Strong asymptotic freeness of Haar unitaries in quasi-exponential dimensional representations
Michael Magee Mikael de la Salle

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Strong asymptotic freeness of Haar unitaries in quasi-exponential
dimensional representations
    Michael Magee and Mikael de la Salle
    September 9, 2024
=============================================================================================

§ ABSTRACT 
We prove almost sure strong asymptotic freeness of i.i.d. random unitaries
with the following law: sample a Haar unitary matrix of dimension
n and then send this unitary into an irreducible representation
of (n). The strong convergence holds as long as the irreducible
representation arises from a pair of partitions of total size at most
n^1/24-ε and is uniform in this regime. 

Previously this was known for partitions of total size up to ≍log n/loglog n
by a result of Bordenave and Collins. 





§ INTRODUCTION


Let (n) denote the group of complex n× n unitary matrices.
For each k,ℓ∈ there is a unitary representation

    π_k,ℓ^0:(n)→((^n)^⊗ k⊗((^n)^∨)^⊗ℓ).

This representation has a non-zero invariant vector for (n) if
and only if k=ℓ, and in that case the space of invariant vectors
is understood[The space of invariant vectors is the image of [S_k] in ((^n)^⊗ k)≅(^n)^⊗ k⊗((^n)^∨)^⊗ k.
If n≥ k this is isomorphic to [S_k] and we will always
be in this regime in this paper. If n<k the dimension of the space
is still understood as the number of permutations in S_k with
longest increasing subsequence ≤ n <cit.>.]. Let π_k,ℓ be the restriction of π_k,ℓ^0 to
the orthocomplement to the invariant vectors.

Fix r∈[n]. The main theorem of this paper is the following.

Let U_1^(n),…,U_r^(n) denote i.i.d.
Haar distributed elements of (n). For any A<1/24 the
following holds almost surely. For any non-commutative *-polynomial[That is, a polynomial in r non-commuting indeterminates X_1,…,X_r
and their formal adjoints X_1^*,…,X_r^*.] p

    sup_k+ℓ≤ n^A|‖π_k,ℓ(p(U_1^(n),…,U_r^(n)))‖ -p(x_1,…,x_r)|=o(1)

where x_1,…,x_r are generators of a free group _r
and the norm on the right is the one in C_^*(_r).


The analog of Theorem <ref> with (n) replaced by S_n
will appear in forthcoming work of E. Cassidy.

Theorem <ref> gives a clean statement about strong convergence
of i.i.d. Haar elements of (n) in representations of quasi-exponential
dimension in the following form:

Let U_1^(n),…,U_r^(n) denote i.i.d.
Haar distributed elements of (n). For any A<1/24
the following holds almost surely. For any non-commutative *-polynomial
p

    sup_π∈(n)\
    (π)≤exp(n^A)
    |‖π(p(U_1^(n),…,U_r^(n)))‖ -p(x_1,…,x_r)|=o(1).



Our proof of Theorem <ref> relies on two things:


  * a recent breakthrough of Chen, Garza-Vargas, Tropp, and van Handel
<cit.> who found a new (and remarkable) approach to strong
convergence based on `differentiation with respect to n^-1'.
We add to this method in various ways in the sequel. One of the conceptual
differences is a new criterion that we find for temperedness of a
unitary representation of a free group (or more generally a group
with the rapid decay property), and therefore for strong convergence
towards the regular representation, see <ref>.
This allows us to bypass the use of Pisier's linearization argument,
which was essential in many proofs of strong convergence so far (a
notable exception is the work of Paraud and Collins-Guionnet-Parraud
<cit.>),
and replace it by considerations on random walks on free groups, see
<ref>.

  * The method above is given as input rapid decay estimates for expected
values of stable characters of word maps on (n). The key feature
of these estimates is that, up to a point, they actually improve with
the complexity of the representations that we consider. This feature
is intimately related to the concept of stable commutator length
in free groups as uncovered in <cit.>.


Theorem <ref> was proved when k=1,ℓ=0 by Collins and
Male in <cit.>. This resolved a problem left open in
Haagerup and Thorbjørnsen's breakthrough work <cit.>.
Bordenave and Collins <cit.> prove Theorem <ref>
in the regime 

    k+ℓ≤ clog(n)/loglog(n)

for some positive c. 

Theorem <ref> can be written in the following two alternate
ways.


  * Let V_i^(n)⊕_k+ℓ≤ n^Aπ_k,ℓ(U_i^(n)).
The random matrices V_i^(n) are (almost surely) strongly
asymptotically free.

  * The random unitary representations of _r described by x_i↦ V_i^(n)
almost surely strongly converge to the regular representation
of _r.


The following simple case of Theorem <ref> is both new and
important.

For any A<1/24, almost surely,
for all k,ℓ∈∪{0} such that k+ℓ≤ n^A

    ‖∑_i=1^rπ_k,ℓ(U_i^(n))+π_k,ℓ(U_i^(n))^-1‖ =2√(2r-1)+o_n→∞(1).



We highlight this corollary in connection with the following well
known question of Gamburd, Jakobson, and Sarnak <cit.>
— do generic in (Haar) measure (u_1,…,u_r)∈SU(2)^r
have some ϵ>0 such that for all k>0

    ‖∑_i=1^rπ_k(u_i)+π_k(u_i)^-1‖≤2r-ϵ?


It was proved by Bourgain and Gamburd in <cit.>
for u_i with algebraic entries that generate a dense subgroup
of SU(2). This was extended to SU(n) for general
n≥3 by Bourgain and Gamburd <cit.> —
in this case the spectral gap is uniform over all representations
π_k,ℓ with k+ℓ>0. The analogous result for all compact
simple Lie groups was obtained by Benoist and de Saxce <cit.>.

Of course, Corollary <ref> does not advance these
questions for any fixed n as it deals only with infinite sequences
of random matrices of dimension n→∞ but it does make significant
progress on a relaxation of the problem. The motif of Theorem <ref>
is that we have only a small amount of randomness of our matrices
compared to their dimensions.

The method of proof also allows us to obtain estimates for large matrix coefficients. For example, we have the following result, which relies on the ideas from <cit.> and the results of <cit.>. 
 Let k_n≤exp(n^1/2 (log n)^-4). For every q and every sequence P_n of non-commutative *-polynomial of degree ≤ q and with coefficients in M_k_n(), almost surely
  
    lim_n P_n(U_1^(n),…,U_r^(n))/P_n(x_1,…,x_r)=1.

  
Bordenave and Collins <cit.> proved the same theorem for q=1 and k_n≤exp(n^1/32r+160) instead of k_n=exp(n^1/2 + o(1)); this result gives yet another proof of the result from <cit.>, which has important consequences in the theory of von Neumann algebras <cit.>. It is an intriguing question whether the exponent 1/2 is optimal or not. As observed by Pisier <cit.>, the optimal exponent is necessarily at most 2.


 §.§ Acknowledgments


We give huge thanks to Ramon van Handel who generously explained the
methods of the work <cit.> to us during several enjoyable
meetings at the I.A.S. in April/May 2024. Without these meetings,
this paper would not exist.

Funding: 

M. M. This material is based upon work supported by the National Science
Foundation under Grant No. DMS-1926686. This project has received
funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme
(grant agreement No 949143).

M. S. Research supported by the Charles Simonyi Endowment at the Institute
for Advanced Study, and the ANR project ANCG Project-ANR-19-CE40-0002.



§ PRELIMINARIES




 §.§ Representation theory of (n)


For partitions λ=(λ_1,…,λ_p)⊢ k,μ=(μ_1,…,μ_q)⊢ℓ,
for n≥ p+q let s_λ,μ:(n)→ denote the character
of the representation 

    (π_λ,μ,V^λ,μ)

with dominant weight

    (λ_1,λ_2,…,λ_p,0,0,…,0    ,-μ_q,-μ_q-1,…,-μ_1).
    
    n-(p+q)

Because this character interpolates for all n≥ p+q, we refer
to it as a stable character. In the paper we write |λ|=k
for the size of λ.



 §.§ Representation theory of (n)


The irreducible unitary representations of (n) are closely related
to those of (n): for every partitions λ,μ as before
with n≥ p+q, the restriction of π_λ,μ to (n)
is an irreducible representation; all irreducible representations
of (n) appear in this way; and the restrictions of π_λ,μ to
(n) and π_λ',μ' to (n) coincide if and only
if the dominant weights differ by a multiple of (1,…,1). In
other words, the irreducible unitary representations of (n)
are indexed by the sets of integral dominant weights of the root system
A_n-1, which can be parametrized by the set of n-tuples
of integers Λ=(Λ_1,…,Λ_n)∈^n with
Λ_1≥…≥Λ_n, modulo the subgroup (1,…,1). 

Write Λ_1 for the norm on ^n/(1,…,1)


    Λ_1=inf_t∈∑_i|Λ_i-t|.


It is the natural norm in the dual of the subspace of (^n,·_∞)
for which the coordinates sum to 0. In what follows, whenever Λ
belongs to the quotient ^n/((1,…,1)), we will
chose a representative (Λ_1,…,Λ_n)∈^n
such that ∑_i|Λ_i|=Λ_1. If Λ
belongs to the image of ^n, we can and will assume that (Λ_1,…,Λ_n)
has integer coordinates.

By Weyl's dimension formula, if π is an irreducible representation
of (n) with highest weight Λ, 

    (π)=∏_1≤ i<j≤ nj-i+Λ_i-Λ_j/j-i.

Lower bounds for (π) in terms of Λ have been obtained
in <cit.>, but we will need a bound that is
more precise[For example if Λ=(k,0,…,0), our bound gives (π)≥exp(ck),
whereas <cit.> gives (π)≥(k+1)^log(n-1).] in the regime Λ_1=o(n). 

There is a constant c>0 such that for every integer n≥2 and
every irreducible representation π of (n), 

    exp(cmin(Λ_1,n))≤(π)≤exp(Λ_1log n),

where Λ is the highest weight of π. 



The upper bound is because π appears as a sub-representation
of the representation of (n) on (^n)^k⊗(^n)^⊗ℓ),
where k is the sum of the positive Λ_i's and -ℓ
is the sum of the negative Λ_i's. And (^n)^k⊗(^n)^⊗ℓ)
has dimension (k+ℓ)n=exp(Λ_1log n).

For the lower bound, by Weyl's dimension formula (<ref>)
we have to prove 

    cmin(Λ_1,n)≤∑_i<jlog(1+Λ_i-Λ_j/j-i).

We will prove this inequality for every Λ∈^n with Λ_1≥…≥Λ_n.
In that case, since the right-hand side increases if we replace Λ
by tΛ for t>1, it is enough to consider the case when
Λ_1≤ n. Finally, replacing Λ by (-Λ_n,…,-Λ_1),
we can assume that the sum of the positive entries of Λ is
at least Λ_1/2.

Let n_1=⌊n/2⌋ and n_2=⌊3n/4⌋.
Expressing that the ℓ_1 norm of (Λ_1,…,Λ_n)
is at most the ℓ_1 norm of (Λ_1-t,…,Λ_n-t)
for every t≠0, we see that Λ has at most n/2
positive entries, and at most n/2 negative entries. In
particular, we have 

    ∑_i=1^n_1Λ_i≥1/2Λ_1,

and Λ_j≤0 for every n_2<j≤ n. If i≤ n_1
and j>n_2, we have 

    Λ_i-Λ_j/j-i≥Λ_i/n.

