RANDOM WALKS IN NEGATIVE CURVATURE AND BOUNDARY REPRESENTATIONS. KEVIN BOUCHER Abstract. In this paper we establish a version of the Margulis Roblin equidistribution theorem’s [28] [32] for harmonic measures. As a conse- quence a von Neumann type theorem is obtained for boundary actions and the irreducibility of the associated quasi-regular representations [3] is deduced. 1. Introduction Given a non-elementary discrete group, Γ, acting properly by isometries on a proper roughly geodesic hyperbolic space X,d , two main classes of measures can be constructed on the Gromov boundaryp q X of X,d : the harmonic measures and quasiconformal measures. Let usB brieflyp recallq these constructions. A measure ν on X is called Γ-quasiconformal if Γ preserves the class of ν and one can find BC 1, for all g Γ and ν-almost every ξ X: ě P PB ´1 dg ν ´1 C´1eδΓβpo,g .o;ξq ˚ ξ CeδΓβpo,g .o;ξq ď dν p qď where β denote the Buseman cocycle on X. Given a point o X in X,d , the criticalB exponent of Γ is defined as: P p q 1 δΓ lim sup ln BX o,r Γ.o arXiv:2001.03932v1 [math.DS] 12 Jan 2020 “ r r | p qX | It is strictly positive when Γ is non-elementary. The Patterson-Sullivan procedure consists to take a weak limit in Prob X p Y X when s goes to δΓ of the sequence of probabilities: B q 1 e´sdpg.o,oqδ H s g.o gPΓ p q ÿ ´sdpg.o,oq for s δΓ and H s gPΓ e . These Patterson-Sullivan measures are quasiconformal.ą p q Moreover “ these are Hausdorff measures for a visual dis- tance on X when Γ actsř cocompactly on X,d [15]. B p q 1 2 KEVIN BOUCHER One the other hand given a probability measure, µ, with 1st moment and support not contained in a elementary group. If Xn n denote the random walk on Γ starting at the identity e Γ, the sequencep q of X-valued random P variables Xn.o n converges almost surely to a random variable Z with values in X. Thep lawq of Z is by definition the harmonic measure on X for the randomB walk generated by µ. B The dynamical systems associated to these measures was extensively in- vestigated these last years. Initiated by U. Bader and R. Muchnik in [28] a particular interest grown around the quasi-regular representations associ- ated to these systems called boundary representations. These representations generalized the well known parabolic inductions L2 G P for semisimple Lie groups G and P minimal parabolic subgroups of G.p In{ theirq work the authors proved the irreducibility of boundary representations arising from Patterson- Sullivan measures and conjectured the following: Conjecture [Bader-Muchnik] Let G be a locally compact group and µ a admissible probability measure on G, the boundary representation of G on the Poisson boundary P G,µ is irreducible. p q Using the interplay between quasiconformal and harmonic measures in negative curvature [8] [14] we propose another approach of these questions that morally attempt to understand boundary dynamics in terms of spectral properties instead of geometric ones. This led us to investigate probabilistic analogues of certain geometric phenomenons. 1.1. Harmonic equidistribution and its consequences. Let µ be a sym- metric probability measure on Γ. The Markov operator Pµ associated to a continuous action α :Γ K K of Γ on a compact space K is the positive ˆ Ñ 0 contraction defined on the space of continuous functions C K , . 8 given by: p p q } } q Pµϕ ξ µ g ϕ gξ p q“ gPΓ p q p q ÿ for ϕ C0 K . A probability measure, ν, on K is µ-stationary if µ ν α µ Pν p νq. Moreover the measure µ is called ν-ergodic if ν is stationary˚ “ ˚p b q“ and every Pµ-invariant function is ν-almost everywhere constant. The principal result of this paper is the following probabilistic version of the Margulis Roblin equidistribution theorem’s [32] [28] for stationary mea- sures on the Gromov boundary of X,d : p q BOUNDARY REPRESENTATIONS 3 Theorem 1 Let ν be a probability measure on the Gromov boundary, X, of X,d . Assume µ Prob Γ is a 1st moment, non-elementary ν-ergodicB probabilityp q measure onP Γ. p q Then the following equidistribution holds: Eµ ´1 ´1 ˚n nÑ`8 ϕ Xn.ξ ψ Xn .