Boundary unitary representations - irreducibility and rigidity

Let $M$ be compact negatively curved manifold, $\Gamma =\pi_1(M)$ and $\tilde{M}$ be its universal cover. Denote by $B =\partial \tilde{M}$ the geodesic boundary of $\tilde{M}$ and by $\nu$ the Patterson-Sullivan measure on $X$. In this note we prove that the associated unitary representation of $\Gamma$ on $L^2(B,\nu)$ is irreducible. We also establish a new rigidity phenomenon: we show that some of the geometry of $M$, namely its marked length spectrum, is reflected in this $L^2$-representations.


Introduction
Quite often ergodic properties of a measurable dynamical system are reflected in its associated quasi-regular unitary representation. This is the case, for example, with the notions of ergodicity, mixing and weak-mixing associated to (probability) measure preserving group actions. They all have an equivalent representation theoretic (or spectral) formulation. The property of irreducibility of the quasi-regular representation, however, can not occur for probability measure preserving actions, as the constants always form an invariant line.
Problem. Given a discrete group can one "classify" its measure class preserving (measurable) actions for which the quasi-regular representations are irreducible?
In [Mac76][ §3.5, Corollary 7] Mackey gives a full solution to the problem when one restricts himself to (infinite) homogeneous actions (see also [BdlH97]). The general case is still not well understood. The only existing examples (to the best of the author's knowledge) for measurable nonhomogeneous group actions with irreducible quasi-invariant representations were given in two cases: • Actions of free groups on their boundaries (see [FTP83], [FTS94]).
• Actions of lattices of Lie-groups (or algebraic-groups) on their Furstenberg boundaries (see [CS91], [BC02] We emphasize that in the above mentioned results one considers boundary actions of the group in question, and asks whether there is a general phenomenon yet to be discovered being hinted at by this coincidence. In particular, we state the following conjecture: Conjecture. For a locally compact group G and a spread-out probability measure µ on G, the quasi-regular representation associated to a µ-boundary of G is irreducible.
In this paper we prove this conjecture for the action of the fundamental group of a compact negatively curved manifold on its boundary endowed with the Paterson-Sullivan measure class. 1 We call the associated quasi-regular representation a boundary representation. In particular, we prove: Theorem 1 (Irreducibility). Every boundary representation is irreducible.
For every negatively curved manifold, there is a naturally attached class function of its fundamental group, called the marked length spectrum. The Irreducibility Theorem gives rise to a rigidity statement based on the marked length spectrum. The following theorem suggests that we view this class function as a character associated to the boundary representation.
Theorem 2 (Rigidity). Two boundary representations are unitary equivalent if and only if the two associated negatively curved manifolds give rise to proportional marked length spectra.
1.1. Structure of the paper: In the next section we introduce the notations that are used in the paper, and recall some standard results and constructions related to negatively curved manifolds. We will also state the important Theorems 3 and 4 that are needed for the proof of Theorem 1.
Section 3 is dedicated to proofs of our preparatory lemmas, in particular, there we introduce the "chopped" functions ((q|·) and its derivatives), which definitions are motivated by [CM]. Section 4 is devoted to the proof of the uniform boundedness of a certain family of functions, which later on (section 6) will be translated into the pre-compactness of a certain family of measures. In section 5 we use a geodesic counting result of Margulis to compute the asymptotic decay of some matrix coefficients. In section 6 we combine the results of sections 4 and 5, into proofs of the theorems stated in section 2, and then show how the latter imply Theorem 1 and Theorem 2.
We add three appendices to this short paper. The first two -concerning with metric measure spaces and functional analysis -are devoted to the recollection of some basic definitions and proofs of easy results of general nature, to be used in the body of the paper, without breaking the principle development of the paper. The third appendix concerns with a certain (not very well known) counting geodesics theorem of Margulis. who should have been a third author for the manuscript. We also would like to thank François Ledrappier who contributed tremendously by sharing his observation about the geometric meaning of the equidistribution result we needed, and pointing out Margulis' thesis as a reference. Also we wish to express our thanks to Vadim Kaimanovich for pointing us toward the work of Figa-Talamanca and Alex Furman for explaining the various aspects of the marked length spectrum and rigidity results.