We deduce the obvious bound 

    ∑_i<jlog(1+Λ_i-Λ_j/j-i)≥∑_i≤ n_1,n_2<jlog(1+Λ_i/n).

By the bound log(1+t)≥ tlog(2) for every 0≤ t≤1 and
the inequality 0≤Λ_i≤Λ_1=n, this is at
most 

    log(2)∑_1≤ i≤ n_1∑_n_2<j≤ nΛ_i/n≥log(2)/4∑_i=1^n_1Λ_i.

By (<ref>), we obtain
(<ref>) with c=log(2)/8.


For every 0<A≤1 and
every n≥ N, for every irreducible representation π of (n):

    (π)<exp(cn^A)

implies 

    π⊂⊕_k+ℓ≤ n^Aπ_k,ℓ,

which in turn implies 

    (π)≤exp(n^Alog n).






§ MATRIX INTEGRAL RESULTS


Let _r denote the free group on a fixed generating set X{x_1,…,x_r}.
For w∈_r let 

    w:(n)^r→(n)

denote the induced word map defined by substituting in an r-tuple
of unitaries for the elements of X appearing in a reduced expression
of w. For example if w=x_1x_2x_1^-1x_2^-1 then w(u_1,u_2,…,u_r)=u_1u_2u_1^-1u_2^-1.
Let |w| denote the word length of w w.r.t. X.

Let 

    _n[s_λ,μ(w)]∫_(n)^rs_λ,μ(w(u_1,u_2,…,u_r))du_1⋯ du_r.

For L∈ let 

    g_L(x)∏_c=1^L(1-c^2x^2)^⌊L/c⌋.


The input into our analysis is the following result about expected
values of stable characters of word maps, building on the works <cit.>.

Let w∈_r be not the identity
with |w|≤ q.


  * There is a polynomial P_λ,μ,w∈[x]
such that for

 n≥(k+ℓ)|w|,

    _n[s_λ,μ(w)]=P_λ,μ,w(1/n)/g_(k+ℓ)q(1/n),

with 

    (P_λ,μ,w)≤3(k+ℓ)q(1+log((k+ℓ)q)).


  * If w is not a proper power in _r,
then _n[s_λ,μ(w)] = O(n^-1/3 (k+ℓ)).

  * If w is not a proper power in _r and μ=∅, then  _n[s_λ,∅(w)] = O(n^-(k+ℓ)).



It is probable that the conclusion of Part <ref>
also holds with O(n^-(k+ℓ)), but we are unsure how to prove
this at the moment, and it just changes constants in our theorems.

Part <ref> of Theorem <ref> follows
from combining a result of <cit.> with one of Duncan
and Howie <cit.> as explained in <ref>.

Theorem <ref> will be proved in the later part
of the paper (<ref>-<ref>).

In <ref>-<ref>
we prove Theorem <ref> from Theorem <ref>.



§ MARKOV BROTHERS INEQUALITY


We use the following fundamental inequality by Andrey and Vladimir
Markov, see <cit.> for references. If P
is a degree ≤ D polynomial in one variable, then for every integer
k, 

    sup_[-1,1]|P^(k)|≤D^2(D^2-1)…(D^2-(k-1)^2)/(2k-1)!!sup_[-1,1]|P|

where (2k-1)!!=1·3…(2k-1). By an affine change of variable,
for a general interval (<ref>) becomes: 

    sup_[a,b]|P^(k)|≤2^k/(b-a)^kD^2(D^2-1)…(D^2-(k-1)^2)/(2k-1)!!sup_[a,b]|P|.


We deduce the following, which is essentially <cit.>.

 For every polynomial P of degree ≤ D
and every integer N≥ D^2

    sup_[0,1/N]|P|≤1/1-D^2/N+1sup_n≥ N|P(1/n)|.




Every element of the interval [0,1/N] is at distance at
most 1/2N(N+1) from {1/n| n≥ N}. Therefore,
by the fundamental inequality of calculus, we have 

    sup_[0,1/N]|P|≤sup_n≥ N|P(1/n)|+1/2N(N+1)sup_[0,1/N]|P'|.

By the Markov brothers inequality (<ref>) with k=1,
this is less than 

    sup_n≥ N|P(1/n)|+1/2N(N+1)·2ND^2sup_[0,1/N]|P|.

The lemma follows. 


As a consequence, 

 For every polynomial P of
degree ≤ D and every integer k≤ D, 

    sup_[0,1/2D^2]|P^(k)|≤2^2k+1D^4k/(2k-1)!!sup_n≥ D^2|P(1/n)|.




By (<ref>) with a=0 and b=1/2D^2, 

    sup_[0,1/2D^2]|P^k|≤(4D^2)^kD^2(D^2-1)…(D^2-(k-1)^2)/(2k-1)!!sup_[0,1/2D^2]|P|.

The lemma follows from the bound D^2(D^2-1)…(D^2-(k-1)^2)≤ D^2k
and Lemma <ref>. 








§ A CRITERION FOR STRONG CONVERGENCE

In this section, Γ will be a finitely generated group with a
fixed finite generating set and corresponding word-length. We denote by 
_≤ q[Γ] the subspace of the elements of the group
algebra of Γ supported in the ball of radius q. 


  A function u Γ→ is tempered if
  
    lim sup_n →∞ |u((x^*x)^n)|^1/2n≤λ(x)
 for every x ∈[Γ]. Here λ is the left-regular representation of Γ on ℓ_2(Γ) and the norm is the operator norm.

  A function u Γ→ is completely tempered
  if 
    lim sup_n →∞ |⊗ u((x^*x)^n)|^1/n≤(id⊗λ)(x)
 for every integer k ≥
  1 and every x ∈ M_k()⊗[Γ], where  is the trace on M_k().

For example, if u(γ) = ⟨π(γ)ξ,ξ⟩ is a matrix coefficient of a unitary representation of Γ, then u is tempered if and only if it is completely tempered, if and only if the cyclic representation generated by ξ is weakly-contained in the left-regular representation. So Definition <ref> can be seen as an adaptation to functions that are not related to representations of the classical notion of tempered representation.

This section is devoted to the proof of the following criterion for strong convergence in probability of a sequence of random representations. 

Let π_n be a sequence of random unitary representations of finite nonrandom dimension, u_n Γ→ be functions and ε_n>0. Assume that
  
  
  * u_n is tempered and there is a polynomial P_n such that for every q and every x ∈_≤ q[Γ], |u_n(x)| ≤ P_n(q) x_C^*(Γ),
  
  * |(π_n(x)) - u_n(x)| ≤ε_n exp(q/log(2+q)^2) x_C^*(Γ) for every q and every x ∈_≤ q[Γ],
  
  * lim_n ε_n = 0.

  Then for every y ∈[Γ], and every δ>0,
  
    lim_n ( π_n(y)≥λ(y)+δ) = 0.


  If C^*_λ(Γ) has a unique trace, then the conclusion becomes that
  
    lim_n ( | π_n(y) - λ(y)| ≥δ) = 0.



  The choice of w(q) = exp(q/log(2+q)^2) is an arbitrary choice that is convenient for our applications, the crucial property that is used is that ∑_q 1/1+q^2log w(q)<∞, reminiscent of the Beurling-Malliavin Theorem. It could be replaced by exp( q^2 u_q) for an arbitrary decreasing and summable sequence (u_q)_g ≥ 0.
  

  Proposition <ref> is a variant with high derivatives of the criterion for strong convergence that appeared implicitely, for polynomial w, in <cit.> with u_n(γ) = 1_γ=1, and <cit.> for general u_n.


  The condition on the existence of P_n is probably not needed, but will be trivially satisfied in the applications. It allows to use standard results on distributions rather than ad-hoc proofs.


We have the following variant for matrix coefficients. 
 Let π_n,d_n,u_n,ε_n be as in Proposition <ref>. Assume moreover that u_n is completely tempered. Let k_n be a sequence of integers such that lim_n ε_n k_n=0.

  Then for every integer q, δ>0 and every sequence y_n ∈_≤ q[Γ] ⊗ M_k_n() with y_n_C^*(Γ)⊗ M_k_n≤ 1,
  
    lim_n ( π_n(y_n)≥λ(y_n)+δ) = 0.


  If C^*_λ(Γ) has a unique trace and is exact, then the conclusion becomes that
  
    lim_n ( | π_n(y_n) - λ(y_n)| ≥δ) = 0.



We first collect a few ingredients that we need for the proof.



































































 §.§ Bump functions with small Fourier coefficient

A nonzero compactly supported continuous function cannot have a Fourier transform that decays too fast at infinity, as it has to satisfy ∫log |f̂(t)|/1+t^2 dt > -∞. In other words, the condition ∫_0^∞φ(t)/1+t^2dt <∞ on a function φ^+ →^+ is a necessary condition for the existence of a nonzero compactly supported continuous function f such that |f̂(t) | ≤exp(-φ(|t|) for every t. The Beurling-Malliavin theorem asserts that this is often also a sufficient condition, see for example <cit.>. The following elementary fact is an illustration of the same phenomenon. The small nuance compared to the Beurling-Malliavin theorem is that we need that f be nonnegative. The same proof works for φ(t) = t/(log(2+|t|))^1+ε replaced by any continuous function [0,∞) → [0,∞) such that φ(t)/t^2 is non-increasing and integrable on [1,∞).

























  






















 For every ε>0, there is a real
number M>0 and an even function C^∞ function f→
that is strictly positive on (-1,1), zero outside of (-1,1), and such
that 

    |f̂(t)|≤exp(-M|t|/(log(2+|t|))^1+ε).




The proof is a standard construction of a smooth bump function as
infinite convolution product of indicator functions <cit.>.
Define a_j=c/j(log2+j)^1+ε, where c>0
is chosen such that ∑_j≥1a_j=1. Let (X_j)_j≥1
be a sequence of independent random variables, all uniform in [-1,1].
Let μ be the law of ∑_ja_jX_j. It is a measure with
full support in [-∑_ja_j,∑_ja_j]=[-1,1]. Then 

    μ(t)=exp(-it∑ a_jX_j)=∏_j≥1sin(ta_j)/ta_j.

The inequality 

    |μ(t)|≤∏_j≤t/(log2+t)^1+ε1/|t|a_j≤exp(-M|t|/(log2+t)^1+ε)

is a computation. It implies that μ is absolutely continuous
with respect to the Lebesgue measure and that its Radon-Nikodym derivative,
which is our f, is C^∞. f does not vanish on the interval
(-1,1) because f is a log-concave function as a limit of convolutions
of log-concave functions. 



In this lemma, we have tried to optimize the rate of decay of the
Fourier transform of f, because this is what allows to take A
the largest in Theorem 1.1. But this is reflected by the fact that
f(t) is extremely small when t<1 is close to 1, something
like exp(-exp(O(1/1-t))), which gives rather bad quantitative
estimates for the speed of the convergence. For smaller values of
K≪ n^A, it would better to use functions with slower decay
of f̂. 


We can periodize the above function:

 For every 0<α<π, there is a nonnegative
real-valued even function φ_α∈^∞(/2π) that is strictly positive on (-α,α) +2π, zero outside of (-α,α) +2π 
with support equal to [α,α] and such that 

    |φ̂_α(k)|≤exp(-M|k|α/log(2+|k|α|)^1+ε)

for every k∈. 



