The Poisson boundary of CAT(0) cube complex groups

Amos Nevo, Michah Sageev

Research output: Contribution to journalArticlepeer-review

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 Gλ subset UNT(X) of B(X) which supports every stationary measure, and on which the action of G is Borel-amenable. We describe the relation of UNT(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 particularweanalyze 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.

Original languageEnglish
Pages (from-to)653-695
Number of pages43
JournalGroups, Geometry, and Dynamics
Volume7
Issue number3
DOIs
StatePublished - 2013

Keywords

  • Boundary space
  • CAT(0) cube complex
  • Co-compact action
  • Cube complex group
  • Poisson boundary
  • Strong proximality
  • Ultrafilters

All Science Journal Classification (ASJC) codes

  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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