Non-triviality of the Poisson boundary of random walks on the group of Monod

Abstract

We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on HpZq and its subgroups. The group HpZq is the group of piecewise projective homeomorphisms over the integers defined by Monod. For a finitely generated subgroup H of HpZq, 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.