Set φ_α(t+2π)=f(t/α) for every t∈[-π,π]. 





 §.§ Sobolev algebras in the full group C^*algebra


Let (w(q))_q∈ be a non-decreasing sequence of positive real
numbers satisfying w(q_1+q_2)≤ w(q_1)w(q_2). Set

    𝒮_w(Γ)={ x∈ C^*(Γ)| x=∑_qx_q,x_q∈_≤ q[Γ],∑_qw(q)x_q_C^*(Γ)<∞}.

It is a Banach algebra for the norm 

    x_w:=inf{∑_qw(q)x_q_C^*(Γ)| x=∑_qx_q,x_q∈_≤ q[Γ]}.



For w(q)=(1+q)^i we denote the corresponding space 𝒮_i(Γ). 

We have the following elementary fact:
 A function u Γ→ extends by linearity to a linear map 𝒮_w(Γ)→ of norm ≤ N if and only if for every q and every x ∈_≤ q[Γ],
  
    |u(x):=∑_γ x(γ) u(γ)| ≤ N w(q) x_C^*(Γ).

 


The next lemma is useful and suggests the terminology of Sobolev algebras for the algebras 𝒮_i(Γ).
 Let y=y^*∈_≤ q[Γ] and f:[-y_C^*(Γ),y_C^*(Γ)]→ be a continuous function. Let w → as above. Assume that the function φ(θ) = f(y_C^*(Γ)cosθ) satisfies ∑_n ∈ w(q |n|) |φ̂(n)|<∞. Then f(y)∈𝒮_w(Γ) and
  
    f(y)_w ≤∑_n ∈ w(q |n|) |φ̂(n)|.


In particular, if f is C^i+1, then f(y) ∈𝒮_i(Γ) with norm ≤ C_i (1+y_C^*(Γ))^i+1 (1+q)^i f_C^i+1.

 In the proof, we write y for y_C^*(Γ). The assumption implies that the Fourier coefficients of φ are absolutely summable, which gives rise to a norm-converging expansion

    f(y)=∑_n ∈φ̂(n)T_n(y/y)

where T_n is the n-th Chebyshev polynomial T_n(cosθ)=cos(nθ). So T_n(y/y) belongs to _≤ q|n|[Γ], has norm ≤ 1 in C^*(Γ) 
and 

    ∑_nw(q|n|)|φ̂(n)|T_n(y/y)_C^*(Γ)≤∑_nw(q|n|)|φ̂(n)|.

The proves the first part of the lemma. For the second part, if f is C^i+1, then so is φ, and by the Cauchy-Schwarz inequality

    ∑_n (1+q|n|)^i |φ̂(n)| ≤ (∑_n(1+q|n|)^-2)^1/2(∑_n(1+q|n|)^2i+2|φ̂(n)|^2)^1/2,

which is finite.

 Let u Γ→. Assume that u extends to a continuous linear map u𝒮_i(Γ) →. Then u is tempered if and only if there is n such that for every selfadjoint x ∈𝒮_i(Γ) ∩λ, u(x^n)=0. In that case, this holds for all n≥ i+2.

Furthermore, u is completely tempered if and only if there is n such 
that for every k and every selfadjoint x ∈ M_k()⊗ 
(𝒮_i(Γ) ∩λ), (Tr⊗ 
u)(x^n)=0. In that case, this holds for all n≥ i+2.

 We prove the equivalence for temperedness, the proof for complete temperedness is identical.

  We start with a preliminary observation. Lemma <ref> tells us that for every self-adjoint y ∈_≤ q(Γ), the function f ∈ C^∞() ↦ u(f(y)) is a compactly supported distribution of order ≤ i+1, and the largest symmetric interval containing its support is [-lim sup|u(y^k)|^1/k,lim sup|u(y^k)|^1/k], see <cit.>).

 Given this observation, the if direction is direct. Let ε>0, and pick f be a C^∞ function equal to 0 on the interval [-λ(y)^2,λ(y)^2] and to 1 outside of the ε-neighbourhood of it.  For k large enough (so that t↦ |t|^k/n is C^i+2), x=(y^*y)^k/n f(y^*y) belongs to 𝒮_i(Γ) ∩λ. As a consequence, u(x^n)=0 and therefore
  
    u( (y^*y)^k) = u( (y^*y)^k(1-f(y^*y)^n)).

  But the C^i+1 norm of t↦ t^k(1-f(t)^n) is ≤ C(i,ε,y) (λ(y)^2+ε)^k, which implies that
  
    lim sup_k |u( (y^*y)^k)|^1/2k≤√(λ(y)^2+ε).

  This proves that u is tempered.

  For the converse, consider a self-adjoint x ∈𝒮_i(Γ)∩λ. Decompose x= ∑_q x_q as in the definition of 𝒮_i(Γ). We can moreover assume that each x_q is self-adjoint. Let y_q = ∑_s≤ q x_q ∈_≤ q[Γ]. Then we have
  
    λ(y_q) = λ(y_q-x)≤∑_q>nx_s = o((1+q)^-1),

  because ∑_q (1+q)^ix_q<∞. Let φ be a C^∞ function equal to 1 on [-1,1] and 0 outside of [-2,2], and let φ_q(t) = φ(t/λ(y_q)). We know from the preliminary observation that Λ_q(f) = u(f(y_q)) is a distribution with support inside [-λ(y_q),λ(y_q)] and such that |Λ_q(f) |≤ C (1+q)^i f_C^i+1() for some C independent from q. Therefore, for every n we have
    
    u(y_q^n) = u(y_q^n φ_q(y_q))=(1+q)^i O_q →∞(λ(y_q)^n-i-1).

    If n≥ i-2 this goes to 0. Therefore, we have
    
    u(x^n) = u(lim_q y_q^n) = lim_q u(y_q^n)=0.

The proposition is proved. 



 §.§ Unique trace and exactness

We use the following easy consequence of the operator-space reformulation of exactness, in the line of <cit.>.
 Let A be an exact C^*-algebra. For every finite family a_1,…,a_n in A and every ε>0, there is an integer d such that for every k and every b_1,…,b_n ∈ M_k(), there are norm ≤ 1 matrices u,v ∈ M_d,k() such that
  
    (1-ε)∑_i a_i ⊗ b_i≤∑_i a_i ⊗ u b_i v^*.



If a_i belong to M_N(), the lemma is easy with d=N, because the 
norm of X=∑ a_i ⊗ b_i is the supremum of ⟨ X 
ξ,η⟩ over norm 1 elements of  ^N ⊗^k, and 
every element of ^N ⊗^k belongs to ^N ⊗ E for a 
subspace E ⊂^k of dimension ≤ N. The general case 
follows by Kirchberg's characterization of exact C^*-algebras, which 
implies that for every ε>0, there is N such that the 
operator space spanned by a_1,…,a_n is at completely bounded 
distance ≤ 1+ε from a subspace of M_N() 
<cit.>.

 Let Γ be a finitely generated group. Assume that C^*_λ(Γ) has a unique trace. Then for every integer q and ε>0, there exist y_1,…,y_n ∈[Γ] and δ>0 such that, for every finite dimensional unitary representation π of Γ, if
  
    max_i π(y_i)/λ(y_i)≤ 1+δ

  then
  
    inf_y ∈_≤ q[Γ]π(y)/λ(y)≥ 1-ε.

If moreover C^*_λ(Γ) is exact, the conclusion (<ref>) can be strengthened to  
    inf_d≥ 1
     y ∈ M_d()⊗_≤ q[Γ](id⊗π)(y)/(id⊗λ)(y)≥ 1-ε.


We use the following fact: C^*_λ(Γ) has a unique trace if 
and only if for every γ∈Γ∖{1} and every 
ε, there is δ>0 and a finite family y_1,…,y_k 
∈[Γ] such that, for every unitary representation πΓ→ A to a C^*-algebra A with a tracial state σ, 
max_i π(y_i)/λ(y_i)≤ 1+δ implies 
|σ(π(γ))| ≤ε. The only if direction is 
direct: if σ is an arbitrary tracial state on 
C^*_λ(Γ), by applying the criterion to 
A=C^*_λ(Γ) and π the left-regular representation, we 
obtain that |σ(λ(γ))|≤ε for every 
γ≠ 1 and ε>0, that is σ is the standard 
trace. The converse is proved by standard ultraproduct arguments (see 
<cit.> for the details). 




The first conclusion of the lemma follows quite easily from this 
characterization of the unique trace property. Indeed, if y ∈[Γ], there is an integer p such that λ(y)≤ 
(1+ε) τ((y^*y)^p)^1/2p, and by compactness the 
same p can be taken for every y ∈_≤ q[Γ]. Now 
applying the criterion for every γ in the ball of radius 2qp, 
we obtain that there is a finite family y_1,…,y_k ∈[Γ] 
such that, for π,A,σ as above, max_i 
π(y_i)/λ(y_i)≤ 1+δ implies that for 
every y ∈_≤ q[Γ], τ((y^*y)^p)^1/2p≤ 
(1+ε)σ(π((y^*y)^p))^1/2p, and in particular 
λ(y)≤ (1+ε)^2 π(y). This is 
(<ref>), up to a change of ε.

Observe that exactly the same argument also shows that, without any 
further assumption on C^*_λ(Γ), for any fixed d, 
(<ref>) can be replaced by
   
    inf_y ∈ M_d() ⊗_≤ q[Γ](id⊗π)(y)/(id⊗λ)(y)≥ 1-ε.

   But of course, y_i and δ a priori depend on d. If 
C^*_λ(Γ) is exact, 
Lemma <ref> allows to remove 
this dependence and to conclude the proof of the lemma.




 §.§ Proof of Proposition <ref> and Proposition <ref>



  Let i_n be the degree of P_n and w(q) = exp(q/log (2+q)^2). Lemma <ref> implies that u_n extends by continuity to 𝒮_i(Γ), and that on its subspace 𝒮_w(Γ),
  
    |(π_n(x)) - u_n(x)| ≤ε_n x_w.

  
  Without loss of generality, we can take y≠ 0 and normalize it so that y_C^*(Γ)=1. 

By Lemma <ref>, the map f ↦ u_n(f(y^*y/2)), is continuous for the
 
C^i_n+1([-1/2,1/2]) topology, so it extends to a distribution. The assumption that u_n is tempered in turn implies
that its support in contained in
[-λ(y)^2/2,λ(y)^2/2] (see <cit.>). In particular, using λ(y)>0, we have that u_n(f(y^*y))=0 for every f that vanishes on [-λ(y)^2/2,λ(y)^2/2] <cit.>.
  
Let α=arccos(λ(y^*y)/2)∈[π/3,π/2). Let φ_α be the function given by
Lemma <ref>, but for ε = 1/2. It is even,
so it is of the form φ_α(θ)=f(cos(θ)) for
a continuous function f[-1,1]→ that is zero on [-λ(y)^2/2,λ(y)^2/2], strictly positive outside of this interval, and C^∞ on (-1,1). We take x=f(y^*y). By the preceding discussion, u_n(x)=0 for every n. Moreover, by Lemma <ref>, x belongs to 𝒮_w(Γ), because

    ∑_N≥1exp(qN/log(1+qN)^2-Mα N/(log(1+α N))^1+1/2)<∞.


So the inequality (<ref>) together with the obvious inequality π_n(x)≤π_n(x) gives

    π_n(x)≤ε_n x_w,

which goes to zero by the last assumption.

