Random walks on convergence groups (Q2102161)

Summary: 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 \(G\).