Ergodic Theorems for Actions of Hyperbolic Groups

In this note we give a short proof of a pointwise ergodic theorem for measure preserving actions of word hyperbolic groups, also obtained recently by Bufetov, Khristoforov and Klimenko. Our approach also applies to infinite measure spaces and one application is to linear actions of discrete groups on the plane. 0. Introduction The well-known Birkhoff ergodic theorem for ergodic transformations T : X → X on a probability space (X,B, μ) gives that the averages of an L function along almost every orbit is equal to the integral of the function. If we view the transformation as a Z-action, then it is natural to ask about pointwise ergodic theorems for other group actions. For Z, or nilpotent groups, pointwise ergodic theorems for L functions are described in the books of Krengel [16] and Templeman [23]. For groups with exponential growth, results are considerably harder to obtain. For free groups, a mean ergodic theorem for L functions was proved by Guivarc’h in the 1960s [13] but it was not until the work of Nevo and Stein [19] in the 1990s that pointwise ergodic theorems were obtained. Their result applied to L functions with p > 1 but subsequently Bufetov proved the stronger pointwise ergodic theorem for L functions using an approach based on Markov operators [4]. (The latter was recently extended to the fundamental groups of compact surfaces with negative Euler characteristic by Bufetov and Series [6].) The extension of these pointwise theorems to more general word hyperbolic groups for L functions was carried out by Fujiwara and Nevo, under strong mixing assumptions on the action [11]. Very recently, Bufetov, Khristoforov and Klimenko [5] showed that, independent of any mixing condition, there is pointwise convergence of appropriate Cesàro averages for L∞ functions. (They also establish L convergence for L functions, for any p ≥ 1.) In this note, we give a short proof of the almost everywhere convergence of (slightly modified) averages for Cesàro averages for L∞ functions. A key ingredient of our approach is an observation of Calegari and Fujiwara [7] on the structure of the Markov matrix encoding a word hyperbolic group. Let Γ be a finitely generated group and let Γ0 be a finite generating set. (For convenience, we always assume our generating set is symmetric, i.e. γ ∈ Γ0 implies that γ−1 ∈ Γ0.) Typeset by AMS-TEX 1 2 MARK POLLICOTT AND RICHARD SHARP Definition. We define the word length |γ| of an element γ ∈ Γ− {e}, with respect to the generators Γ0, by |γ| = inf {k ≥ 1 : γ = γ1 . . . γk where γi ∈ Γ0, 1 ≤ i ≤ k} . Definition. We say that Γ is word hyperbolic (or Gromov hyperbolic) if, for some finite generating set Γ0, the following holds. Let |·| denote word length with respect to Γ0 and define a metric by d(g, h) = |g−1h|. Then there exists δ ≥ 0 such that every geodesic triangle in this metric is δ-thin, i.e., every point on one side of the triangle is within δ of the other two sides. We say that a word hyperbolic group is non-elementary if it does not contain a finite index cyclic subgroup. For n ≥ 1, let N(n) = #{γ : |γ| = n} denote the word length counting function. We let ρ denote the exponential growth rate with respect to word length, i.e., ρ = limn→+∞N(n) . This limit always exists for word hyperbolic groups and, by [9], one has the stronger estimate C1ρ n ≤ N(n) ≤ C2ρ, for some C2 > C1 > 0. Let (X,B, μ) be a probability space and let Γ × X → X be an action with preserves μ. We then have the following general result. Theorem 1. Let Γ be a non-elementary word hyperbolic group and let Γ0 be a symmetric set of generators. For f ∈ L∞(X,μ) we have that