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

TL;DR

This paper establishes conditions under which random walks on certain groups, including Thompson's group F, have non-trivial Poisson boundaries, revealing new insights into their probabilistic and algebraic structure.

Contribution

It provides sufficient conditions for non-trivial Poisson boundaries on groups of piecewise projective homeomorphisms, including Thompson's group F, extending understanding of their boundary behavior.

Findings

Non-trivial Poisson boundary for measures with finite first moment on $H(\

Non-trivial Poisson boundary for measures on Thompson's group F that generate it as a semigroup.

Solvability criterion for subgroups of $H(\

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.