# An ergodic theorem for the quasi-regular representation of the free group.

1 Introduction

In this paper, we consider the action of the free group [F.sub.r] on its boundary B, a probability space associated to the Cayley graph of [F.sub.r] relative to its canonical generating set. This action is known to be ergodic (see for example [FTP82] and [FTP83]), but since the measure is not preserved, no theorem on the convergence of means of the corresponding unitary operators had been proved. Note that a close result is proved in [FTP83, Lemma 4, Item (i)].

We formulate such a convergence theorem in Theorem 1.2. We prove it following the ideas of [BM11] and [Boy15] replacing [Rob03, Theorem 4.1.1] by Theorem 1.1.

1.1 Geometric setting and notation

We will denote [F.sub.r] = <[a.sub.1],...,[a.sub.r]> the free group on r generators, for r [greater than or equal to] 2. For an element [gamma] [member of] [F.sub.r], there is a unique reduced word in {[a.sup.[+ or -]1.sub.1],...,[a.sup.[+ or -]1.sub.r]} which represents it. This word is denoted [[gamma].sub.1] ... [[gamma].sub.k] for some integer k which is called the length of [gamma] and is denoted by |[gamma]|. The set of all elements of length k is denoted [S.sub.n] and is called the sphere of radius k. If u [member of] [F.sub.r] and k [greater than or equal to] |u|, let us denote [Pr.sub.u] (k) := {[gamma] [member of] [F.sub.r] | |[gamma]| = k, u is a prefix of [gamma]}.

Let X be the geometric realization of the Cayley graph of [F.sub.r] with respect to the set of generators {[a.sup.[+ or -]1.sub.1],...,[a.sup.[+ or -]1.sub.r]}, which is a 2r-regular tree. We endow it with the (natural) distance, denoted by d, which gives length 1 to every edge ; for this distance, the natural action of [F.sub.r] on X is isometric and freely transitive on the vertices. As a metric space, X is CAT(-1). In particular, it is uniquely geodesic, the geodesics between vertices being finite sequences of successive edges. We denote by [x, y] the unique geodesic joining x to y.

We fix, once and for all, a vertex [x.sub.0] in X. For x [member of] X, the vertex of X which is the closest to x in [[x.sub.0], x], is denoted by [??]x[??] ; because the action is free, we can identify [x] with the element [gamma] that brings [x.sub.0] on it, and this identification is an isometry.

The Cayley tree and its boundary

As for any other CAT(-1) space, we can construct a boundary of X and endow it with a distance and a measure. For a general construction, see [Bou95]. The construction we provide here is elementary.

Let us denote by B the set of all right-infinite reduced words on the alphabet {[a.sup.[+ or -]1.sub.1],..., [a.sup.[+ or -]1.sub.r]}. This set is called the boundary of X.

We will consider the set [bar.X] := X [union] B.

For u = [u.sub.1] ... [u.sub.l] [member of] [F.sub.r] \ {e}, we define the sets

[X.sub.u] := {x [member of] X | u is a prefix of [??]x[??]} [B.sub.u] := {[xi] [member of] B | u is a prefix of [xi]} [C.sub.u] := [X.sub.u] [union] [B.sub.u]

We can now define a natural topology on [bar.X] by choosing as a basis of neighborhoods

1. for x [member of] X, the set of all neighborhoods of x in X

2. for [xi] [member of] B, the set {[C.sub.u] | u is a prefix of [xi]}

For this topology, [bar.X] is a compact space in which the subset X is open and dense. The induced topology on X is the one given by the distance. Every isometry of X continuously extends to a homeomorphism of [bar.X].

Distance and measure on the boundary

For [[xi].sub.1] and [[xi].sub.2] in B, we define the Gromov product of [[xi].sub.1] and [[xi].sub.2] with respect to [x.sub.0] by

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

Then d defines an ultrametric distance on B which induces the same topology ; precisely, if [xi] = [u.sub.1] [u.sub.2] [u.sub.3] ..., then the ball centered in [xi] of radius [e.sup.-k] is just [mathematical expression not reproducible].

