The Poisson boundary of CAT(0) cube complex groups

Abstract

We consider a finite dimensional, locally finite CAT(0) cube complex $X$ admitting a co-compact properly discontinuous countable group of automorphisms $G$. We construct a natural compact metric space $B(X)$ on which $G$ acts by homeomorphisms, the action being minimal and strongly proximal. Furthermore, for any generating probability measure on $G$, $B(X)$ admits a unique stationary measure, and when the measure has finite logarithmic moment, it constitutes a compact metric mean-proximal model of the Poisson boundary. We identify a dense invariant $G_{\delta }$ subset ${\mathcal{U}}_{\mathrm{NT}}(X)$ of $B(X)$ which supports every stationary measure, and on which the action of $G$ is Borel-amenable. We describe the relation of ${\mathcal{U}}_{\mathrm{NT}}(X)$ and $B(X)$ to the Roller boundary. Our construction can be used to give a simple geometric proof of property A for the complex. Our methods are based on direct geometric arguments regarding the asymptotic behavior of halfspaces and their limiting ultrafilters, which are of considerable independent interest. In particular we analyze the notions of median and interval in the complex, and use the latter in the proof that $B(X)$ is the Poisson boundary via the strip criterion developed by V. Kaimanovich.