Now, since f is strictly positive on [-1,1]∖ [-λ(y)^2/2,-λ(y)^2/2], there is c>0 such that f(t/2)>c if |t|≥ (λ(y)+δ)^2. In particular, we have

    ( π_n(y)>λ(y)+δ) ≤(π_n(x) >c) ≤1/cε_n x_w,

which goes to 0. This proves the first part of the proposition.

The second half follows from the first half of Lemma <ref>.



 The first part is identical to that of Proposition <ref>, considering the distribution f↦⊗ u_n(f(y_n^* y_n)).

  The second part follows, using the full statement of Lemma <ref>. 
  




§ RANDOM WALKS ON FREE GROUPS


 The content of this section will play a key rôle in
the proof of Theorem <ref> (see Lemma <ref>).



 §.§ Proper powers

Let μ be a symmetric probability measure on _r, whose
support is finite, contains the identity element and generates _r.
To save space, we will call such a measure reasonable. Let
(g_n)_n≥0 be the corresponding random walk on _r,
that is g_n=s_1s_2… s_n for iid s_i distributed
as μ. Let ρ=ρ(μ) be the spectral radius, that is the
norm of λ(μ) on ℓ_2(_r). We have the easy bound

    ∀ g∈_r,(g_n=g)=⟨λ(μ)^nδ_e,δ_g⟩≤ρ^n.


We say that an element of _r is a proper power if it is of
the form h^d for h∈_r and d≥2. 

 There is a constant C=C(μ) such that

    (g_n is a proper power )≤ Cn^5ρ^n.



The proof uses that the Cayley graph of _r is a tree, in the
following form:

 There is a constant C and an integer a
such that for every h∈_r and every h' on the segment from
the identity to h,

    (g_n=h)≤ C∑_k=0^n+a(g_n+a=h and g_k=h'}).




When μ is supported on the generators, this is clear with C=1
and a=0: any nearest-neighbor path from e to h has to pass
through h'. In the general case, the idea is that such a path has
to pass not too far from h', and forcing it to pass exactly through
h' does not cost too much, neither in time nor in probability.
Let us make this idea precise. By our assumptions on μ, there
is an integer r such that the support of μ is contained in
the ball of radius r, and another integer a_0 such that c:=inf_g∈ B(e,r)(g_a_0=g)>0.

Let T be the first hitting time of the ball of radius r around
h', so that if g_n=h then T≤ n. The lemma will follow
from the following observation: if we add 2a_0 steps to the random
walk after time T, with probability at least c^2 we will be
at h' at time T+a_0 and back to g_T at time T+2a_0,
and then we can run the random walk as before. More formally, define
another realization of the random walk as follows: let g'_i be
an independent copy of the random walk, and define 

    g̃_n=
    g_n    if n≤ T
    
    g_Tg'_n-T    if T≤ n≤ T+2a_0
    
    g_Tg'_2a_0g_T^-1g_n-2a_0    if T+2a_0≤ n.

By the Markov property (T is a stopping time) (g̃_n)_n≥0
is distributed as the random walk with step distribution μ.

Conditionally to T, the event A={g'_2a_0=e and g'_a_0=g_T^-1h'}
happens with probability ≥ c^2, and when it happens we have
g̃_n+2a_0=g_n for every n≥ T and g̃_T+a_0=h'.
Therefore, we can bound the probability of the event 

    B={g̃_n+2a_0=h and h'∈{g̃_̃k̃,0≤ k≤ n+2a_0}}

as follows 

    (B)≥(g_n=h and A)≥ c^2(g_n=h).

By the union bound we obtain 

    c^2(g_n=h)≤∑_k=0^n+2a_0(g̃_n+2a_0=h and g̃_̃k̃=h'),

which is the content of the lemma with C=c^-2 and a=2a_0. 



Denote by p_n the probability that g_n is a cyclically
reduced proper power, that is g_n=h^d for a cyclically reduced
h and d≥2. We shall prove the following two inequalities,
from which the proposition follows immediately: there are constants
C_1,C_2 such that 

    (g_n is a proper power )≤ C_2n^2max_k≤ n+2ap_n+2a-kρ^k.



    p_n≤ C_1n^3ρ^n,


We start with the proof of (<ref>). Let h
be cyclically reduced and d≥2. The assumption that h is cyclically
reduced means that h belongs to the segment between 1 and h^d,
and that h^d-1 belongs to the segment between h and h^d
(tautologically if d=2). By two applications of Lemma <ref>,
we have 

    (g_n=h^d)    ≤ C^2∑_k_1+k_2+k_3=n+2a(g_k_1=h,g_k_1+k_2=h^d-1,g_k_1+k_2+k_3=h^d)
        =C^2∑_k_1+k_2+k_3=n+2a(g_k_1=h)(g_k_2=h^d-2)(g_k_3=h).

The inequality (<ref>)
allows us to bound the middle term by ρ^k_2, and by symmetry
we have 

    (g_k_1=h)(g_k_3=h)=(g_k_1=h)(g_k_3=h^-1)=(g_k_1=h and g_k_1+k_2=e).

Summing over all cyclically reduced words, we get 

    ∑_h cyclically reduced(g_n=h^d)≤ C^2∑_k_1+k_2+k_3=n+2aρ^k_2(g_k_1+k_3=e).

By (<ref>), this is less
than C^2(n+2a+1)^2ρ^n+2a. Summing over all d≤ n
we obtain (<ref>).

We now move to (<ref>). Let
h∈_r, not necessarily cyclically reduced. Write h=wh̃w^-1
its reduced expression with h̃ cyclically reduced. Then
h^d=wh̃^dw^-1 is the reduced expression of h^d.
In particular w and wh̃^d are on the segment from the
identity to h^d, and by two applications of Lemma <ref>
we obtain 

    (g_n=h^d)≤ C^2∑_k_1+k_2+k_3≤ n+2a(g_k_1=w)(g_k_2=h̃^d)(g_k_2=w^-1).

If we first sum over all h̃ such that wh̃w^-1
is reduced , and then over w, we obtain 

    (g_n is a proper power)≤ C^2∑_k_1+k_2+k_2≤ n+2ap_k_2∑_w(g_k_1=w)(g_k_3=w^-1).

The formula (<ref>) follows
because ∑_w(g_k_1=w)(g_k_3=w^-1)=(g_k_1+k_3=e)≤ρ^k_1+k_2,
by (<ref>). 




 §.§ Random walks and tempered functions

The following is a useful criterion for a function on a group with the rapid decay property to be tempered. Recall a group is said to have the rapid decay property if it admits a finite generating set and a polynomial P such that, for every a∈_≤ q[Γ],

    λ(a)≤ P(R) (∑_γ |a(γ)|^2)^1/2.

Haagerup's inequality <cit.> asserts that free groups with their standard generating sets have the rapid decay property, with P(R)=3(1+R^2).
 Let Γ be a finitely generated group with the rapid decay property, and u Γ→ a function. Assume that for every reasonable probability measure μ on Γ, if (γ_n)_n ≥ 0 is the associated random walk on Γ,
  
    lim sup_n ( |u(γ_n)|)^1/n≤ρ(μ).

  Then u is tempered, and even completely tempered.

 Before we start, let us observe that the assumption (<ref>) in fact holds for every symmetric finitely supported probability measure μ, even if it not reasonable in the sense that its support is not generating or does not contain the identity. Indeed, let ν be a fixed reasonable probability measure. Then for every ε>0, μ_ε = 1/1+ε(μ+εν) is a reasonable probability measure, so that (<ref>) holds for μ_ε. If _ε and  denote the law of the random walk with transition μ_ε and μ respectively, the inequality μ≤ (1+ε) μ_ε gives
  
    |u(γ_n)| ≤ (1+ε)^n _ε |u(γ_n)|,

  so we obtain 

    lim sup_n ( |u(γ_n)|)^1/n≤ (1+ε)ρ(μ_ε) ≤ρ(μ) + ερ(ν).

This proves that (<ref>) holds for μ by taking ε→ 0.

We now move to the proof of the proposition. We claim that for every k, q ≥ 1 and every x=x^* ∈ M_k() ⊗_≤ q[Γ],
  
    lim sup_n →∞ |⊗ u(x^n)|^1/n≤ P(q) √(k)(id⊗λ)(x).

  Here P is the polynomial satisfying (<ref>), given by the assumption that Γ has the rapid decay property.
  
By a standard tensor-power trick, (<ref>) implies that u is completely tempered. Indeed, given an arbitrary x ∈ M_k() ⊗_≤ q[Γ], applying (<ref>) to y=(x^*x)^m (which belongs to M_k() ⊗_≤ 2m q[Γ] and satisfies (id⊗λ)(y) = (id⊗λ)(x)^2m) and raising to the power 1/2m, it implies

    lim sup_n →∞ |⊗ u((x^*x)^n)|^1/2n≤ (P(2mq) √(k))^1/2m(id⊗λ)(x),

and in the m →∞ limit 

    lim sup_n →∞ |⊗ u((x^*x)^n)|^1/2n≤(id⊗λ)(x).


So it remains to prove (<ref>). We can normalize x so that ∑_γx(γ)=1, where · is the operator norm on M_k(). Let μ(s) = x(s); it is a symmetric finitely supported probability measure on Γ. By the triangle inequality and the inequality |(x(s_1)… x(s_n))| ≤ k x(s_1)…x(s_n), we have 

    |⊗ u(x^n)| ≤ k ∑_s_1,…,s_n ∈Γμ(s_1)…μ(s_n) |u(s_1… s_n)|  = k  |u(γ_n)|.

By the assumption in the extended form above, we obtain

    lim sup_n →∞ |⊗ u(x^n)|^1/n≤ρ(μ).

By the rapid decay assumption (<ref>), this is less than

    P(q) (∑_s μ(s)^2)^1/2.

By by the basic fact μ(s)^2=x(s)^2 ≤(x(s)^* x(s)), we have

    ∑_s μ(s)^2 ≤ (⊗λ)(x^*x) ≤ k (id⊗λ)(x)^2,
which proves the claimed inequality (<ref>).





§ PROOF OF THEOREM <REF>
FROM THEOREM <REF>


Let K≥1 be an integer. Most of the objects will depend on K
but our notation will not reflect this. The only exception is the
letter C, which will always denote a constant that does not depend
on anything, but that can change from one line to the next.

The aim of this section is to explain why Theorem <ref>
implies the strong convergence result from Theorem <ref>.

Let 

    σ_n⊕_|λ|+|μ|=Kπ_λ,μ.

It is a sub-representation of ⊕_k=0^ℓπ_k,K-k^0,
therefore 

    dim(σ_n)≤(K+1)n^K.

Recall the definition of g_L from <ref>.
Theorem <ref> immediately implies the following
result about σ_n. Throughout this section and as in <ref> _n or simply  will denote
the integral with respect to the Haar measure on (n)^r.

 For every w∈_r, there
is a rational function φ_w∈(x) such that 


  *  For every integer n≥ Kmax(|w|,1),

    φ_w(1/n)=1/n^K_n[(σ_n(u_1,…,u_r))].


  *  If w≠ e has length ≤ q,
then g_Kqφ_w is a polynomial of degree ≤ D_q=3Kqlog(1+Kq).

  *  If w is not a proper
power, then φ_w^(i)(0)=0 for all i<K+K/3. 



