We consider a finite dimensional, locally finite CAT(0) cube complex admitting a co-compact properly discontinuous countable group of automorphisms . We construct a natural compact metric space on which acts by homeomorphisms, the action being minimal and strongly proximal. Furthermore, for any generating probability measure on , 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 subset of which supports every stationary measure, and on which the action of is Borel-amenable. We describe the relation of and 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 is the Poisson boundary via the strip criterion developed by V. Kaimanovich.
Cite this article
Amos Nevo, Michah Sageev, The Poisson boundary of CAT(0) cube complex groups. Groups Geom. Dyn. 7 (2013), no. 3, pp. 653–695