η ϕ gξ ψ g η µ g ϕdν ψdν r p q p qs “ gPΓ p q p q p q ÝÝÝÝÑ BX BX ÿ ż ż for all ϕ, ψ C0 X continuous functions and ξ, η X. P pB q PB Let X be a Hadamard manifold with pinched curvature. This means that X is a complete simply connected Riemannian manifold of dimension greater 2 2 than 2 and curvature, KX , which satisfies: b KX a . The Brownian ´ ď ď´ motion Bt t on X, that is the diffusion process generated by the Laplace- p q Beltrami operator on X, converges almost surely to a random variable B8 with values in the geometric boundary X. When the action Γ on X has fi- nite covolume W. Ballman and F. LedrappierB in [5] refined the discretization procedure introduced by T. Lyons and D. Sullivan in [27] and earlier dis- cussed by H. Furstenberg in [19] which allows to interpret the distribution of B8, also called harmonic, has a stationary measure for a random walk performed on Γ: Theorem [Ballman, Ledrappier [5]] There exists a admissible sym- metric probability measure on Γ with 1st moment such that the harmonic measure of the random walk generated by µ and the distribution of the limit random variable B8 coincide. This result guarantees the ergodicity of the measure µ and as a direct consequence of Theorem 1 one has: Corollary Let X be a Hadamard manifold with pinched negative curvature and Γ a discrete group of isometries with finite covolume. Let ν be the unique harmonic measure on the visual boundary X of X for the Brownian motion B Bt t. Thenp q Γ carries a admissible probability measure with 1st moment, µ, such that: Eµ ´1 ´1 ˚n nÑ`8 ϕ Xn.ξ ψ Xn .η ϕ gξ ψ g η µ g ϕdν ψdν r p q p qs “ gPΓ p q p q p q ÝÝÝÝÑ BX BX ÿ ż ż for all continuous functions ϕ, ψ C0 X and ξ, η X. P pB q PB Back to a more general situation where X,d is a proper roughly geodesic hyperbolic space. Assuming the action ofp Γ onq X,d is cocompact and µ p q 4 KEVIN BOUCHER is finitely supported symmetric and admissible it was proved in [8] that the harmonic measure, ν, can be interpreted as a quasiconformal measure coming for the Green distance on Γ that satisfies hyperbolic properties. Conversely when X,d is a CAT( 1) space [11] every quasiconformal measure for Γ can be interpretedp q as a stationary´ measure for a symmetric probability measure with 1st moment on Γ [14]. This interplay motivate the investigation on quasiconformal measures that can be regarded as stationary measures. A measure ν on a compact metric space K,d is called Ahlfors-regular if there exist α 0 and C 1 such that: p q ą ě C´1rα ν B x, r Crα ď p p qq ď for 0 r Diam K . Examplesă ă of suchp measuresq in our setting are given by quasiconformal mea- sures on X associated to discrete groups which act geometrically on X,d . B p q Considering Ahlfors-regular quasiconformal measures on X for the action of Γ on X,d , Theorem 1 together with techniques inspiredB by [14] [3] led us to thep followingq von Neumann type result for measure class preserving actions: Theorem 2 Let µ Prob Γ be a admissible finitely supported symmetric probability on Γ and νP its harmonicp q measure on X. B Denote Pn, with n 0, the operator defined as: ě πo Xn πo g P Eµ p q p q µ˚n g n Ξ X Ξ g “ r o n s“ gPΓ o p q p q ÿ p q where Ξo πo1 X 1 X 2 denotes the Harish-Chandra function of the bound- “p B | B qL ary representation πo on Γ. Then for all ϕ, ψ L2 X, ν , square integrable functions on X, the follow- ing von NeumannP typerB convergences holds: B Pnϕ ψ L2 ϕdν ψdν p | q ÝÝÝÑnÑ8 żBX żBX P In other words if 1BX denote the one dimensional projector on the con- stant function 1BX given by: P1 ϕ ϕ 1 X 1 X BX p q“p | B q B for ϕ L2 X, ν . The Theorem 2 corresponds to the weak convergence of P rB s the sequence of operators Pn n to P1 . p q BX BOUNDARY REPRESENTATIONS 5 In opposite to the measure preserving situation, the vector 1BX appears to 2 be cyclic for the boundary representation Γ, πo, L X, ν and a standard argument leads to the following: p rB sq Corollary 4.1 Let µ Prob Γ be a admissible finitely supported symmet- ric probability on Γ and Pν its harmonicp q measure on X. Then the associated 2 B boundary representation Γ, πo, L X, ν is irreducible. p rB sq 1.2. Outlines. In Section 2, we remind the reader standard facts about random walk on groups as well as some part of the theory of Patterson-Sullivan on quasicon- formal measures.