On B, there is at most one Borel regular probability measure which is invariant under the isometries of X which fix [x.sub.0]; indeed, such a measure [mathematical expression not reproducible] must satisfy

[mathematical expression not reproducible]

and it is straightforward to check that the ln(2r - 1)-dimensional Hausdorff measure associated to the distance [mathematical expression not reproducible] (normalized to give measure 1 to B) verifies this property, so we will denote this measure by [mathematical expression not reproducible].

If [xi] = [u.sub.1] ... [u.sub.n] ... [member of] B, and x, y G X, then the sequence (d(x, [u.sub.1] ... [u.sub.n]) - [d(y, [u.sub.1] ... [u.sub.n])).sub.n [member of] N] is stationary. We denote this limit [[beta].sub.[xi]](x, y). The function [[beta].sub.[xi]] is called the Busemann function at [xi].

Let us denote, for [xi] [member of] B and [gamma] [member of] [F.sub.r] the function

[mathematical expression not reproducible]

The measure [mathematical expression not reproducible] is, in addition, quasi-invariant under the action of [F.sub.r]. Precisely, the Radon-Nikodym derivative is given for [gamma] [member of] [GAMMA] and for a.e. [xi] [member of] B by

[mathematical expression not reproducible]

where [mathematical expression not reproducible] for any Borel subset A [subset] B.

The quasi-regular representation

Denote the unitary representation, called the quasi-regular representation of [F.sub.r] on the boundary of X by

[pi] : [F.sub.r] [right arrow] U([L.sup.2](B)) [gamma] [??] [pi]([gamma])

defined as

([pi]([gamma])g) ([xi]) := P[([gamma], [xi]).sup.1/2] g([[gamma].sup.-1][xi])

for [gamma] [member of] [F.sub.r] and for g [member of] [L.sup.2] (B). We define the Harish-Chandra function

[mathematical expression not reproducible], (1.1)

where 1B denotes the characteristic function on the boundary.

For [Florin] [member of] C([bar.X]), we define the operators

[mathematical expression not reproducible] (1.2)

We also define the operator

[mathematical expression not reproducible] (1.3)

where m([f.sub.|B]) is the multiplication operator by [f.sub.|B] on [L.sup.2](B), and [mathematical expression not reproducible] is the orthogonal projection on the subspace of constant functions. So, for g [member of] [L.sup.2](B), M(f)g : = <g, [1.sub.B]>[f.sub.|B].

1.2 Results

We have the following equidistribution theorem.

Theorem 1.1. We have, in C([bar.X] x [bar.X])*, the weak-* convergence

[mathematical expression not reproducible]

where [D.sub.x] denotes the Dirac measure on a point x.

We use the above theorem to prove the following convergence of operators.

Theorem 1.2. We have, for all f in C([bar.X]), the weak operator convergence

[mathematical expression not reproducible]

In other words, we have, for all f in C([bar.X]) and for all g, h in [L.sup.2](B), the convergence

[mathematical expression not reproducible]

We deduce the irreducibility of [pi], and give an alternative proof of this well known result (see [FTP82, Theorem 5]).

Corollary 1.3. The representation [pi] is irreducible.

Proof. Applying Theorem 1.2 to [Florin] = [1.sub.[BAR.X]] shows that the orthogonal projection onto the space of constant functions is in the von Neumann algebra generated with [pi]. Then applying Theorem 1.2 to g = [1.sub.B] shows that the vector [1.sub.B] is cyclic. Let F [less than or equal to] [L.sup.2](B) be a closed nonzero invariant subspace. Suppose that [for all]h [member of] F, <h, [1.sub.B]> = 0. Then if h [member of] F, by assumption, for all [gamma] [member of] [F.sub.r], 0 = <[pi]([gamma])h, [1.sub.B]> = <h, [pi]([[gamma].sup.-1])[1.sub.B]>, so by cyclicity of [1.sub.B], h = 0. So there is a vector h [member of] F such that [mathematical expression not reproducible]. But [mathematical expression not reproducible] is in the von Neumann generated by [pi], so [mathematical expression not reproducible]. So F contains the cyclic vector <h,[1.sub.B]>[1.sub.B], so F = [L.sup.2](B).

