Non-triviality of the Poisson boundary of random walks on the group $H(\mathbb{Z})$ of Monod

Abstract

We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by Monod. For a finitely generated subgroup $H$ of $H(\mathbb{Z})$, we prove that either $H$ is solvable, or every measure on $H$ with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson's group $F$ that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich.

27 pages, 6 figures. Changes from previous version: Split Section 5 in two, added a remark (Remark 6.3) on amenability of Schreier graphs, minor changes to the introduction

References in corpus (6)

• Amenability and paradoxical decompositions for pseudogroups and for discrete metric spaces
• Choquet-Deny groups and the infinite conjugacy class property
• The Liouville property and random walks on topological groups
• Boundary of the action of Thompson group F on dyadic numbers
• Chain groups of homeomorphisms of the interval
• Thompson's group $F$ is not Liouville

Cited by in corpus (1)

• Convergence towards the end space for random walks on Schreier graphs