Random walks on weakly hyperbolic groups

TL;DR

This paper proves that random walks on groups with weakly hyperbolic actions converge to the Gromov boundary almost surely and characterizes the Poisson boundary under certain conditions, extending understanding of random walks in hyperbolic geometry.

Contribution

It establishes convergence of random walks on weakly hyperbolic groups and identifies the Poisson boundary under acylindrical actions with finite entropy and moments.

Findings

Random walks converge to the Gromov boundary almost surely.

Linear progress and growth of translation length are shown.

Poisson boundary is identified under specific conditions.

Abstract

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.