Random walks on Convergence Groups

TL;DR

This paper generalizes properties of random walks from hyperbolic groups to convergence groups, showing almost sure convergence to boundary points and identifying the Poisson boundary under certain conditions.

Contribution

It extends the theory of random walks to convergence groups, establishing convergence to boundary points and characterizing the Poisson boundary.

Findings

Random walks on convergence groups converge almost surely to boundary points.

The boundary with the hitting measure models the Poisson boundary.

Results generalize hyperbolic group properties to 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 $G$.