant from Lemma <ref>, while the constant S=S(E,κ(0),λ',) will be defined in the proof of Claim <ref> below (more precisely, in the paragraph named Case 1).



Now fix any i=2, …, k-1. Recall that _W_i(C_i, C_i+1)≥ M by Assumption <ref> of Theorem <ref>. Combining this with Lemma <ref>, we get that

    _W_i(1,w)≥_W_i(C_i, C_i+1)-_W_i(C_i, 1)-_W_i(C_i+1, w)≥ 4M/5-10E.

Notice that, by our choice of M in Notation <ref>, we have that 
    4M/5-10E≥ 2M/5+2λ'.
 In other words, the balls of radius M/5 around π_W_i(1) and π_W_i(w) are at distance at least 2λ'. Then, as the projection of γ' to W_i is a (λ', λ')-quasigeodesic (after reparametrisation), there must be a point ℓ_i∈γ' such that 

    min{_W_i(1,ℓ_i), _W_i(w,ℓ_i)}>M/5.


Now, the core of the proof is the following:


     For every i=2…, k-1, the distance between ℓ_i and A_i is bounded by some constant Ψ, depending on K, κ, T, R, and (G,).

Before proving the Claim, we show that it implies Lemma <ref>. Indeed, we can decompose γ' as a union of λ'-hierarchy paths γ'_i with endpoints {ℓ_i, ℓ_i+1}, where we set ℓ_1=1 and ℓ_k=w. Every γ'_i thus connects two points on some coset of N_Ψ(A∪ B). As A and B T-fill all squares, by Theorem <ref> there exists κ, depending on κ and T, such that A∪ B is κ-hierarchically quasiconvex; then Remark <ref> implies that N_Ψ(A∪ B) is κ'-hierarchically quasiconvex, where κ' depends on κ and Ψ. Finally, Lemma <ref> implies that each γ'_i is contained in a neighbourhood of a coset of N_Ψ(A∪ B), whose radius only depends on κ' and λ'. Summing everything up, we just proved that, if M≥ M(K), then γ' decomposes as a union of subpaths, whose distance from A *_C B is bounded above only in terms of K, κ, T, R, and (G,).




Recall that, for every i=2,…, k-1, we have a point ℓ_i such that 
    min{_W_i(1,ℓ_i), _W_i(w,ℓ_i)}>M/5,
 and we want to prove that ℓ_i and A_i are uniformly close. To do so, it is enough to prove that, for every U∈, _U(A_i, ℓ_i) is uniformly bounded in terms of  K, κ, T, R, and (X,), because then we can apply the realisation property of the κ-HQC A_i. There are five cases to analyse, depending on the relation between U and W_i.




Case 1: U=W_i. Let c∈ C_i be such that 
    _W_i(1,c)≤_W_i(1,C_i)+E≤ M/10+6E,
 where we invoked Lemma <ref>. Similarly, let c'∈ C_i+1 be such that 
    _W_i(w,c')≤_W_i(w,C_i+1)+E≤ M/10+6E.
 Fix three geodesics [π_W_i(1),π_W_i(c)]∪ [π_W_i(c),π_W_i(c')]∪ [π_W_i(c'),π_W_i(w)] ⊂ W_i. As W_i is E-hyperbolic and π_W_i(γ') is a (λ', λ')-quasigeodesic (after reparametrisation), there exists a constant S', depending on E and λ', such that π_W_i(ℓ_i) is S'-close to one of the three geodesics (this is a consequence of e.g. <cit.>, plus the fact that geodesic quadrangles in E-hyperbolic spaces are 2E-thin). 

Now, if π_W_i(ℓ_i) is (2S'+6E)-close to [π_W_i(c),π_W_i(c')], then by κ(0)-quasiconvexity of A_i we have that _W_i(ℓ_i, A_i)≤ S, where S 2S'+6E+ κ(0) (this is the constant we use in Notation <ref> to choose M). 

Thus suppose by contradiction that π_W_i(ℓ_i) is at least (2S'+6E)-far from the geodesic [π_W_i(c),π_W_i(c')], and without loss of generality we can assume that π_W_i(ℓ_i) is S'-close to some point r on the geodesic [π_W_i(1),π_W_i(c)], as in Figure <ref>. In particular _W_i(r,c)>S'+6E, because otherwise π_W_i(ℓ_i) would be at distance at most (2S'+6E) from π_W_i(c). But then

    _W_i(1,ℓ_i)≤_W_i(1,r)+_W_i(r,ℓ_i)=


    =_W_i(1,c)- _W_i(r,c)+_W_i(r,ℓ_i)<


    < (M/10+6E)- (S'+6E)+S'=M/10<M/5,

contradicting Equation(<ref>).