We express our appreciation to the following mathematicians for their help: Chris Connell -for many helpful discussions about negative curvature, Benson Farb -for his support and the reference he supplied, Alex Gorodnik -for explaining his equidistribution results and Nicolas Monod -for various discussions of operator algebras. We thank the anonymous referee for a careful reading and some helpful suggestions.

Preliminaries
In this section we set the notation, and state some facts, to be used throughout the paper. We will resist the temptation of stating things in a greater generality than needed. A good reference for Patterson-Sullivan theory for CAT(-1) space is [BM96].
Let (M, g) be a (strictly) negatively curved compact Riemannian manifold. Up to a rescaling of the metric we can and will assume that M is a CAT(-1) metric space. Denote the diameter of M by R. We set Γ = π 1 (M) to be the fundamental group of M. Denote by X the universal cover of M, and by pr : X → M the obvious projection map. The space X is endowed with the lifted Riemannian metricg. We denote the associated metric of X by d. The ball in X of radius t ≥ 0, centered at p ∈ X is denoted X(p, t). Thus, for example, we have Γ · X(p, R) = X. It is known that Γ acts isometrically, properly discontinuously and co-compactly on X; with the quotient being Γ\X = M. Every point p ∈ X gives rise to length function on Γ, defined by |γ| p = d(p, γp). The Gromov product on X is defined for every triplep, q, r ∈ X by It is well known that (X, d) is a δ-hyperbolic space. That is, there exists some δ ≥ 0, such that for all p, q, r, w ∈ X.
(hyp) (p|q) r ≥ min((p|w) r , (q|w) r ) − δ Consider the quotient space of complex valued continuous functions on X modulo the constant functions, C(X)/C. This space is endowed with the quotient Frechet structure, coming from the topology of uniform convergence on compact sets in C(X). The map is a homeomorphism on its image, which is relatively compact. The closure X is called the horofunctions compactification of X. We set B = ∂X. This is a topological sphere. The Γ action on X extends continuously to X, with B being the unique minimal closed invariant subset. The Gromov-product (·|·) p continuously extends to X × X (for a fixed p ∈ X), and the inequality (hyp) stays valid. For a fixed a point p ∈ X, the function is a metric, denoted the Busemann metric, on B (see [Bou95, Chapter 2]). The ball in B with respect to σ p , of radius t ≥ 0, centered at b ∈ B will be denoted B p (b, t). For every two distinct points r ∈ X, and q ∈ X denote by ℓ r,q : [−∞, ∞] → X the unique unit length geodesic passing through r and q, and satisfying ℓ r,q (0) = p and ℓ r,q (d(r, q)) = q. Denote ℓ r,q (∞) by z q r (observe that if q ∈ B then z q r = q). We denote B p (q) = B p (z q p , e −d(p,q) ), and (for completeness) B p (p) = B. When p is fixed and q varies the map q → B p (q) is a surjective continuous map from X to the collection of balls of (B, σ), taken with the Hausdorff distance. The quantity is defined for all p, q ∈ X and b ∈ B. It is called the Busemann cocycle, and satisfies the cocycle equation for every p, q, r ∈ X, b ∈ B, β b (p, q) + β b (q, r) = β b (p, r).
Denote by η the critical exponent associated to (M, g), that is (for a fixed, yet insignificant, p ∈ X) The critical exponent is positive and finite. It is known that η equals the Hausdorff dimension of the metric space (B, σ p ). The associated Hausdorff measure is denoted ν p . The collection of measures ν p , p ∈ X is an η-conformal density for Γ, i.e.,the associated map ν : X → Meas(B) is Γ-equivariant, and for every p, q ∈ X, dν q dν p (b) = e −ηβ b (p,q) .
where Meas(B) denotes the space of radon measures on B.