1.3 Remarks

The study of such averages of unitary operators has first been carried out in [BM11], where an ergodic theorem is proved, in the context of the action of the fundamental group of a compact negatively curved manifold on its universal cover, using an equidistribution result due to Margulis. This work has been generalized in [Boy15] to the context of certain discrete groups of isometries of CAT(-1) spaces, where the equidistribution result is replaced by one of Roblin [Rob03, Theorem 4.1.1]. The Cayley graph of the free group with respect to the standard symmetric set of generators is, itself, a CAT(-1) space, but the quotient (a wedge of circles of length 1) dramatically lacks the property of having a non-arithmetic spectrum, which forces us to prove an analog of Roblin's equidistribution theorem in this setting : this is Theorem 1.1.

It would have been possible to define the length of the edges of X labelled by [a.sup.[+ or -].sub.1] to be [alpha] ([alpha] being a real positive number) instead of 1. Let us denote by [X.sub.[alpha]] the obtained metric space. The quotient has a non-arithmetic spectrum if and only if [alpha] [??] Q. According to [Gar14], the Hausdorff measures on the boundaries of [mathematical expression not reproducible] and [mathematical expression not reproducible] would have been unequivalent, as well as the associated unitary representations, as soon as [[alpha].sub.1] = [[alpha].sup.[+ or -]1.sub.2]. It would be interesting to prove, in this context, analogs of Theorems 1.1 and 1.2, for [alpha] [member of] [Q.sup.*.sub.+] \ {1}.

2 Proofs

2.1 Proof of the equidistribution theorem

For the proof of Theorem 1.1, let us denote

[mathematical expression not reproducible]

The subspace E is clearly closed in C([bar.X] x [bar.X]) ; it remains only to show that it contains a dense subspace of it.

Let us define a modified version of certain characteristic functions : for u [member of] [F.sub.r] we define

[mathematical expression not reproducible]

It is easy to check that he function [[chi].sub.u] is a continuous function which coincides with [[chi]C.sub.u] on [F.sub.r][x.sub.0] and B.

The proof of the following lemma is straightforward.

Lemma 2.1. Let u [member of] [F.sub.r] and k [greater than or equal to] |u|, then [[chi].sub.u] - [summation over ([gamma] [member of] [Pr.sub.u](k))] [[chi].sub.[gamma]] has compact support included in X.

Proposition 2.2. The set [chi] := {[[chi].sub.u] | u [member of] [F.sub.r] / {e}} separates points of B, and the product of two such functions of [chi] is either in [chi], the sum of a function in [chi] and of a function with compact support contained in X, or zero.

Proof. It is clear that [chi] separates points. It follows from Lemma 2.1 that [[chi].sub.u][[chi].sub.v] = [[chi].sub.v] if u is a proper prefix of v, that [[chi].sup.2.sub.u] - [[chi].sub.u] has compact support in X, and that [[chi].sub.u][[chi].sub.v] = 0 if none of u and v is a proper prefix of the other.

Proposition 2.3. The subspace E contains all functions of the form [[chi].sub.u] [cross product] [[chi].sub.v].

Proof. Let n [greater than or equal to] |u| + |v|. We make the useful observation that

1/|[S.sub.n]| [summation over ([gamma] [member of] [S.sub.n])] ([[chi].sub.u] [cross product] [[chi].sub.v]) ([gamma][x.sub.0], [[gamma].sup.-1][x.sub.0]) = |[S.sup.u,v.sub.n]|/|[S.sub.n]|

where [S.sup.u,v.sub.n] is the set of reduced words of length n with u as a prefix and [v.sup.-1] as a suffix. We easily see that this set is in bijection with the set of all reduced words of length n - (|u| + |v|) that do not begin by the inverse of the last letter of u, and that do not end by the inverse of the first letter of [v.sup.-1]. So we have to compute, for s, t [member of]{[a.sup.[+ or -]1.sub.1],...,[a.sup.[+ or -]1.sub.r]} and m [member of] N, the cardinal of the set [S.sub.m](s, t) of reduced words of length m that do not start by s and do not finish by t.