Case 2: U W_i. The projection ρ^U_W_i is well-defined. Furthermore, by Case 1 there is some q∈ A_i such that _W_i(ℓ_i, q)≤ S.

Suppose first that _W_i(ℓ_i, U)> S+2E. Then any geodesic [π_W_i(ℓ_i), π_W_i(q)] inside W_i cannot pass through the E-ball around ρ^U_W_i, so the bounded geodesic image Lemma <ref> tells us that _U(A_i, ℓ_i)≤_U(q, ℓ_i)≤ E, and we are done.

Thus suppose that _W_i(ℓ_i, U)≤ S+2E. As _W_i(1,ℓ_i)>M/5, the triangle inequality yields that 

    _W_i(1,U)≥_W_i(1,ℓ_i)-_W_i(ℓ_i, U)-_W_i(ρ^U_W_i)> M/5-S-3E=


    =M/10+5E + (M/10-S-8E)≥_W_i(1, C_i) + 2E,

where we used that, by our choice of M in Notation <ref>, M/10≥ S+10E, and that _W_i(1, C_i)≤ M/10+5E by Lemma <ref>. This means that any geodetic connecting π_W_i(1) to the closest point in π_W_i(C_i) cannot pass E-close to ρ^U_W_i, and the bounded geodesic image Lemma <ref>, applied to the domains U W_i, yields that _U(1, C_i)≤ E. Symmetrically, one gets that _U(w, C_i+1)≤ E. The situation in U is depicted in Figure (<ref>). 



Now, the projection of A_i inside U is a κ(0)-quasiconvex subset, and π_U(γ') is a (λ', λ')-quasigeodesic (after reparametrisation) whose endpoints π_U(1) and π_U(w) are within distance at most E from π_U(A_i). Therefore, again as a consequence of e.g. <cit.>, the distance between π_U(ℓ_i) and π_U(A_i) is bounded in terms of κ(0), λ', and E.












Case 3: U W_i. We claim that both A_i and ℓ_i project uniformly close to ρ^W_i_U inside U. Indeed, _W_i(C_i,C_i+1)≥ M≥ 4E, and in particular one between C_i and C_i+1 is E-far from ρ^U_W_i. Then the Behrstock inequality yields _U(A_i, W_i)≤ E.

In order to bound _U(ℓ_i, W_i), first notice that the projections of 1, ℓ_i, and w to W_i are all at distance at least M/5≥ 4E from each other. Thus, again by Behrstock inequality, at least two of these points project E-close to ρ^W_i_U inside U. If _U(ℓ_i, W_i)≤ E we are done; otherwise

    _U(1, w)≤_U(1, W_i)+_U(ρ^W_i_U)+_U(W_i, w) ≤ 3E.

As π_U(ℓ_i) lies on a (λ',λ')-quasigeodesic between π_U(1) and π_U(w), we get that _U(ℓ_i, W_i) is bounded in terms of λ' and _U(1, w)≤ 3E. 

Case 4: W_i U. Again, we claim that both A_i and ℓ_i project uniformly close to ρ^W_i_U inside U. As pointed out above _W_i(C_i, C_i+1)≥ 4E, so by the bounded geodesic image Lemma <ref> every geodesic in U with endpoints on π_U(C_i) and π_U(C_i+1) must pass E-close to ρ^W_i_U. But π_U(A_i) is κ(0)-quasiconvex, hence _U(W_i, A_i)≤ E+κ(0).

Now, both _W_i(1, ℓ_i) and _W_i(ℓ_i, w) are greater than M/5 ≥ E. Thus any two geodesics [π_U(1), π_U(ℓ_i)] and [π_U(ℓ_i), π_U(w)] inside U must pass E-close to ρ^W_i_U. In turn, π_U(γ') is a (λ',λ')-quasigeodesic (after reparametrisation); therefore, again by e.g. <cit.>, there exists a constant ω≥ 0, depending only on λ' and E, such that the segment of π_U(γ') between π_U(1) and π_U(ℓ_i) is ω-close to the geodesic [π_U(1), π_U(ℓ_i)]. Summing the two facts, we can find a point p∈π_U(γ') such that _U(p,W_i)≤ω+E. Arguing similarly for the other segment of π_U(γ'), we get a point q∈π_U(γ') such that _U(q,W_i)≤ω+E. Now, π_U(ℓ_i) lies on the segment of π_U(γ') between p and q, as in Figure <ref>, and _U(p,q)≤ 2ω+3E. This implies that the distance between π_U(ℓ_i) and ρ^W_i_U is controlled in terms of E, ω, and λ'.