It is known that ν is the unique (up to a constant multiplicative factor) η-conformal density. In particular it coincides (up to a scale) with the Patterson-Sullivan measures. Therefore, we choose the representation such that ν p (B) = 1 for some fixed point p ∈ X.
Denote H = L 2 (B, ν p ) the space of complex valued square integrable functions on B. The vector space H is independent of p. Endow H with the inner product We denote by U p (H) the group of unitary operators of (H, ·, · p ). The map is a unitary representation of Γ, called the quasi-regular representation, or the boundary representation associated to (B, ν p ). For every two points p, q ∈ X, the boundary representations ρ p and ρ q are unitary equivalent by the Observe that this is a closed cone in H. We denote by End(H) the algebra of bounded operators on H and endow it with the weak operator topology. We denote by End + (H) the cone of all positive bounded operators, namely all operators T ∈ End(H) that satisfy T H + ⊂ H + . We set VN(ρ p ) to be the closure of span(ρ p (Γ)) < End(H), and VN + (ρ p ) to be the closed cone generated by ρ(Γ), that is, the closure of the cone of linear combinations with positive coefficients of elements of ρ p (Γ). Obviously, VN + (ρ p ) ⊂ End + (H). The following is the main theorem of this paper.
We now present a family of group algebra elements which play a central role in our analysis. Fix p ∈ X. For every t > 0 set and to every bounded Borel function on on f : B → C we associate the group algebra element The representation ρ p extends linearly to the group algebra CΓ. We set T t (f ) = T f t . Thus, ρ p • T t might be seen as the End(H)-valued measure on B. The space of End(H)-valued measures carries a natural weak * -topology (see appendix B, and in particular, lemma B.1). An example of an End(H) valued measure is given by the measure m p defined by Theorem 4 (measure convergence). With respect to the weak * topology, we have lim In particular, for f ∈ C(B) and g, h ∈ L 2 (B, ν p ) we have

Some Lemmas
In this section we start the analysis of some functions on the boundary, B. Before starting we will establish our point of view. We fix once, and for the rest of the paper, a point p ∈ X. We set |q| = d(p, q).
We now proceed by defining some family of important functions on the boundary, and their "chopped" analogs. Set λ q (b) = e − 1 2 ηβ b (p,q) , and λ q 1 = λ q , 1 . Sometimes we use λ γ = λ γp = ρ(γ)1 and λ γ 1 = λ γp 1 = ρ(γ)1, 1 . Define the functions (q|·) on B by (q|b) = min{(z q p |b), |q|}, and accordingly, . The justification for the notation is given by Lemma 3.1. For every p, q ∈ X and for every b ∈ B, Proof. Parts (2),(3) and (4) follow from (1), which we will prove. By (hyp) and (z q On the other hand as (q|b) ≤ |q| For the reader convenience the definition of a regular metric measure space and a short discussion of its properties is given in appendix A (definition A.1). We set σ = σ p .
We will denote the associated multiplicative constants by 0 < k ≤ k ′ . Recall our definition B r (q) = B r (z q r , e −d(r,q) ) and set also B(q) = B p (q).
Before proving the proposition we need to prove the following lemma.
Proof of the lemma. We first claim that for every γ ∈ Γ, Recall now that Γ·X(p, R) = X, and choose γ ∈ Γ such that γq ∈ X(p, R). From the definition of the Gromov product we get that for every x, y ∈ X, ). By compactness argument, using lemma A.1 in the appendix This proves the lemma.
Proof of the proposition. Any ball in B equals B(q) for some q ∈ B. By the η-conformality of the measure, We get k e −|q| η ≤ ν(B(q)) ≤ k ′ e −|q| η This proves η-regularity.
We will also need an estimate of the L 1 norm of λ q and λ q .
