The Poisson boundary of CAT(0) cube complex groups

By A. Nevo, Michah Sageev. 2011

54 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8e4f22f4d6c2459acab7e06d3c023b1bede0d1f7

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Citations in the corpus

The median class and superrigidity of actions on CAT(0) cube complexes (2012, background): “A for the definition) as a metric model for a Poisson boundary of Γ (see [43]).”
Hierarchically hyperbolic spaces I: curve complexes for cubical groups (2014, result): “Theorem D and the results of [MT14] combine to yield the following, which is related to work of Nevo-Sageev on Poisson boundaries of cube complexes [NS13]:”
Probability on Trees and Networks (2017, background): “a tree? This question is due to Newman and Stein (1996), who conjecture that the answer is d < 8 or d ≤ 8.”
Boundaries and automorphisms of hierarchically hyperbolic spaces (2016, result): “26 should be compared to the results of [NS13], in which Nevo-Sageev construct the Poisson boundary for a cubical group using the Roller boundary of the cube complex.”
Contracting boundaries of CAT(0) spaces (2013, background): “Other notions of boundaries for CAT(0) cube complexes have been introduced by Roller [26], Guralnik [16], Nevo-Sageev [24], Hagen [17] and Behrstock-Hagen [5].”