Case 5: U W_i. This case is itself split into several subcases, as it also involves the relation between U and both W_i-1 and W_i+1. 

Case 5.1: First, notice that neither W_i-1 nor W_i+1 can be nested in U, as otherwise one of them would be orthogonal to W_i.

Case 5.2: Suppose that W_i-1 is also orthogonal to U. As A and B have no drift in the orthogonals (Definition <ref>), one of the following happens:

    
  * If π_U(A_i) is R-dense in U, then in particular _U(ℓ_i, A_i)≤ R, and we are done.
    
  * Otherwise, π_U(A_i-1) is R-dense in U. If _U(A_i-1)≤ T then U has diameter at most T+2R, and again we conclude as _U(ℓ_i, A_i)≤ R+2T. Otherwise, as A_i-1 and A_i fill all squares, we must have that π_U(_A_i-1(A_i)) is T-dense in π_U(A_i-1), and therefore (T+R)-dense in U. Then, as π_U(C_i) coarsely coincides with π_U(_A_i-1(A_i)) by Lemma <ref>, we get that the distance between π_U(ℓ_i) and π_U(C_i)⊆π_U(A_i) is uniformly bounded.





Case 5.3: We are left with the cases when both ρ^U_W_i-1 and ρ^U_W_i+1 are well-defined. First, we notice that

    M≤_W_i-1(C_i-1,C_i)≤_W_i-1(C_i-1,U)+_W_i-1(ρ^U_W_i-1∪ρ^W_i_W_i-1) +_W_i-1(C_i, W_i)≤


    ≤_W_i-1(C_i-1,U)+4E +M/10+E,

where we used Lemma <ref> to bound the projections of the two orthogonal domains U W_i, and Claim <ref> to bound _W_i-1(C_i, W_i). Therefore 
    _W_i-1(C_i-1,U)≥ 9/10M-5E.

Furthermore _W_i-1(1,C_i-1)≤ M/10+5E by Lemma <ref>, so

    _W_i-1(1,U)≥_W_i-1(C_i-1,U)-_W_i-1(C_i-1)-_W_i-1(1,C_i-1)≥


    ≥9/10M-5E-M/10-M/10-5E=7/10M-10E.

Notice that both _W_i-1(1,U) and _W_i-1(C_i-1,U) are greater than 2E, by our choice of M in Notation <ref>.

Now we claim that _U(1, C_i-1)≤ 3E. Indeed, if W_i-1 U, then the Behrstock inequality (<ref>) yields that both π_U(1) and π_U(C_i-1) are E-close to ρ^W_i-1_U. If instead U W_i-1 we notice that any geodesic connecting π_W_i-1(1) to π_W_i-1(C_i-1) lies in the (M/10+5E)-neighbourhood of π_W_i-1(C_i-1), and in particular it cannot pass E-close to ρ^U_W_i-1 as 

    _W_i-1(C_i-1, U)-(M/10+5E)≥ 4M/5-10E≥ 2E;

thus the bounded geodesic image Lemma <ref> tells us that _U(1, C_i-1)≤ E.







Arguing the exact same way, one gets that _U(w, C_i+2)≤ 3E, so the picture inside U is as in Figure (<ref>).




Now, the projections of A_i-1, A_i, and A_i+1 inside U are all κ(0)-quasiconvex, so their union is (κ(0)+4E+2)-quasiconvex by Lemma <ref>. As π_U(γ') is a (λ', λ')-quasigeodesic (after reparametrisation), whose endpoints π_U(1) and π_U(w) are within distance at most 3E from π_U(C_i-1) and π_U(C_i+1), respectively, we get that π_U(ℓ_i) must be ξ-close to π_U(A_i-1∪ A_i∪ A_i+1), for some constant ξ depending on λ', κ(0), and E. 

Now, if π_U(ℓ_i) is (ξ+T)-close to π_U(A_i) then we are done. Otherwise, suppose that _U(ℓ_i, A_i)>ξ+T, so that we can assume without loss of generality that π_U(ℓ_i) is ξ-close to some point q∈π_U(A_i-1). Then by triangle inequality 

    _U(A_i-1)≥_U(C_i,q)≥_U(C_i,ℓ_i)-_U(ℓ_i, q)≥_U(A_i, ℓ_i)-_U(ℓ_i, q)≥ T.