Now we have

[S.sub.m] = [S.sub.m](s, t) [union] {x | |x| = m and starts by s} [union] {x | |x| = m and ends by t}.

Note that the intersection of the two last sets is the set of words both starting by s and ending by t, which is in bijection with [S.sub.m-2] ([s.sup.-1], [t.sup.-1]).

We have then the recurrence relation :

[mathematical expression not reproducible]

We set C := 2(r-1)([(2r-1).sup.2]+1)/[(2r-1).sup.3], n = 4k + j with 0 [less than or equal to] j [less than or equal to] 3 and we obtain

[mathematical expression not reproducible]

Now we can compute

[mathematical expression not reproducible]

when k [right arrow] [infinity], and this proves the claim.

Corollary 2.4. The subspace E is dense in C([bar.X] x [bar.X]).

Proof. Let us consider E', the subspace generated by the constant functions, the functions which can be written as f [cross product] g where f, g are continuous functions on [bar.X] and such that one of them has compact support included in X, and the functions of the form [[chi].sub.u] [cross product] [[chi].sub.v]. By Proposition 2.2, it is a subalgebra of C([bar.X] x [bar.X]) containing the constants and separating points, so by the Stone-Weierstra[beta] theorem, E' is dense in C([bar.X] x [bar.X]). Now, by Proposition 2.3, we have that E' [??] E, so E is dense as well.

2.2 Proof of the ergodic theorem

The proof of Theorem 1.2 consists in two steps:

Step 1: Prove that the sequence Mn is bounded in [??](C([bar.X]), B([L.sup.2](B))).

Step 2: Prove that the sequence converges on a dense subset.

2.2.1 Boundedness

In the following [1.sub.[bar.X]] denotes the constant function 1 on [bar.X]. Define

[F.sub.n] := [[M.sub.n]([1.sub.[bar.X]])][1.sub.B].

We denote by [XI](n) the common value of [XI] on elements of length n.

Proposition 2.5. The function [zeta] [??] [summation over ([gamma] [member of] [S.sub.n])] [(P([gamma], [zeta])).sup.1/2] is constant equal to |[S.sub.n]| x [XI](n).

Proof. This function is constant on orbits of the action of the group of automorphisms of X fixing [x.sub.0]. Since it is transitive on B, the function is constant. By integrating, we find

[mathematical expression not reproducible]

Lemma 2.6. The function [F.sub.n] is constant and equal to [1.sub.B].

Proof. Because [XI] depends only on the length, we have that

[mathematical expression not reproducible]

and the proof is done.

It is easy to see that [M.sub.n] (f) induces continuous linear transformations of [L.sup.1] and [L.sup.[infinity]], which we also denote by [M.sub.n] (f).

Proposition 2.7. The operator [M.sub.n](l[bar.X]), as an element of [??]{[L.sup.[infinity]], [L.sup.[infinity]]), has norm 1; as an element of B([L.sup.2](B)), it is self-adjoint.

Proof Let h [member of] [L.sup.[infinity]](B). Since [M.sub.n](1[bar.X) is positive, we have that

[mathematical expression not reproducible]

so that [mathematical expression not reproducible].

The self-adjointness follows from the fact that [pi]([gamma])* = [pi]([[gamma].sup.-1]) and that the set of summation is symmetric.

Let us briefly recall one useful corollary of Riesz-Thorin's theorem :

Let (Z, [micro]) be a probability space.

