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 Gδ 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 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.
1
Technion - Israel Institute of Technology, Haifa, Israel
Amos Nevo; Michah Sageev. The Poisson boundary of CAT(0) cube complex groups. Groups, geometry, and dynamics, Tome 7 (2013) no. 3, pp. 653-695. doi: 10.4171/ggd/202
@article{10_4171_ggd_202,
author = {Amos Nevo and Michah Sageev},
title = {The {Poisson} boundary of {CAT(0)} cube complex groups},
journal = {Groups, geometry, and dynamics},
pages = {653--695},
year = {2013},
volume = {7},
number = {3},
doi = {10.4171/ggd/202},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/202/}
}
TY - JOUR
AU - Amos Nevo
AU - Michah Sageev
TI - The Poisson boundary of CAT(0) cube complex groups
JO - Groups, geometry, and dynamics
PY - 2013
SP - 653
EP - 695
VL - 7
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/202/
DO - 10.4171/ggd/202
ID - 10_4171_ggd_202
ER -
%0 Journal Article
%A Amos Nevo
%A Michah Sageev
%T The Poisson boundary of CAT(0) cube complex groups
%J Groups, geometry, and dynamics
%D 2013
%P 653-695
%V 7
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/202/
%R 10.4171/ggd/202
%F 10_4171_ggd_202