Moreover, 
    _W_i(A_i)≥_W_i(C_i, C_i+1)≥  M≥ T.

As A and B fill all squares (Definition <ref>) and W_i U, 
we have that either π_U(_A_i-1(A_i)) is T-dense in π_U(A_i-1), or π_W_i(_A_i(A_i-1)) is T-dense in π_W_i(A_i). Combining this with Lemma <ref>, which states that the gates coarsely coincide with the intersection, and the fact that projection maps are (E,E)-coarsely Lipschitz, we get that either π_U(C_i) is (T+LE+E)-dense in π_U(A_i-1), or π_W_i(C_i) is (T+LE+E)-dense in π_W_i(A_i).

However _W_i(C_i, C_i+1)≥ M>T+LE+E, again by our choice of M; so we must have that π_U(C_i) is (T+LE+E)-dense in π_U(A_i-1). In turn, this means that

    _U(ℓ_i, A_i)≤_U(ℓ_i, C_i)≤_U(ℓ_i, A_i-1)+(T+LE+E)≤ξ+T+LE+E,

and we are done.

The proof of Lemma <ref>, and in turn of Theorem <ref>, is now complete.




















 §.§ Why no drift?

We now show that the conclusion of Theorem <ref> might not hold if one removes the hypothesis of having no drift in the orthogonals, Definition <ref>. Let ℱ_a,b be the free group on two generators a and b, and let D_x,y⟨ x,y | x^2=y^2=1⟩ be a copy of D_∞ generated by the involutions x and y. Let

    G=ℱ_a,b× D_x_1,y_1× D_x_2,y_2,

and let X be the Cayley graph for G with respect to the generators {a,b, x_1, y_1, x_2, y_2}. The G-action on X, which is a direct product of hyperbolic spaces, makes G into a HHG; in particular, combining <cit.>, we can find a HHG structure  whose only domains with unbounded coordinate spaces are the following:

    
  * For every g∈ℱ_a,b, there is a domain gL_a whose coordinate space is g⟨ a⟩a, and a domain gL_b defined analogously;
    
  * The Bass-Serre tree T of the splitting ℱ_a,b=⟨ a⟩ * ⟨ b⟩ is a domain, whose coordinate space is the tree itself;
    
  * Finally, there are two domains W_1 and W_1, whose coordinate spaces are, respectively,  D_x_1,y_1{x_1,y_1} and D_x_2,y_2{x_2,y_2}.

The relations between the above domains are as follows: T, W_1, and W_2 are pairwise orthogonal; for every g∈ℱ_a,b, gL_a and gL_b are nested inside T; every two domains which are nested inside T are transverse.

Now let A=⟨ a^N x_1x_2⟩ and B=⟨ b^N y_1y_2⟩, where N is a positive integer to be chosen later. For every g∈ A-{1} let Y_g=L_a, and similarly for every g∈ B-{1} let Y_g=L_b. One can choose N large enough that A and B satisfy the assumptions of Theorem <ref>. Moreover, notice that the projection of A to every domain which is not L_a has diameter bounded by some constant K, while π_L_a(A) is coarsely dense in L_a. In particular, A is hierarchically quasiconvex, and similar considerations hold for B. We also notice that A and B (K+1)-fill all squares, as for every two orthogonal domains U and V we have that min{_U(A),_V(B)}≤ K. However, A and B do not satisfy Definition <ref>, as for example the unbounded domain W_1 is orthogonal to both L_a and L_b but both A and B have bounded projection to W_1. 

Finally, A*B is not hierarchically quasiconvex. Indeed, the projection of A*B to the product D_∞^2 D_x_1,y_1× D_x_2,y_2 is within finite distance from the diagonal ⟨ (x_1y_1,x_2y_2)⟩; hence the projection of A * B to both  W_1 and W_2 is coarsely dense, but a point p∈ D^2_∞ can be arbitrarily far from the diagonal. Thus A*B does not satisfy the realisation property from Definition <ref>. 





§ COMBINATION OF STRONGLY QUASICONVEX SUBGROUPS

We conclude the paper by studying when our amalgamation procedure preserves the following notion of quasiconvexity:

Let X be a geodesic metric space. A subspace Y⊆ X is Q-strongly quasiconvex, for some function Q [0,+∞)→ [0,+∞) called the strong convexity gauge of Y, if, given any λ≥ 0, every (λ, λ)-quasigeodesic with endpoints on Y lies in the Q(λ)-neighbourhood of Y.

