JournalsggdVol. 7, No. 3pp. 653–695

The Poisson boundary of CAT(0) cube complex groups

  • Amos Nevo

    Technion - Israel Institute of Technology, Haifa, Israel
  • Michah Sageev

    Technion - Israel Institute of Technology, Haifa, Israel
The Poisson boundary of CAT(0)  cube complex groups cover
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Abstract

We consider a finite dimensional, locally finite CAT(0) cube complex XX admitting a co-compact properly discontinuous countable group of automorphisms GG. We construct a natural compact metric space B(X)B(X) on which GG acts by homeomorphisms, the action being minimal and strongly proximal. Furthermore, for any generating probability measure on GG, B(X)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 GδG_\delta subset UNT(X)\mathcal{U}_{\,\rm NT}(X) of B(X)B(X) which supports every stationary measure, and on which the action of GG is Borel-amenable. We describe the relation of UNT(X)\mathcal{U}_{\,\rm NT}(X) and B(X)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)B(X) 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

DOI 10.4171/GGD/202