Year: 2012
Type: preprint
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 $Γ$ with the set of open ends ${\rm Ends}(Γ)$, then for every non-degenerate probability measure $μ$ on $G$ there exists a unique $μ$-stationary probability measure $ν_μ$ on ${\rm Ends}(Γ)$, and the space $({\rm Ends}(Γ), ν_μ)$ is a $μ$-boundary. Moreover, if $μ$ has finite first moment with respect to the word metric on $G$ (induced by a finite generating set), then the measure space $({\rm Ends}(Γ), ν_μ)$ is isomorphic to the Poisson--Furstenberg boundary of $(G, μ)$.
Source: arXiv (Cornell University)
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2733218737