The above notion is equivalent to quasiconvexity in hyperbolic spaces (see e.g. <cit.>), but it is stronger in general. 

We also need the following definition from <cit.>:

    For Θ≥ 0, a subset A of an HHS (X,) has the Θ–orthogonal projection dichotomy if for all U,V∈ with U V, if _U(A)≥Θ then π_V(A) is Θ-dense in V.


The following Lemma shows how strong quasiconvexity and hierarchical quasiconvexity are related:


    Let (X,) be a hierarchically hyperbolic space. A subspace Y is Q-strongly quasiconvex, for some gauge Q, if and only if it is κ-hierarchically quasiconvex and has the Θ-orthogonal projection dichotomy, for some κ and Θ. Moreover, the gauge Q and the pair (κ,Θ) each determine the other.




 Let (G,) be a HHG, let A,B≤ G be two Q-strongly quasiconvex subgroups of G, and let C=A∩ B. Suppose that there exists M≥0 such that A and B satisfy the hypotheses of Theorem <ref>, for some choice of Y_a,Y_b∈ for every a∈ A-C and every b∈ B-C.

There exists a positive constant 𝔐≥ 0 and a function 𝔔[0,+∞)→ [0, +∞), both depending on Q and (G,), such that if M≥𝔐 then ⟨ A,B⟩_G≅ A*_C B is 𝔔-strongly quasiconvex in G.



In view of Lemma <ref>, Theorem <ref> can be rephrased in the following form, which is the one we shall prove:


 Let (G,) be a HHG, let A,B≤ G be two κ-hierarchically quasiconvex subgroups of G, and let C=A∩ B. Suppose that:
 
     
  * There exists M≥100E such that A and B satisfy the hypotheses of Theorem <ref>;
     
  * There exists Θ≥ 0 such that A and B have the Θ-orthogonal projection dichotomy (Definition <ref>).
 
Then there exist positive constants 𝔐, 𝔗≥ 0 and a function 𝔨[0,+∞)→ [0, +∞), all depending on κ, Θ, and (G,), such that if M≥𝔐 then ⟨ A,B⟩_G≅ A*_C B is 𝔨-hierarchically quasiconvex in G, and has the 𝔗-orthogonal projection dichotomy.







    Firstly, we prove that, if M≥Θ, then the orthogonal projection dichotomy for A and B implies the second and third hypotheses of Theorem <ref> (Definitions <ref> and <ref>), for some constants T and R depending on κ, Θ, and (G,).

A and B fill all squares. Let U,V∈ be such that U V, and suppose that 
    min{_U(A), _V(B)}≥ 2Θ+1.

By the orthogonal projection dichotomy, π_U(B) is Θ-dense in U. This means that, for every a∈ A, there is some b∈ B such that _U(a,b)≤Θ, and in particular _U(a, _A(b))≤ 2Θ+1 as π_U(_A(b)) is defined by taking the coarse closest point projection of π_U(b) onto π_U(A). Hence π_U(_A(B)) is (2Θ+1)-dense in π_U(A), that is, we proved that A and B T-fill all squares for T=2Θ+1. 

A and B have no drift in the orthogonals. Next, notice that the Θ-orthogonal projection dichotomy for A and B implies that A and B have no drift in the orthogonals, for R=Θ. Indeed, if b∈ B-C and U∈ is orthogonal to Y_b, then π_U(B) is Θ-dense in U, as we know that _Y_b(B)≥_Y_b(C,bC)≥ M≥Θ.


Orthogonal projection dichotomy. Now assume that 

    M≥Θ+4E+𝔐,

where 𝔐 is the constant from Theorem <ref> (which in turn depends on κ, E, and the constants T and R from the previous paragraphs). This choice of M grants the hierarchical quasiconvexity of A*_C B. Our final goal is to prove that A*_C B satisfies the 𝔗-orthogonal projection dichotomy, where 
    𝔗=.


Let U V be such that _U(A*_C B)≥𝔗. Up to the action of the group, we can assume that _U(1,w)≥𝔗/2, where w=g_1… g_k c is some element of A*_C B, with every g_i in a different factor than g_i+1. Define A_i, C_i, and W_i as in Notation <ref>. 


If _U(A_i)≥Θ for some i then, by the orthogonal projection dichotomy for either A or B, we have that π_V(A_i) is Θ-dense in V. In particular π_V(A*_C B) is 𝔗-dense in V, and we are done. 

