The Poisson boundary of hyperbolic groups without moment conditions

Abstract

We prove that the Poisson boundary of a random walk with finite entropy on a non-elementary hyperbolic group can be identified with its hyperbolic boundary, without assuming any moment condition on the measure. We also extend our method to groups with an action by isometries on a hyperbolic metric space containing a WPD element; this applies to a large class of non-hyperbolic groups such as relatively hyperbolic groups, mapping class groups, and groups acting on CAT(0) spaces.

23 pages. Changes in the new version: we included Theorem 1.2, which now applies to all groups with a WPD element, and its applications in Corollary 1.3. The proof is in Section 5. For the sake of exposition, we also added Section 3.1, explaining the concept of pivots for free groups