Proposition 2.8. Let T be a continuous operator of [L.sup.1] (Z) to itself such that the restriction [T.sub.2] to [L.sup.2](Z) (resp. [T.sub.[infinity]] to [L.sup.[infinity]] (Z)) induces a continuous operator of [L.sup.2] (Z) to itself (resp. [L.sup.[infinity]] (Z) to itself).

Suppose also that [T.sub.2] is self-adjoint, and assume that [mathematical expression not reproducible]. Then [mathematical expression not reproducible].

Proof. Consider the adjoint operator T* of ([L.sup.1])* = [L.sup.[infinity]] to itself. We have that

[mathematical expression not reproducible]

Now because [T.sub.2] is self-adjoint, it is easy to see that T* = [T.sub.[infinity]]. This implies

[mathematical expression not reproducible].

Hence the Riesz-Thorin's theorem gives us the claim.

Proposition 2.9. The sequence [([M.sub.n]).sub.n[member of]N] is bounded in [??](C([BAR.X]), B([L.sup.2](B))).

Proof. If f is real-valued, we have, for every positive g [member of] [L.sup.2](B), the pointwise inequality

-[||[Florin]||.sub.[infinity]][[M.sub.n]([1.sub.[bar.X]])]g [less than or equal to] [[M.sub.n](f)]g [less than or equal to] [||[Florin]||.sub.[infinity]][[M.sub.n]([1.sub.[bar.X]])]g

from which we deduce, for every g [member of] [L.sup.2](B)

[mathematical expression not reproducible]

which allows us to conclude that

[mathematical expression not reproducible]

This proves that [mathematical expression not reproducible].

Now, it follows from Proposition 2.7 and Proposition 2.8 that the sequence [([M.sub.n](1[bar.X])).sub.n [member of] N] is bounded by 1 in B([L.sup.2]), so we are done.

2.2.2 Estimates for the Harish-Chandra function

The values of the Harish-Chandra are known (see for example [FTP82, Theorem 2, Item (iii)]). We provide here the simple computations we need.

We will calculate the value of

[mathematical expression not reproducible]

Lemma 2.10. Let [gamma] = [s.sub.1] ... [s.sub.n] [member of] [F.sub.r]. Let l [member of] {1,...,|[gamma]|}, and u = [s.sub.1]***[s.sub.l-1][t.sub.l][t.sub.l+1]*** [t.sub.l+k] (1), with [t.sub.l] [not equal to] [s.sub.l] and k [greater than or equal to] 0, be a reduced word. Then

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

Proof. The function [xi] [right arrow] [[beta].sub.[xi]]([x.sub.0], [[gamma]x.sub.0]) is constant on [B.sub.u] equal to 2(l - 1) - |[gamma]|.

So [mathematical expression not reproducible] is the integral of a constant function:

[mathematical expression not reproducible]

The value of [mathematical expression not reproducible] is computed in the same way.

Lemma 2.11. (The Harish-Chandra function)

Let [gamma] = [s.sub.1] ... [s.sub.n] in [S.sub.n] written as a reduced word. We have that

[XI]([gamma]) = (1+r-1/r|[gamma]|)[(2r-1).sup.-|[gamma]|/2].

Proof. We decompose B into the following partition:

[mathematical expression not reproducible]

and Lemma 2.10 provides us the value of the integral on the subsets forming this partition. A simple calculation yields the announced formula.

The proof of the following lemma is then obvious :

Lemma 2.12. If [gamma], w [member of] [F.sub.r] are such that w is not a prefix of [gamma], then there is a constant [C.sub.w] not depending on 7 such that

[mathematical expression not reproducible]

2.2.3 Analysis of matrix coefficients

The goal of this section is to compute the limit of the matrix coefficients [mathematical expression not reproducible].

Lemma 2.13. Let u, w [member of] [F.sub.r] such that none of them is a prefix of the other (i.e. [B.sub.u] [intersection] [B.sub.w] = [empty set]). Then

[mathematical expression not reproducible]

Proof. Using Lemma 2.12, we get

[mathematical expression not reproducible]

Lemma 2.14. Let u, v [member of] [F.sub.r]. Then

[mathematical expression not reproducible]

Proof.