We collect in the next lemma some of the basic properties of g_L
that we need. 

Let L≥1 be an integer. The polynomial

g_L(t)=∏_c=1^L(1-c^2t^2)^⌊L/c⌋
has the following properties, for every 0≤ t≤1/2L^2
and every integer i≥0: 

    1/2≤ g_L(t)≤1,


    |g_L^(i)(t)|≤(3iL^3/2)^i,


    |(1/g_L)^(i)(t)|≤2· i!·(C√(i)L^3/2)^i.




Fix 0≤ t≤1/2L^2. The inequality g_L(t)≤1
is clear and only uses |t|≤1/L. For the converse, use
that (1-u)≥ e^-2u for 0<u<1/4 to bound 

    g_L(t)≥exp(-2∑_c=1^LL/cc^2t^2)=exp(-L^2(L+1)t^2)≥1/2.

We now prove (<ref>). Let N=∑_c=1^L⌊L/c⌋
and write g_L(t)=∏_k=1^N(1-c_k^2t^2). Each of the
factors h_k(t)=(1-c_kt^2) is a degree 2 polynomial, so
when we differentiate i times g_L using the Leibniz rule,
we obtain a big sum of terms, in which some factors are derived twice
(so equal to h”_k(t)=-2c_k^2), some are derived once (so
equal to h'(t)=-2c_k^2t) and all the other are not derived.
Gathering the terms according to how many factors are derived twice,
we can write 

    g_L^(i)/g_L=∑_s=0^⌊i/2⌋i2s(2s-1)!!∑_k_1,…,k_s,ℓ_1,…,ℓ_i-2s all distinct ∏_α=1^sh_k_α”/h_k_α·∏_β=1^i-2sh'_ℓ_β/h_ℓ_β.

The term i2s(2s-1)!! appears as the number of ways to
partition a set of size i (the steps of derivation) into s sets
of size 2 (the steps when a factor h_k is derived twice) and
i-2s sets of size 1 (the steps when a factor h_k is derived
once). Forgetting the condition that the k_α and
ℓ_β are distinct, and bounding (1-c_k^2t^2)≤1
and (2s-1)!!≤ i^i/2, we can bound the preceding, in
absolute value, as follows:

    |g_L^(i)(t)|≤ i^i/2∑_s=0^⌊i/2⌋i2st^i-2s(∑_k=1^N2c_k^2)^i-s≤ i^i/2∑_j=0^iijt^i-j(∑_k=1^N2c_k^2)^i-j/2.

Using 

    ∑_k=1^N2c_k^2=∑_c=1^L2⌊L/c⌋ c^2≤ L^2(L+1)≤2L^3,

we obtain 

    |g_L^(i)(t)|≤ i^i/2∑_j=0^iij(1/2L^2)^i-j(2L^3)^i-j/2=i^i/2(L+√(2)L^3/2)^i≤(3√(i)L^3/2)^i.

This proves (<ref>).

The last inequality (<ref>) is
a formal consequence of the first two. Indeed, by induction we see
that, for any function g we can write the ith
derivative of 1/g as a product of (2i-1)!! terms, each
of which is of the form 

    ± g^(α_1)·…· g^(α_i)/g^i+1

where (α_1,…,α_i) are nonnegative integers that
sum to i. For g=g_L, by (<ref>) and (<ref>),
each of these terms is bounded above that (3√(i)L^3/2)^i2^i+1. 

We extend the map w↦φ_w by linearity, setting φ_x=∑_w∈_rx(w)φ_w
for x∈[_r].

 If x∈_≤ q[_r],
then for any i, 

    sup_t∈[0,1/2D_q^2]|φ_x^(i)(t)|/i!≤ p(i,q)x_C^*(_r),


where

    p(i,q)=
    (2K+2)(CD_q^4/i^2)^i   if  i≤ D_q,
    
    (2K+2)(CD_q^2)^D_q(C√(i)D_q^3/2)^i-D_q   if  i>D_q.




Let P=g_Kqφ_x. We know that P is a polynomial of degree
≤ D_q, so to bound its derivatives using the Markov brothers
inequality, we need to derive an a priori bound on the values of P.
For n≥ Kq, we have g_Kq(1/n)∈[0,1] and therefore

    |P(1/n)|≤|φ_x(1/n)|=n^-K|[(σ_n(x(u_1,…,u_r)))]|.

For every (u_1,…,u_r)∈(n)^r, the map w↦σ_n(w(u_1,…,u_r))
is a unitary representation of _r (hence of C^*(_r)),
so σ_n(x(u_1,…,u_r))≤x_C^*(_r)
almost surely. Bounding the trace of a matrix by its norm times its
size, we obtain |[(σ_n(x(u_1,…,u_r)))]|≤dim(σ_n)x_C^*(_r),
and we conclude by the dimension bound (<ref>)
that 

    |P(1/n)|≤(K+1)x_C^*(_r).

We deduce by Lemma <ref> that for
any integer j, 

    sup_t∈[0,1/2D_q^2]|P^(j)(t)|/j!≤(2K+2)(CD_q^4/j^2)^jx_C^*(_r).

By the Leibniz rule, remembering that P is a polynomial of degre
≤ D_q,

    φ_x^(i)=∑_j=0^min(i,D_q)ijP^(j)(1/g_Kq)^(i-j).

We see from Lemma <ref> and (<ref>)
that the leading term is for j=min(i,D_q). The lemma follows
directly if i≤ D_q. If i>D_q, we obtain the lemma by
bounding Kq≤ D_q and i-D_q≤ i, so that 

    sup_t∈[0,1/2D_q^2]1/(i-D_q)!|(1/g_Kq)^(i-D_q)(t)|≤(CD_q^2)^i(i/D_q)^i-D_q/2.



We deduce two facts from this lemma. In the first, the temperedness of v_i is where most of the results of this paper are used. It is worth noting here that, in the particular case of K=1, a much stronger result is known: all the v_i's are tempered, as proven by Parraud <cit.>. We will see below that they are even completely tempered, see Proposition <ref>.

   For every integer i≥0, let v_i w ∈_r  ↦φ_w^(K+i)(0)/(K+i)!. There is a polynomial P of degree 4K+Ki+1 such that |v_i(x)| ≤ P_n(q) x_C^*(_r) for every q and every x ∈_≤ q[_r].
  Moreover, v_i is tempered if i<K/3.


The existence of P_n is a consequence of Lemma <ref> and the bound D_q≤ CKqlog(1+K)log(1+q), which implies sup_q≥1p(i,q)/(1+q)^4i+1<∞.

Let us explain why v_i is tempered if i<K/3. The proof combines several ingredients: the first is Haagerup's inequality, which tells us that we can apply the criterion for temperedness given in Proposition <ref>. The second is the polynomial growth of v_i: as a particular case of what we have just shown, there is a constant C_i such that for every q and every w ∈_r in the ball of radius q, |v_i(w)| ≤ C_i (1+q)^j, where j=4i+4K+1. The third ingredient is the small support of v_i given by item <ref> in Theorem <ref>. The last is the random walk results from Section <ref>. 

Let us put all these ingredients together. Let μ be a reasonable probability measure on _r, and (γ_n)_n be the associated random walk on _r. If μ is supported in the ball of radius q, we know that γ_n belongs to the ball of radius qn, so that

    |v_i(γ_n)| ≤ C_i (1+qn)^j ( v_i(γ_n) ≠ 0) ≤ C_i C(μ) (1+qn)^j n^5 ρ(μ)^n.

We deduce

    lim sup_n ( |v_i(γ_n)|)^1/n≤ρ(μ),

so v_i is tempered by Proposition <ref>.

The second consequence is an analogue in our context to the master
inequality in <cit.>. In the particular case
K=1, a similar result was obtained in <cit.>.


 
Let
For every integers r,q,n≥ 1 and every x∈_≤ q[Γ]

    |(σ_n(x(u_1,…,u_r))-τ(x)Id)-∑_i=0^r-1v_i(x)/n^i|
    ≤(C(K+r)^2 log(1+K+r)^12 K^4 log (1+K)^4)^K+r/n^rw(q)x_C^*(_r).




If n≥ Kq, by (<ref>) in Theorem <ref>,
we know that the left-hand side of (<ref>)
is equal to 

    n^K|φ_x(1/n)-∑_i=0^K+r-1φ_x^(i)(0)/i!n^i|.

By Taylor's inequality and Lemma <ref>,
if we moreover assume that n≥2D_q^2, this is less than 

    1/n^rp(K+r,q)x_C^*(_r).

If n≤2D_q^2, we claim that this bound is still valid. Indeed, we can bound the left-hand side of (<ref>) by 

    2dim(σ_n)x_C^*(_r)+∑_i=0^r-1|v_i(x)|/n^i.

By Lemma <ref> and (<ref>), this is less than

    n^K(p(0,q) + ∑_i=0^r-11/n^K+ip(K+i,q)) x_C^*(_r),

which is indeed bounded
above by (<ref>), at least if the
constant C appearing there is large enough : for example C≥ 2 e^2 guarantees  that 1/n^j+1p(j+1,q) ≥ 2 1/n^jp(j,q) for every j.

The statement of the lemma follows because

    sup_qp(i,q)exp(-q/log(2+q)^2)≤(CK^4i^2log(1+K)^4log(1+i)^12)^i.



We can now deduce the main result of this section. Now we make K
vary, so we write σ_K,n the representation that was denoted
σ_n so far, and v_K,i the function denoted v_i so far.


 Let A<1/24. Then
for every x∈[_r], 

    lim_nsup_K≤ n^A|σ_n,K(x(u_1,…,u_r))-x_C_λ^*(_r)|=0.




  Let 1 ≤ K_n ≤ n^A for every n, and let π_n be the random representation of _r: π_n(w) = σ_K,n(w(u_1,…,u_r)). Apply Lemma <ref> for the smallest integer r ≥K/3: for every q and every x ∈_≤ q[Γ],

    | (π_n(x)) - u_n(x)| ≤ε_n  w(q)x_C^*(_r).

where

    ε_n = (C K_n^6 (log K_n)^12+8ε)^K_n+r/n^r≤ (C n^24A-1 (log n)^12+8ε)^1/3→ 0

and

    u_n(x) = (σ_n,k)τ(x) + ∑_i=0^r-1v_K_n,i(x)/n^i.

By Lemma <ref>, the function u_n is tempered as a finite sum of tempered function, and it satisfies

    |u_n(x)| ≤ C(n) (1+q)^4r+4K_n-3.


Moreover, C^*_λ(Γ) has a unique trace (it is even simple <cit.>). So all the hypotheses of Proposition <ref> are satisfied; its conclusion proves the lemma. 




 §.§ From convergence in probability to almost sure convergence


Lemma <ref> is not exactly our main
result, because we have convergence in expected value.
However, by the concentration of measure phenomenon in the groups
(n) (see <cit.>), we can improve our
results to almost sure convergence. 

 For every x∈_≤ q[_r],
there is a constant C(x)=∑_w|x(w)||w| such that 

    (σ_n,K(x(U_1^(n),…,U_r^(n)))≥σ_n,K(x(U_1^(n),…,U_r^(n)))+ε))
       ≤exp(-(n-2)ε^2/24C(x)^2K^2).




Equip (n)^r with the L_2-sum of Hilbert-Schmidt distances.
The function (U_1^(n),…,U_r^(n))↦σ_n,K(x(U_1^(n),…,U_r^(n)))
is KC(x)-Lipschitz, so the bound is <cit.>
(for the original see <cit.>).



