Boundaries of ${\mathbb{Z}}^n$-free groups

Abstract

In this paper we study random walks on a finitely generated group $G$ which has a free action on a ${\mathbb{Z}}^n$-tree. We show that if $G$ is non-abelian and acts minimally, freely and without inversions on a locally finite ${\mathbb{Z}}^n$-tree $\Gamma$ with the set of open ends $\mathrm{Ends}(\Gamma )$, then for every non-degenerate probability measure $\mu$ on $G$ there exists a unique $\mu$-stationary probability measure ${\nu }_{\mu }$ on $\mathrm{Ends}(\Gamma )$, and the space $(\mathrm{Ends}(\Gamma ),{\nu }_{\mu })$ is a $\mu$-boundary. Moreover, if $\mu$ has finite first moment with respect to the word metric on $G$ (induced by a finite generating set), then the measure space $(\mathrm{Ends}(\Gamma ),{\nu }_{\mu })$ is isomorphic to the Poisson--Furstenberg boundary of $(G,\mu )$.

29 pages