The Furstenberg Poisson Boundary and CAT(0) Cube Complexes

TL;DR

This paper demonstrates that the Roller boundary of a finite dimensional CAT(0) cube complex serves as the Furstenberg-Poisson boundary for certain random walks on groups acting on the complex, under weak hypotheses.

Contribution

It establishes the boundary identification for random walks on groups acting on CAT(0) cube complexes, extending previous results to weaker conditions.

Findings

The Roller boundary is the Furstenberg-Poisson boundary under specified conditions.

Existence of a stationary measure on the boundary for the random walk.

Support of the measure is within the closure of regular points exhibiting strong contraction.

Abstract

We show under weak hypotheses that $\partial X$, the Roller boundary of a finite dimensional CAT(0) cube complex $X$ is the Furstenberg-Poisson boundary of a sufficiently nice random walk on an acting group $\Gamma$. In particular, we show that if $\Gamma$ admits a nonelementary proper action on $X$, and $\mu$ is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a $\mu$-stationary measure on $\partial X$ making it the Furstenberg-Poisson boundary for the $\mu$-random walk on $\Gamma$. We also show that the support is contained in the closure of the regular points. Regular points exhibit strong contracting properties.