[mathematical expression not reproducible]

Hence, by taking the lim sup and using Theorem 1.1, we obtain the desired inequality.

Proposition 2.15. For all u, v, w [member of] [F.sub.r], we have

[mathematical expression not reproducible]

Proof. We first show the inequality

[mathematical expression not reproducible]

If none of u and w is a prefix of the other, we have nothing to do according to Lemma 2.13. Let us assume that u is a prefix of w (the other case can be treated analogously). According to Lemma 2.1,

[mathematical expression not reproducible]

and according to Lemma 2.13, for all [gamma] [member of] [Pr.sub.u](|w|) \ {w}, [mathematical expression not reproducible].

[mathematical expression not reproducible]

We now compute the expected limit. Let us define

[S.sub.u,v,w] := {(u', v',w') [member of] [F.sub.r] | |u| = |u'|, |v| = |v'|, |w| = |w'|}

so that

[mathematical expression not reproducible].

To simplify the calculation, let us denote

[mathematical expression not reproducible]

It is obvious that A [less than or equal to] B ; we have that B [less than or equal to] C and D [less than or equal to] E because of the inequality we just proved. We also have that C + E = 1 (it is the sum of the measures of members of a partition), and finally, we have that 1 = lim inf<[M.sub.n]([1.sub.[bar.X]])[1.sub.B], [1.sub.B]) [less than or equal to] A + D, because lim [inf.sub.n[right arrow][infinity]] ([a.sub.n] + [b.sub.n]) [less than or equal to] lim [inf.sub.n[right arrow][infinity]] [a.sub.n] + lim [sup.sub.n[right arrow][infinity]] [b.sub.n] for every bounded real sequences [([a.sub.n]).sub.n] and [([b.sub.n]).sub.n].

In conclusion, we have that 1 [less than or equal to] A + D [less than or equal to] C + E [less than or equal to] 1, A [less than or equal to] B [less than or equal to] C and D [less than or equal to] E, from which we deduce A = B = C.

Proof of Theorem 1.2. Because of the boundedness of the sequence [([M.sub.n]).sub.n[member of]N] proved in Proposition 2.9, it is enough to prove the convergence for all {f, [h.sub.1], [h.sub.2]) in a dense subset of C([bar.X]) x [L.sup.2] x [L.sup.2], which is what Proposition 2.15 asserts.

References

[BM11] U. Bader and R. Muchnik. Boundary unitary representations - irre-ducibility and rigidity. Journal of Modern Dynamics, 5(1):49-69, 2011.

[Bou95] M. Bourdon. Structure conforme au bord et flot geodesique d'un CAT(-1)-espace. Enseign. Math, 2(2):63-102, 1995.

[Boy15] A. Boyer. Equidistribution, ergodicity and irreducibility in CAT(-1) spaces. arXiv:1412.8229v2, 2015.

[FTP82] A. Figa-Talamanca and M. A. Picardello. Spherical functions and harmonic analysis on free groups. J. Functional Anal., 47:281-304, 1982.

[FTP83] A. Figa-Talamanca and M. A. Picardello. Harmonic analysis on free groups. Lecture Notes in Pure and Applied Mathematics, 87, 1983.

[Gar14] L. Garncarek. Boundary representations of hyperbolic groups. arXiv:1404.0903, 2014.

[Rob03] T. Roblin. Ergodicite et Equidistribution en courbure negative. Memoires de la SMF 95, 2003.

Weizmann Institute of Science, email: aadrien.boyer@gmail.com

Universite d'Aix-Marseille, CNRS UMR7373, email :a.p.lobos@outlook.com

Received by the editors in February 2016 - In revised form in September 2016.

Communicated by A. Valette.

2010 Mathematics Subject Classification : Primary 37; Secondary 43, 47.

Key words and phrases : boundary representations, ergodic theorems, irreducibility, equidis-tribution, free groups.

(1) For l = 1, [s.sub.1] ... [S.sub.L-1] is e by convention.