Thus suppose that max_i=1, …, k{_U(A_i)}< Θ. Our current goal is to find an index j such that, inside U, the factor A_r projects far from both 1 and w whenever the difference |j-r| is sufficiently small, as we will clarify in Equation (<ref>) below.

Let j≤ k be the first index for which 
    _U(1, A_j)≥𝔗/4-Θ
 (such an index exists as, for example, _U(1, A_k)≥_U(1, w)-_U(A_k)≥𝔗/2-Θ). Notice that j>3, as 1∈ A_1 and _U(A_1∪ A_2∪ A_3)≤ 3Θ, which is strictly less than 𝔗/4-Θ by our choice of 𝔗, Equation (<ref>). Furthermore _U(1, A_j-1)≤𝔗/4-Θ, by minimality of j. Hence

    _U(A_j, w)≥_U(1, w)-_U(1, A_j-1)-_U(A_j-1∪ A_j)≥𝔗/4-3Θ,

where we used that A_j-1 intersects A_j and therefore _U(A_j∪ A_j-1)≤ 2Θ. Summarising, we found an index j such that

    min{_U(1, A_j), _U(A_j,w)}≥𝔗/4-3Θ.

Now notice that, whenever |j-r|≤ 3, we have that 

    min{_U(1, A_r), _U(A_r,w)}≥𝔗/4-7Θ.

Indeed

    _U(1, A_r)≥_U(1, A_j)-_U(⋃_t=j^rA_t),

and as each A_t intersects the next one we have that

    _U(⋃_t=j^rA_t)≤∑_t=j^r_U(A_r)≤ (|j-r|+1)Θ≤ 4Θ.
 
The same argument shows that _U(A_r,w)≥𝔗/4-7Θ, hence the situation in U is as in Figure (<ref>).



Now we look at the relation between U and any W_r for which |j-r|≤ 3. 

Containment implies the conclusion. Suppose first that W_r U, for some W_r as above. Then V, which was orthogonal to U, is also orthogonal to W_r. As 
    _W_r(A_r)≥_W_r(C_r, C_r+1)≥ M≥Θ,
 the orthogonal projection dichotomy for A_r implies that π_V(A_r) is Θ-dense in V, and we are done. Then from now on assume that U does not contain any W_r as above.

No transversality. Next, we argue that U cannot be transverse to any W_r. Indeed _W_r(C_r, C_r+1)≥  M≥ 4E. Therefore, if U were transverse to W_r, then at least one between C_r and C_r+1 would be at distance greater than E from ρ^U_W_r. By the Behrstock inequality, this would imply that _U(W_r, A_r)≤ E, and in turn

    _U(1, W_r)≥_U(1, A_r)-_U(ρ^W_r_U)-_U(W_r, A_r)≥


    ≥𝔗/4-7Θ-2E≥ 2E,

by how we chose 𝔗 in Equation (<ref>). The same proof gives that _U(W_r,w)≥ 2E. But then the Behrstock inequality would again imply that 1 and w both project E-close to ρ^U_W_r inside W_r, which is impossible.


No consecutive orthogonality. Then we can assume that U is either orthogonal to, or nested in, each W_r with |j-r|≤ 3. 
Suppose that both |j-r|≤ 3 and |j-(r-1)|≤ 3. We claim that U cannot be orthogonal to both W_r-1 and W_r. If this was the case, then π_U(C_r) would be R-dense inside U, as A and B have no drift in the orthogonals (Definition <ref>). But this cannot happen, as _U(C_r)≤_U(A_r)≤Θ while ( U)≥_U(1,w)≥𝔗/2, which is strictly greater than 2R+Θ by our choice of 𝔗 in Equation (<ref>).

No alternated nesting. We are left with the case when, whenever |j-r|≤ 3 and |j-(r-1)|≤ 3, U is nested inside one between W_r-1 and W_r. As there are seven domains between W_j-3 and W_j+3, we can find three indices j-3≤ a<b<c≤ j+3 such that U is nested in W_a, W_b, and W_c. The consistency Axiom (<ref>) yields that _W_b(W_a, U)≤ E and _W_b(W_c, U)≤ E. Hence

    _W_b(W_a,W_c)≤_W_b(W_a, U)+(ρ^U_W_b)+_W_b(W_a, U)+≤ 3E.

This gives a contradiction, because a<b<c and Claim <ref> tells us that _W_b(W_a,W_c) must be at least 6E.
The proof of Theorem <ref> is now complete.















 



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