Theorem <ref> now follows by combining Proposition <ref>
and Lemma <ref>. 




 §.§ Variants and operator coefficients

If we only consider the representation π_k,ℓ from the introduction with ℓ=0, we have the stronger conclusion <ref> in Theorem <ref> instead of <ref>. Taking this improvement into account, with the notation of Theorem <ref>, the conclusion becomes that if A < 1/12,

    sup_k≤ n^A|‖π_k,0(p(U_1^(n),…,U_r^(n)))‖ -p(x_1,…,x_r)|=o(1).

This improvement illustrates that the more of the higher derivatives v_K,i from Lemma <ref> are shown to be tempered, the stronger the conclusion. For example, if we knew that v_K,i is tempered for every i and every K (which Parraud has proved in the case K=1 <cit.>), or at least for every i ≤ K/o(1), then the condition on A in Theorem <ref> would be A<1/6.


 Let A<1/6 and for every n, let K_n ≤ n^A. Assume that v_K_n,i is tempered for every n and every i. Let k_n = exp(n^1/2 - 2A(log n)^-4).

  For every q and every sequence y_n ∈ M_k_n⊗_≤ q(_r), almost surely
  
    lim_n (id⊗σ_K_n,n)(y_n)/(id⊗λ)(y_n)=1.


If the assumption was that v_K_n,i is completely tempered, the proof of the theorem would be a straightforward adaptation of the proof of Theorem <ref>. So our main task is to prove the following:
 For every K,I, if v_K,0,…,v_K,I are all tempered, then they are all completely tempered.

For the proof, we need the following result. Here if 𝒜 is an algebra, 𝒜^n denotes its subalgebra equal to the linear span of {x_1 … x_n | x_i ∈𝒜}.
 Let 𝒜 be a *-algebra and u 𝒜→ a positive linear map: u(x^*x)≥ 0 for every x ∈𝒜. If there is i such that u(x^i)=0 for every self-adjoint x, then u= 0 on 𝒜^3.

 The form ⟨ a,b⟩ = u(a^*b) is a scalar product. By the Cauchy-Schwarz inequality, if k ≥ 2, if a^* = x^*xx^* … (k+1) terms and b= … x (k-1) terms,
  
    |u((x^*x)^k)|^2 = |⟨ a,b⟩|≤ u((x^*x)^k+1) u((x^*x)^k-1),

  so by induction we have u((x^*x)^2)=0, and by the Cauchy-Schwarz inequality again, u(x^* x z)=0 for every z ∈𝒜. By polarization we deduce u(xyz)=0 for every x,y,z.


  We use the notation and results from Section <ref>. In particular, Lemma <ref> tells us that v_K,i extends by continuity to a linear map 𝒮_4K+4i+1(_r) →. Let 𝒜_i = 𝒮_4K+4i+1(_r) ∩λ. We prove by induction that v_K,i vanishes on (𝒜_i)^3^i+1 for every i≤ I. This readily implies that ⊗ v_K,i vanishes on M_k ⊗ (𝒜_i)^3^i+1 for all k. In particular, v_K,i is completely tempered by Proposition <ref>.

So let us assume that v_K,j vanishes on (𝒜_j)^3^j+1 for every j<i (we assume nothing if i=0), and let us show that it is true for j=i.
 
Using the induction hypothesis, it follows from Lemma <ref> that on the *-algebra (𝒜_i)^3^i (which is contained in τ and (𝒜_j)^3^j+1 for every j<i),
  
    v_K,i(x) = lim_n n^iσ_K,n(x).

  In particular, v_K,i is positive on  (𝒜_i)^3^i as a limit of positive maps. Moreover, the fact that v_K,i is tempered implies that v_K,i(x^n)=0 for all n≥ 4K+4i+3, see Proposition <ref>. By Lemma <ref>, we obtain that v_K,i vanishes on (𝒜_i)^3^i+1. This concludes the proof of the proposition.
  

Thanks to Proposition <ref>, the temperedness assumption is automatically upgraded to complete temperedness. We apply Lemma <ref> with r = n^1/2 - 2A (log n)^-7/2, which is chosen so that
  
    k_n (C(K+r)^2 log(1+K+r)^12 K^4 log (1+K)^4)^K+r/n^r→ 0.

So by Proposition <ref> we deduce the convergence in probability. The almost sure convergence is obtained by the concentration of measure phenomenon from Proposition <ref>.
For K_n=1 (that is, A=0), Parraud proved that v_1,i are tempered for every i. As a consequence, we obtain the unconditional result from Theorem <ref>.



§ PROOF OF COROLLARY <REF> 


Corollary <ref> follows easily from the proof of Theorem
<ref> and the following lemma:

For every partitions λ⊢ k,μ⊢ℓ,
every w∈_r and every n>(k+ℓ)|w|, we have



    _n[s_λ,μ(w)]=∫_(n)^rs_λ,μ(w(v_1,v_2,…,v_r))dv_1⋯ dv_r.



Let z_1,…,z_r,z'_1,…,z'_r,v_1,…,v_r be
independent random variables where z_i are uniform in the center
of (n) (the complex numbers of modulus one), z'_i are uniform
in the center of (n) (the n-th roots of unity) and v_i∈(n)
are Haar distributed, so that (z_1v_1,…,z_rv_r) are
independent Haar-distributed variables in (n), and (z'_1v_1,…,z'_rv_r)
are independent Haar-distributed variables in (n). 

By Schur's lemma, for every z in the center of U(n), π_λ,μ(z)
is a multiple of the identity. On the other hand, we know that (λ_1,…,λ_p,…,-μ_q,…,-μ_1)
is the maximal weight of π_λ,μ, so the corresponding
character of the maximal torus appears in the restriction of π_λ,μ.
This means that the scalar π_λ,μ(z) is z^λ_1+…+λ_p-μ_q-…-μ_1=z^k-ℓ
for every z in the center of (n). Putting these two facts
together, we see that

    _n[s_λ,μ(w)]     =[s_λ,μ(w(z_1v_1,…,z_rv_r))]

decomposes by independence as the product 

    _n[s_λ,μ(w)]=[s_λ,μ(w(v_1,…,v_r))]·[w(z_1,…,z_r)^k-ℓ],

and similarly

    [s_λ,μ(w(z'_1v_1,…,z'_rv_r))]=[s_λ,μ(w(v_1,…,v_r))]·[w(z'_1,…,z'_r)^k-ℓ].

But w(z_1,…,z_r)^k-ℓ is equal to 1 if w^k-ℓ
vanishes in the abelianization ^r of _r, and is equal
to 0 otherwise. Similarly, w(z'_1,…,z'_r)^k-ℓ
is equal to 1 if w^k-ℓ vanishes in (/n)^r and 0
otherwise. These two conditions coincide if n>|k-ℓ| · |w|, and
in particular if n>(k+ℓ)|w|.



 Using Lemma <ref>, we see that Lemma <ref>
also holds if _n denotes the integration over (n)^r:
the validity of (<ref>) when x∈_≤ q[_r]
with n≥2D_q^2 is by Lemma <ref>, the
extension for arbitrary x follows with the same proof. Therefore,
Lemma <ref> also holds for (n).
Moreover, the concentration of measure from Proposition <ref>
also holds for (n) (same reference as there), so Theorem <ref>
holds also for independent variables in (n). Corollary <ref>
follows by Corollary <ref>. 




§ TRANSVERSE MAPS


We now fix the free group we work with _r and its generators
{x_1,…,x_r}. Here we introduce a framework, following
<cit.>, that we use to prove Theorem <ref>.

In the sequel, a marked rose refers to a finite CW-complex
structure R_r whose underlying topological space is the wedge
of r circles, with the base-point being the wedge point, denoted
o, and with a marking π_1(R_r,o)≅_r that identifies
the generators {x_1,…,x_r} with oriented circles of
the rose.

The following definition will not be used in the paper but gives important
background motivation for the definitions to follow.

A transverse map is a manifold M, a based marked rose R_r
and a continuous function

    f:M→ R_r

transverse to all the vertices of R_r. 


In the rest of the paper we work with classes of transverse
maps. Instead of getting into what is an isotopy of transverse maps,
etc, we make the following equivalent combinatorial definition.

An (isotopy) class of (filling) transverse
map from a surface with boundary to a marked rose R_r is the
finitary combinatorial data:


  * a marked base-point and orientation for each boundary component

  * a ribbon graph structure on the surface, where vertices are discs
and edges correspond to interfaces between discs on their boundaries;
such an interface is called an arc

  * information about which arc is in the preimage of each vertex of R_r
other than o, and which sides of the vertex correspond to which
sides of the arc

  * up to isomorphism of the above data under decoration respecting homemorphisms.

We refer to these simply as classes of transverse maps.



A class of transverse map f as in Definition <ref>
is strict if for every two vertices p_1,p_2 of R_r
that are consecutive on some circle (and not o), do not
have parallel preimages in the following sense. Suppose the orientation
of their circle in R_r points from p_1 to p_2. The
preimages of p_1 and p_2 are parallel if for every
arc α in the preimage of p_1, there is a parallel[Cobounding a rectangle.]
arc in the preimage of p_2 on the side of α `towards'
p_2. 


Given a class [f] of transverse map on surface with boundary there
is an obvious way to get a boundary class of transverse map on a union
of circles, denoted ∂[f]. 

Given κ:[r]→∪{0} let R_r^κ be the (isomorphism
class of) rose with κ(i)+1 vertices in the interior of the
circle corresponding to x_i. Let |κ|=∑_i∈[r]κ_i. 

Given every non-identity w∈ F_r, let 𝔴:S^1→ R_r
denote a fixed immersion[Away from the base point of S^1, where there could be backtracking
if w is not cyclically reduced.] from a based oriented circle to the based rose such that if γ
is the generator of π_1(S^1,basepoint) corresponding
to the chosen orientation, 

    𝔴_*(γ)=w∈π_1(R_r,o).





§ PROOF OF THEOREM <REF> PART <REF>


Suppose |w|≤ q, λ⊢ k,μ⊢ℓ as before.
By a result of Koike <cit.> it is possible to write
for g∈(n)

    s_λ,μ(w)=∑_|λ'|≤ k,|μ'|≤ℓα_λ',μ'^λ,μs_λ'(w)s_μ'(w^-1)

where the coefficients α_λ'.μ'^λ,μ are
given in (ibid.) as sums of products of Littlewood–Richardson
coefficients and importantly, do not depend on n. Then by base
change between Schur polynomials as above and power sum symmetric
polynomials, we obtain

    s_λ,μ(w)=∑_|λ'|≤ k,|μ'|≤ℓβ_λ',μ'^λ,μp_λ'(w)p_μ'(w^-1)

 where again the coefficients do not depend on n. This tells us
the poles of [s_λ,μ(w)] are at most the union of every
possible pole of 

    _n[p_λ'(w)p_μ'(w^-1)]

 with k'=|λ'|≤ k,ℓ'=|μ'|≤ℓ. 