Proposition 3.3. There exist constants 0 < C < C ′ and 0 < T such that for every q ∈ X(p, T ) c , Proof. First observe that for every q ∈ X, by the definition of (q|b), Applying η-regularity to the first summand of the right hand side, and corollary A.2 to the second, we get Thus, picking C < k and C ′ > k ′ , there exist T > 0 such that for all |q| > T , C|q|e − 1 2 η|q| ≤ λ q 1 ≤ C ′ |q|e − 1 2 η|q| . The second assertion follows by lemma 3.1.

Uniform boundedness
This section is devoted to the proof that the functions are uniformly bounded in t in L ∞ (B, ν). This is achieved in proposition 4.4. Our main tool is lemma A.3 which, after a minor effort, we show applies here.
The following lemma computes the radius of the inside and outside balls of a shape in X which arises as the intersection of an annulus with a cone based at a boundary ball (see figure).
Lemma 4.1. Let q ∈ X be a point such that q = p. Denote q ′ = ℓ p,q (|q| + 2δ + R), and define the set p P P P P P P P P P P P P P P P P P P P P P P P P P

h h h h h h h h h h h h h h h h h h h h h h h h h h
Proof. First, we prove the left hand inclusion. Fix a point r ∈ X(q ′ , R). Obviously, ||r| − |q ′ || < R. By lemma 3.1(1), Therefore, (z q p |z r p ) ≥ |q| and z r p ∈ B(q). Now we prove the right hand inclusion. Fix a point r ∈ Y q . Then z r p ∈ B(q). By Lemma 3.1(1), (q ′ |c) ≥ |q ′ | − δ. By (hyp) A consequence of lemma 4.1 and the definition of R is that for every q ∈ X, for some fixed integer m (one can take m = |{γ | |γp| < 3R + 4δ}|).  Proof. Fix b ∈ B. Apply Lemma 4.1 with q = ℓ p,b (t − R − 2δ) and q ′ = ℓ p,b (t).
By applying lemma A.3 we get the following global estimate.
Corollary 4.3. Fix t > R + 2δ. For every q ∈ X, with |q| ≤ t + R, We use lemma A.3 to finish the proof.

Analyzing matrix coefficients
In this section the asymptotic of some matrix coefficients of operators of the form T χ U t (for appropriate subsets U ⊂ B) is computed, with the aid of appendix C, thus providing the main technical tool needed in the proof of theorem 4 given in the next section. The outcome of this section will be proposition 5.4. We begin with the more general setting of proposition 5.1.
Given a positive number a > 0 and a subset U ⊂ B we set Thus, ∩ a>0 U(a) = U .
Then for every measurable set V ⊂ B, and every a > 0, lim sup t→∞ γ∈Γ For the proof we will state and prove the following lemma.
Observe that, by lemma 3.1, We are done by proposition 3.3.
Specializing to On the other hand, specializing to we deduce Corollary 5.3. For every measurable subsets U, V ⊂ B and for every a > 0, Finally, we get Proposition 5.4. For every measurable sets U, V, W ∈ B such that ν(∂U) = ν(∂V ) = ν(∂W ) = 0, Proof. The last two corollaries readily imply that for every a, a ′ > 0, By corollary C.1 in the appendix, for every a and a ′ such that ν(∂W (a)) = ν(∂V (a ′ )) = 0, This condition is valid for all, but at most countable many values of a and a ′ . Thus, by taking sequences tending to zero, we get From the definition of T it is obvious that T 1 t (1), 1 = 1 for every t > 0. If we denote U 1 = U, U −1 = B − U, and similarly for V and W , we get

Proofs
We are now in a position to gather the various ingredients supplied in the last two sections into proofs. Schematically, the sequence of proofs goes as follows: prop 4.4 + prop 5.4 ⇒ thm 4 ⇒ thm 3 ⇒ thm 1 + thm 2 For a clarification of some of the notation, we advise the reader to read Appendix B.
