Random walks on weakly hyperbolic groups

By Joseph Maher, G. Tiozzo. Journal für die Reine und Angewandte Mathematik, 2014

164 citations recorded by the source corpus.

Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk. If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with the hitting measure is the Poisson boundary.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9b5914bc7bc19639c82b835d6645dbd75a48a429

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Citations in the corpus

Boundaries and automorphisms of hierarchically hyperbolic spaces (2016, methodology): “We use acylindricity of the action of G on CS and a result of Maher-Tiozzo [MT14]: Theorem 9.27 (Theorem 1.5 in [MT14]).”