For the purpose of finding the poles, we can here use the Weingarten
calculus in the most naive possible way (later, we need to do something
more sophisticated). To this end, <cit.>
yields

    _n[p_λ'(w)p_μ'(w^-1)]=∑_[f:Σ→ R_r^(2)]
    ∂ f≅𝔴∘φ_λ',μ'(∏_1≤ i≤ rWg_(k'+ℓ')L_i(w)(π_f,i))n^N(f)

where R_r^(2)=R_r^(κ) with κ=(2,2,…,2),
π_f,i∈ S_(k'+ℓ')L_i(w) are permutations determined
from the combinatorial structure of f, and N(f)∈ is similar
— neither depend on n. It is a finite sum.



Each term

    Wg_(K+ℓ)L_i(w)(π_f,i)∈(n)

is given by the formula for the Weingarten function <cit.>

    Wg_L(π)=1/(L!)^2∑_λ⊢ Lχ_λ(1)^2/s_λ(1)χ_λ(π);

here we are viewing s_λ(1) as the formal element of (n)

    s_λ(1)∏_□∈λ(n+c(□))/∏_□∈λh_λ(□),

c(□) is the content of the box,
𝐅

    c(□)=j(□)-i(□)

and h_λ(□) is the hook-length of the box. The only
terms in the formula for the Weingarten function that depend on n
are therefore the denominators ∏_□∈λ(n+c(□))

If c≥0 then the factor (n+c(□)) appears at most d(c,L)
times in the denominator of Wg_L(π) where d(c,L)
is the largest natural number d such that 

    d(d+c)≤ L.

Clearly d(c,L)≤L/c for c>0. If c<0 then a similar
argument says (n+c) appears at most d(n,-c) times.

This means that ∏_□∈λ(n+c(□)) can be
written as n^LQ_λ(1/n) where Q_λ is
a polynomial that divides g_L(1/n) where (as in (<ref>))

    g_L(x)∏_c=1^L(1-c^2x^2)^⌊L/c⌋.


Therefore, g_L(1/n)/∏_□∈λ(n+c(□))=1/n^Lg_L/Q_λ(1/n)
is a polynomial in 1/n of degree ≤ L + deg(g_L).
By <ref> the same is true for g_L(1/n)Wg_L(π).
Since 

    ∏_i=1^rg_(k'+ℓ')L_i

divides g_(k+ℓ)q, we see from (<ref>)
that for n≥(k+ℓ)q, the rational function agreeing with g_(k+ℓ)q(1/n)[s_λ,μ(w)]
in this range can be written as a sum of a polynomial in 1/n
of degree ≤(k+ℓ)q+deg(g_(k+ℓ)q) and possibly
a polynomial in n (because of the terms n^N(f)). However,
the polynomial in n is necessarily constant because it is known
(see Theorem 1.7 in <cit.>) that g_(k+ℓ)q(1/n)[s_λ,μ(w)]
remains bounded as n→∞.

To conclude, observe that we can bound the degree of g_L by


    2∑_c=1^L⌊L/c⌋≤2∑_c=1^LL/c≤2L(1+log L),

so that (k+ℓ)q+deg(g_(k+ℓ)q)≤3L(1+log L).




§ PROOF OF THEOREM <REF> PART <REF>
AND <REF>


In the paper <cit.> the integrals 

    ∫_U(n)^4(w̃(u_1,u_2,u_3,u_4))s_λ,μ([u_1,u_2][u_3,u_4])du_1du_2du_3du_4

w.r.t Haar measure are calculated in a very specific way that exhibits
cancellations that cannot be easily seen[In fact, we do not know how to.]
using the formula (<ref>). The initial part
of this calculation is very general and does not depend on w̃
or the word w=[x_1,x_2][x_3,x_4]. So the method generalizes
as-is to the case when the (w̃) term is not present and
[x_1,x_2][x_3,x_4] is replaced by any reduced word w.
That is, to integrals of the form

    _n[s_λ,μ(w)]∫_U(n)^rs_λ,μ(w(u_1,…,u_r))du_1⋯ du_r

 where w is any element of _r, viewed as a reduced word
in the generators. 

The analog of <cit.> here is the following
bound. Let R_r^0 be the based CW-complex structure on R_r
with one interior vertex per circle (and a vertex for the wedge point
o). Recall the immersion 𝔴:(S^1,basepoint)→(R_r^0,o)
from <ref>. Let 

    φ_λ,μ:∐_i=1^ℓ(λ)S^1⊔∐_i=1^ℓ(μ)S^1→ S^1

denote the map whose components are 


  * multiplication by λ_i on the ith
circle, if i≤ℓ(λ), 

  * and multiplication by -μ_i (so orientation reversing) on the
(i-ℓ(λ))thcircle for i>ℓ(λ). 

This map is a convenient way of book-keeping winding numbers.

We have 

    _n[s_λ,μ(w)]=O_k,ℓ,w(n^max_w,k,lχ(Σ))

where max_w,k,ℓ is the maximum over classes of transverse
maps [f] from a surface with boundary to R_r^0 such that


    Σ is an oriented surface with compatibly oriented
boundary

    For some λ'⊢ k and μ'⊢ℓ, ∂[f]
factors as 

    ∂[f]=[𝔴∘φ_λ',μ']

where [𝔴∘φ_λ',μ'] is the (unique)
class of transverse map to R_r^0 obviously associated to the
immersion 𝔴∘φ_λ',μ'. Components
of ∂Σ mapped in an orientation respecting (resp. non-orientation
respecting) way by φ_λ',μ' are called positive
(resp. negative).

  Forbidden Matchings In the factorization of ∂ f above,
no such arc has endpoints in a positive and negative component of
∂Σ that are mapped to the same point under φ_λ',μ'.



This proposition follows formally from <cit.>,
but we give most details in the Appendix, <ref>..
The Forbidden Matchings property is completely crucial and
the main point of the method introduced in (ibid.). The following
topological result is a new input to that method.

If w is not a proper power in _r
and cyclically reduced, then any class of transverse map [f] on
underlying surface Σ satisfying the conditions of Proposition
<ref> has 

    χ(Σ)≤-k+ℓ/3.




If w is not a proper power in _r then

    _n[s_λ,μ(w)]=O_k,ℓ,w(n^-1/3(k+ℓ)).



N.B. One can assume without loss of generality in Corollary <ref>
that w is cyclically reduced, as conjugating w does not change
_n[s_λ,μ(w)].



 §.§ Proof of Proposition <ref>


Suppose w is cyclically reduced and not a proper power or the identity,
with word length |w|. Suppose that some class of transverse map
[f:Σ→ R_r^0] satisfies P1-P4 of Proposition
<ref>.

Let L(e) denote the number of edges in the chain defining e. 

Let V^*,E^* denote the topological vertices and edges respectively.
We have

    2E^*=∑_v∈ V^*d(v)≥3V^*

and hence

    -χ(Σ)=E^*-V^*≥E^*/3.

We now aim to show E^* is large under the previous assumptions.

Each class of strict transverse map f is a (isomorphism class of)
ribbon graph ℛ whose edges correspond to the arcs of
f. Each edge of the ribbon graph inherits a direction and labeling
by {x_1,…,x_r} from the manner in which f crosses
the (sole) interior vertex of the ith circle
in R_r^0 along this edge. 

Say that a vertex v of ℛ is a topological vertex
if it has valence d(v)≥3. A topological edge e is
a maximal chain of edges incident at vertices of valence 2. (As
w is cyclically reduced, there are no vertices of valence 1).

Every topological edge e is bordered by two segments of ∂ℛ
that each spell a subword of w^k_i or w^-ℓ_i for
some i, reading along the fixed orientation of the boundary. 

Suppose that for some topological edge e, L(e)>|w|. For a reduced
word v in the generators of _r, write v for
its mirror (reversing order and replacing generators by inverses).
Then both boundary segments spell words

    u_1w^au_2,   v_1w^bv_2

 with a,b∈\{0}, |u_1|,|u_2|,|v_1|,|v_2|<|w|. 

Case 1. If a,b have same sign. In this case one arrives
at

    wu=vw

with |u|,|v|<w. (see Figure <ref>).

 This is impossible since then v is a prefix of w and v
is too, meaning v=v̅ since they have the same length, so v
is empty. Similarly u is empty and w=w contradicting
w being the identity. 

Case 2. If a,b have opposite signs. So without loss of
generality assume a>0 and b=-B with B>0, then we have 

    u_1w^au_2=v_2w^Bv_1.

Condition P4 implies that |u_1|≠|v_2|.
Again without loss of generality assume |u_1|<|v_2|.
Write v_2=u_1w_1.

If |u_2|>|v_1| then write u_2=w_2
to obtain


    w^aw_2=w_1w^B

as reduced words. This immediately implies w_2=w_1, as both
are prefixes of w of the same length. Since w is not a proper
power, e.g. <cit.> implies that w_1 is a
power of w, which must be empty since its length is less than w.

If |u_2|<|v_1 then write w_2u_2=v_1
to arrive at 

    w^a=w_1w^Bw_2.

 This implies w_1 and w_2 are both prefixes and suffixes
of w. 

Write w=w_1w_3 and obtain

    w_3w^a-1=w^Bw_2.

Since B>0, a>1 here and we argue as we did from (<ref>)
to get a contradiction.

The upshot of this argument is that each topological edge e has

    L(e)≤|w|.

Since the total lengths of topological edges is (k+ℓ)|w| this
gives |E^*|≥ k+ℓ. Hence from (<ref>)

    -χ(Σ)≥k+ℓ/3.





 §.§ Proof of Theorem <ref> Part <ref>


If μ=∅, i.e. s_λ,μ=s_λ is a polynomial
stable character, then instead we can use the following two results.

For w∈_r the commutator length of w, denoted by cl(w)∈∪{∞}
is defined by 

    cl(w)=inf{ g: w=[u_1,v_1]⋯[u_g,v_g] , u_i,v_i∈_r }

 interpreted as ∞ if it is not possible to write w as a
product of commutators, i.e. w∉[_r,_r]. The stable
commutator length of w, denoted scl(w), is defined
as

    scl(w)lim_m→∞cl(w^m)/m.

On _r, scl takes values in  by a result of Calegari <cit.>.
Duncan and Howie proved in <cit.> that 

    (w)≥1/2

 for all w∈_r. This bound can be combined with the following
result of the first named author and Puder following immediately from
<cit.>.

We have 

    _n[s_λ(w)]=O(1/n^2k(w)).



In fact, <cit.> says that for every w∈[_r,_r],
there is some λ such that the bound (<ref>)
is saturated, but we do not use this here.

Combining (<ref>) with (<ref>) gives

    _n[s_λ(w)]=O(n^-k)

as required.



§ APPENDIX: EXTENSION OF THE METHOD OF RANDOM UNITARY REPRESENTATIONS
OF SURFACE GROUPS II


𝖲𝖴
𝔭
𝒩
𝒩̂
Wg
Res
ABG
zd
𝒮𝒮𝒯
ℱ
𝔮
𝖯𝖲𝖴
irr
𝔰𝔲
Ad
𝐦
𝐱
𝖫𝖱
𝐐
𝐙
𝒯
_n^k,ℓ
_n^k,ℓ
𝖬𝖠𝖳𝖢𝖧
𝗌𝗎𝗋𝖿𝖺𝖼𝖾𝗌
𝒯̌̇̌_n^k,ℓ
𝒯̌_n^k,ℓ
ℑ
ȷ𝔧

𝐮̆
𝐯̌
𝐫̊
𝐬
𝐔
𝐕
𝐑
𝒮

In this section we explain the extension of the methodology of <cit.>
to bound the integrals _n[s_λ,μ(w)] defined in (<ref>).
We import some results from <cit.>. 



