The Poisson boundary of CAT(0) cube complex groups

TL;DR

This paper constructs a natural boundary space for CAT(0) cube complex groups, showing it models the Poisson boundary and has applications in understanding group actions and properties like Property A.

Contribution

It introduces a new geometric construction of the Poisson boundary for CAT(0) cube complex groups using ultrafilters and asymptotic analysis of half-spaces.

Findings

B(X) admits a unique stationary measure for any generating measure on G

B(X) is a compact model of the Poisson boundary when the measure has finite logarithmic moment

The construction provides a geometric proof of Property A for the complex

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 model of the Poisson boundary. We identify a dense G-delta subset of B(X) on which the action of G is Borel-amenable, and describe the relation of these two spaces 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 half-spaces 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.