Random walks on convergence groups

Abstract

We extend some properties of random walks on hyperbolic groups to random walks on convergence groups. In particular, we prove that if a convergence group $G$ acts on a compact metrizable space $M$ with the convergence property, then we can provide $G\cup M$ with a compact topology such that random walks on $G$ converge almost surely to points in $M$. Furthermore, we prove that if $G$ is finitely generated and the random walk has finite entropy and finite logarithmic moment with respect to the word metric, then $M$, with the corresponding hitting measure, can be seen as a model for the Poisson boundary of $G$.