 §.§ Background 


In this section, fix λ⊢ k and μ⊢ℓ, and
assume n≥ k+ℓ. Let D_λ,μ(n)=s_λ,μ(1).
Write χ_λ for the character of S_k associated
to the irreducible representation W^λ that corresponds
to λ. Let d_λχ_λ(𝕀)= W^λ.
Given λ⊢ k, the element

    _λd_λ/k!∑_σ∈ S_kχ_λ(σ)σ∈[S_k]

 is the central projection in [S_k] to the W^λ-isotypic
component. We view S_k× S_ℓ as a subgroup of S_k+ℓ
in the standard way. 

Let 

    _λ⊗μ_λ'_μ

 where '_μ is the image of _μ under the inclusion
S_ℓ≤ S_k× S_ℓ≤ S_k+ℓ. We define Young
subgroups, for λ⊢ k 

    S_λ S_λ_1× S_λ_2×⋯× S_λ_ℓ(λ)≤ S_k.

Let 

    (^n)^⊗ k⊗((^n)^∨)^⊗ℓ.

For I=(i_1,…,i_k+ℓ) let 

    I'(I;π)     i_π(1),…,i_π(k),
    
    J'(I;π)     i_π(k+1),…,i_π(k+ℓ).

For π∈ S_k+ℓ let 

    Φ(π)∑_I=(i_1,…,i_k),J=(j_k+1,…,j_k+ℓ)e_I'(I⊔ J;π)^J⊗ě_I^J'(I⊔ J;π)∈()

and extend the map Φ linearly to [S_k+ℓ]. 

Recall the definition of the Weingarten function from (<ref>).
Let


    z    ∑_τ∈ S_k+ℓz(τ)τ
       [S_k:S_λ][S_ℓ:S_μ]/d_λd_μ_λ⊗μ(∑_σ∈ S_λ× S_μσ)_λ⊗μ_n,k+ℓ∈[S_k+ℓ].

One has the following bound on the coefficients of z <cit.>

    z(τ)=O_k,ℓ(n^-k-ℓ-τ_k,ℓ).

 

Let 

    D_λ,μ(n)Φ(z).



 


  * The operator  is an orthogonal projection with (n)-invariant
image that is isomorphic to V^λ,μ as a (n)-representation.

  * For any i_1,…,i_k+ℓ,j_1,…,j_k+ℓ and any
p∈[k+ℓ],

    ∑_u_(i_1⋯ i_p-1ui_p+1⋯ i_k+ℓ),(j_1⋯ j_p-1uj_p+1⋯ j_k+ℓ)=0.




Part 1 is the combination of <cit.>.

Part 2 arises from the fact that  is zero on the orthocomplement
to the contraction free subspace  from <cit.>. 




 §.§ Combinatorial integration


We write w in reduced form:

    w=f_1^ϵ_1f_2^ϵ_2… f_q^ϵ_q,  ϵ_u∈{±1}, f_u∈{x_i},

where if f_u=f_u+1, then ϵ_u=ϵ_u+1. For
f∈{a,b,c,d} let p_f denote the number of occurrences of
f^+1 in (<ref>). 

The expression (<ref>) implies that for u(u_i:i∈[r])∈(n)^r,

    s_λ,μ(w(u))     =_( f_1^ϵ_1 f_2^ϵ_2… f_|w|^ϵ_|w|)
        =∑_I_j∈[n]^k,J_j∈[n]^ℓ∏_u=1^q_K_i⊔ L_i,_I_i+1⊔ J_i+1
        (u_f_1^ϵ_1)_I_1⊔ J_1,K_1⊔ L_1(u_f_2^ϵ_2)_I_2⊔ J_2,K_2⊔ L_2⋯(u_f_q^ϵ_q)_I_q⊔ J_q,K_q⊔ L_q
        =∑_π_1,…,π_1∈ S_k+ℓ∏_i=1^qz(π_i)∑_I_j∈[n]^k,J_j∈[n]^ℓ_K_i⊔ L_i,_I_i+1⊔ J_i+1
        (u_f_1^ϵ_1)_I_1⊔ J_1,K_1⊔ L_1(u_f_2^ϵ_2)_I_2⊔ J_2,K_2⊔ L_2⋯(u_f_q^ϵ_q)_I_q⊔ J_q,K_q⊔ L_q
       1{ K_i⊔ J_i+1=(I_i+1⊔ L_i)∘π_i : i∈[q] } .

In the last line above the indices run mod q. Each product of matrices
here can be integrated using the Weingarten calculus.

We view all I_u etc as functions from indices (the domain)
to [n]. We view all domains for distinct u as disjoint as possible. 

There are however, fixed matchings between the domains of each 

    I_u⊔ J_u and K_u⊔ L_u.

These will come into play later. 

We now define sub-collections of all the indices that are treated
as `the same type' by the Weingarten calculus. 

For each i∈[r], let:


  * _i be all indices of I_u such that f_u=x_i and
ϵ_u=+1 and indices of L_u such that f_u=x_i
and ϵ_u=-1,

  * _i^* be all indices of J_u such that f_u=x_i
and ϵ_u=+1 and indices of K_u such that f_u=x_i
and ϵ_u=-1,

  * _i be all indices of K_u such that f_u=x_i and
ϵ_u=+1 and indices of J_u such that f_u=x_i
and ϵ_u=-1,

  * _i^* be all indices of L_u such that f_u=x_i
and ϵ_u=+1 and indices of I_u such that f_u=x_i
and ϵ_u=-1.

Let  denote the set of data consisting of 


  * for each i∈[r], σ_i a bijection from _i to
_i^*,

  * for each i∈[r], τ_i a bijection from _i to _i^*.


Doing the integral of terms in either (<ref>) or (<ref>)
replaces 

    (u_f_1^ϵ_1)_I_1⊔ J_1,K_1⊔ L_1(u_f_2^ϵ_2)_I_2⊔ J_2,K_2⊔ L_2⋯(u_f_q^ϵ_q)_I_q⊔ J_q,K_q⊔ L_q

by 

    ∑_∈∏_i∈[r]_k+ℓ(σ_iτ_i^-1)1{  `respected' by }.

Before going on, we make a key argument. Suppose that some index of
I_2 is matched to an index of J_2 by , for example
the two first indices. Then the result of integrating (<ref>),
the terms corresponding to  contain all contain factors 

    ∑_a=I_2(1)=J_2(1)_K_1⊔ L_1,_I_2⊔ J_2

where all indices but the first indices of I_2 and J_2 are
frozen (conditioned upon). This is zero by Theorem <ref>
part 2. 

This means, going back to (<ref>), if we define ^*
to be the subset of  such that 


  * σ_i never matches indices I_u to those of J_u
for any u,

  * σ_i never matches indices of L_u to those of K_u
for any u,

  * τ_i never matches indices of K_u to those of L_u
for any u,

  * τ_i never matches indices of J_u to those of I_u
for any u,

then we obtain

    _n[s_λ,μ(w(u))]=∑_π_1,…,π_q∈ S_k+ℓ∑_∈^*∏_i=1^qz(π_i)∏_i∈[r]_k+ℓ(σ_iτ_i^-1)(π_1,…,π_q,)

where (π_1,…,π_q,) is the number of choices of
I,J,K,L such that 

    K_u⊔ J_u+1    =(I_u+1⊔ L_u)∘π_u : u∈[q], 
    indices matched by     have the same value.





 §.§ Surface construction


Now construct a surface as follows. 

Begin with a vertex for every index. Add an edge (called -edge)
between all matched indices (by ). Add an edge (called w-edge)
between indices paired by the fixed identifications

    I_u⊔ J_u≅[k+ℓ]≅ K_u⊔ L_u

and direct this edge from the indices on the left hand side above
to those on the right hand side.

Add also an edge (called π-edge) between indices matched by (<ref>).
We now have a trivalent graph.

We now glue two types of discs to this graph following <cit.>.
The boundaries of the discs are glued along two types of cycles in
the graph:


  Type-I Cycles that alternate between π-edges and  edges.
Such cycles are disjoint.

  Type-II Cycles that alternate between w-edges and -edges.
Again, such cycles are disjoint.

The resulting glued discs therefore meet only along the -edges
and the resulting total object is a topological surface we call 

    Σ(,{π_i}).

The boundary cycles of this surface alternate between w-edges and
π-edges.

For σ∈ S_k+ℓ, let σ_k,ℓ denote the
minimum m for which 

    σ=σ_0t_1t_2⋯ t_m

 where σ_0∈ S_k× S_ℓ and t_1,…,t_m
are transpositions in S_k+ℓ. The analog of <cit.>
is that, after an elementary calculation using (<ref>)
and bounds for the Weingarten function one obtains

    ∏_i=1^qz(π_i)∏_i∈[r]_k+ℓ(σ_iτ_i^-1)(π_1,…,π_q,)≪_w,k,ℓn^-∑_iπ_i_k,ℓn^χ(Σ(,{π_i})),

hence from (<ref>)

    _n[s_λ,μ(w(u))]≪_w,k,ℓ∑_π_1,…,π_q∈ S_k+ℓ∑_∈^*n^-∑_iπ_i_k,ℓn^χ(Σ(,{π_i})).





 §.§ Surfaces with large contribution


It is shown in <cit.> — the same proof
applies without change to the current setting — that given any

    π_1,…,π_q∈ S_k+ℓ, ∈^*,

it is possible to modify these so that

    π'_1,…,π'_q∈ S_k× S_ℓ,     ∑_iπ'_i_k,ℓ=0



    '={σ'_i,τ_i'}∈^* with  for all

and there exists an inequality between corresponding terms

    n^χ(Σ(',{π'_i}))≥ n^-∑_iπ_i_k,ℓn^χ(Σ(,{π_i})).


The condition (<ref>) means all type-II cycles
are now rectangles with two (non-consecutive) edges in the boundary;
we now replace each rectangle with an arc connecting the two boundary
segments of the rectangle.



 §.§ Connection to transverse maps


We now create a class of transverse map on the surface Σ(',{π'_i})
as follows (cf. Definition <ref>).

The ribbon graph structure is the one dictated by the arcs we just
prior constructed. By construction, they cut the surface into discs.
For each boundary component, place a marked point in some (it does
not matter) π-edge that arose from π'_q. Recall the CW-complex
rose R_r^0 with one point (call it z_i) in the interior
of the circle corresponding to x_i. If an arc arose from a rectangle
with edges that arose from σ'_i=τ'_i then we declare
our function to take the constant value z_i on this arc and the
transverse map will traverse the arc from the σ'_i side
to the τ'_i side. 

The property (<ref>) implies that for every boundary
component of the surface, the w-edges are all directed the same
way along this boundary component and hence give an orientation to
the boundary. With respect to this orientation, the isotopy class
of SF transverse map satisfies (<ref>).

The fact that '∈^*, rather than , implies that the
class of transverse map has the crucial Forbidden Matchings
property.

amsalpha




Michael Magee, 

Department of Mathematical Sciences, Durham University, Lower Mountjoy,
DH1 3LE Durham, UK

IAS Princeton, School of Mathematics, 1 Einstein Drive, Princeton
08540, USA

 


Mikael de la Salle, 

Institut Camille Jordan, CNRS, Université Lyon 1, France

IAS Princeton, School of Mathematics, 1 Einstein Drive, Princeton
08540, USA




