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The Poisson boundary of CAT(0) cube complex groups. (English) Zbl 1346.20084

Summary: 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 \(\mathcal U_{\text{NT}}(X)\) of \(B(X)\) which supports every stationary measure, and on which the action of \(G\) is Borel-amenable. We describe the relation of \(\mathcal U_{\text{NT}}(X)\) 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.

MSC:

20P05 Probabilistic methods in group theory
22F10 Measurable group actions
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
57S30 Discontinuous groups of transformations
60J50 Boundary theory for Markov processes
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