Proof of theorem 4. The proof breaks into two parts. In the first part, using the results of section 4, we prove that the maps T t have limit points when t tends to infinity. In the second part, using the results of section 5, we compute the actual limit.
For the first part, observe that the space of End(H) valued measures on B can be naturally identified with the space (C(B) ⊗ π H ⊗ π H * ) * (see corollary B.1). By the Banach-Alaoglu theorem, every ball is compact in the corresponding weak * -topology. Thus, we only have to prove that there is a uniform bound on the norms of T t . The norms of T t are the operator norms of T 1 t . Observe that T 1 t preserve the subspace L ∞ (B) < L 2 (B). Consider T 1 t as transformations of L ∞ (B), using the positivity of all their coefficients as elements of CΓ, their norms are given by T 1 t (1) ∞ , which are uniformly bounded by proposition 4.4. By their self adjointness, the operators T 1 t are uniformly bounded as operators of L 1 (B) as well. It follows, by the Riesz-Thorin interpolation theorem, that they are uniformly bounded on L 2 (B), and the first part of the proof follows.
We define two transformations, p and e, ranging from the algebraic tensor product H ⊗ H, by For the second part of the proof we choose a limit point at infinity of T t , denoted T ∞ , and we wish to show that for every measurable set U ⊂ B, T ∞ (χ U ) = e(1 ⊗ χ U ). It is enough to prove this equation for sets U in a generating set of the Borel σ-algebra. We will do so for the sets with measure zero boundaries. Indeed, for such a set U ⊂ B, and for every two sets W, V ⊂ B with measure zero boundaries, by proposition 5.4, By part 3 of lemma B.2, the span of the set {p(χ V ⊗χ W ) | ν(∂V ) = ν(∂W ) = 0} is weak * dense in End(L 2 (B)) * , and the proof is complete.
Proof of theorem 3. By theorem 4, As the group ρ(Γ) is stable under conjugation, so is VN + (ρ). Then for every measurable U, V ⊂ B, We are done by part 4 of lemma B.2.
The implication to theorem 1 is immediate.
Proof of theorem 1. It follows at once from theorem 3 that VN(ρ) = End(H). This implies that the centralizer of ρ(Γ) is the center of End(H), that is, the group of scalar multiplications. The irreducibility follows by Schur's lemma.
In order to prove theorem 2 we state the following theorem, which in its generality is due to Furman [Fur].  . (B, σ, ν) is called η-regular if there exist constants 0 < k < k ′ such that for every b ∈ B and 0 < t ≤ diam(B), Proof. We prove the right inequality, the proof of the left one being similar. Fix ǫ > 0. Choose a partition u 0 = s < u 1 < . . . < u n = t of the interval [s, t], such that the upper sum associated to the partition u η 0 < u η 1 < . . . < u η n is less than t η s η f (u)du + ǫ. We simplify the notation by setting As ǫ was arbitrary we get the desired inequality.
A particular case of interest is recorded in the following corollary.
Corollary A.2. Assume that (B, σ, ν) is η-regular with multiplicative constants 0 < k ≤ k ′ , and of diameter one. Then for every b ∈ B and 0 < s < 1 we have In a regular space one can estimate an integral by sampling. In order to make this statement precise we define what we mean by a "sample". Definition A.3. Fix r > 0. An r-sampling set for (B, σ), with multiplicity m ∈ N, is a finite set S together with a mapl : S → B, s →l s , such that B = ∪ S B(l s , r), and for all b ∈ B, |l −1 (B(b, r))| ≤ m.
