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

TL;DR

This paper investigates the boundary behavior of random walks on non-abelian groups acting on $ ext{Z}^n$-trees, establishing the uniqueness of stationary measures and their relation to the Poisson--Furstenberg boundary.

Contribution

It proves the existence and uniqueness of stationary measures on the ends of $ ext{Z}^n$-trees for non-abelian groups and links these measures to the Poisson--Furstenberg boundary.

Findings

Unique $ u_$-stationary measure on ends of $ ext{Z}^n$-trees

Boundary measures coincide with Poisson--Furstenberg boundary under finite first moment condition

Random walks exhibit boundary behavior characterized by these measures

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 ${\rm Ends}(\Gamma)$, then for every non-degenerate probability measure $\mu$ on $G$ there exists a unique $\mu$-stationary probability measure $\nu_\mu$ on ${\rm Ends}(\Gamma)$, and the space $({\rm Ends}(\Gamma), \nu_\mu)$ 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 $({\rm Ends}(\Gamma), \nu_\mu)$ is isomorphic to the Poisson--Furstenberg boundary of $(G, \mu)$.