Lemma A.3. Assume that (B, σ, ν) is η-regular with multiplicative constants 0 < k ≤ k ′ . Fix r > 0. Let S be an r-sampling set for (B, σ), with multiplicity m ∈ N. For every L > 0, set C L = m(Le L +1)k ′ k (this constant does not depend on r). Then for every function f on B such that log f is (r, L)-almost continuous, i.e., | log(f (x)) − log(f (y))| ≤ L whenever σ(x, y) ≤ r, we have Proof. We prove the right inequality, the proof of the left one being similar. First observe that, denoting g = log f , for every b ∈ B and b ′ ∈ B(b, r), by almost continuity we have that Using the fact that for every s, t ∈ R, |e s − e t | ≤ |s − t|e max{s,t} , we get Second, observe that 3. The set p(E ⊗ E) is weak * total in End(H) * . 4. The weak * -closed cone in End(H), generated by e(E ⊗E) equals End(H) + .
Proof. 1. This follows from the regularity of ν. Indeed, for every measurable subset A in B, and for every ǫ, we can find compact set K and open set O, with K ⊂ A ⊂ O, and ν(O −A), ν(A−K) < ǫ/2. All, but at most countably many of the sets have zero measure boundary, and all of them have symmetric difference with A of measure less then ǫ, Thus, Every function in h ∈ H + can be approximated by elements of span(E). But, every element of span(E) is measurable with respect to some finite (σ-)algebra, the space of such makes a (finite dimensional) subspace I < span(E), and the best approximation of h in I is given by its so called conditional expectation, which is obviously positive.
3. This follows from 1, using the fact that for every u, v ∈ H, u ⊗ π v = u v . 4. Fix P ∈ End(H) + . By 1, a basis for the neighborhoods of P is given by the sets U h 1 ,...,hn;ǫ = {L | for all i = 1, . . . , n, P h i − Lh i < ǫ} ⊂ End (H) where ǫ > 0, and the functions h 1 , . . . , h n are mutually orthogonal characteristic functions in E. Assume given such a basis set. Let I = span{h 1 , . . . , h n } < span(E). Using 2, we can approximate the functions P h 1 , . . . , P h n by positive elements of span(E), say f 1 , . . . , f n . There is a minimal finite (σ-)algebra, which measures all the f i 's. Denote by h ′ 1 , . . . , h ′ m ∈ E the characteristic functions of the atoms of that algebra. Set I ′ = span{h ′ 1 , . . . , h ′ m } < span(E). Denote by i : I ⊂ H and i ′ : I ′ ⊂ H the inclusions maps. Let L ∈ Hom(I, I ′ ) be the transformation satisfies Lh i = f i . Then i • L • (i ′ ) * is easily seen to be an element of the positive cone generated by e(E ⊗ E), which is in U h 1 ,...,hn;ǫ (for well enough approximating f i 's).
Appendix C. A "geodesics counting" formula We wish to express our deep gratitude to François Ledrappier, for pointing out to us the connection between the expression 1 |S t | γ∈St χ {γ |l γ −1 ∈U } χ {γ |lγ ∈V } , and the number of closed geodesics of a certain kind, as well as the exact reference to its asymptotic, computed in Margulis' thesis, and by that enable us to replace a long argument by a short and elegant one. The price we are willing to pay, is having our presentation considerably less self contained. This section is devoted to the explanation of Ledrappier's observation. • ℓ(s) = m ′ and dℓ(t) dt t=s ∈ U ′ .
We refer the reader to theorem 7 in Margulis' thesis [Mar04, p. 57, Theorem 7]. In a similar fashion to the proof of Theorem (6.2.5) in [Yue96], applying Margulis' theorem to the submanifolds T 1 m M and T 1 m ′ M (which are transversal to both the stable and unstable foliations), yields the following Theorem 6 (Margulis). Let m, m ′ ∈ M be points, U ⊂ T 1 m and U ′ ⊂ T 1 m ′ M measurable subsets satisfying ν m (∂U) = ν m ′ (∂U ′ ) = 0. For every a > 0, there exists C = C a,m,m ′ ∈ (0, ∞), such that lim t→∞ e −ηt n(U, U ′ , (t − a, t + a)) = Cν m (U)ν m ′ (U ′ ) (In particular, C does not depend on the sets U and U ′ ).