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A boundary criterion for cubulation

From: American Journal of Mathematics
Volume 134, Number 3, June 2012
pp. 843-859 | 10.1353/ajm.2012.0020

Abstract

Abstract:

We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a ${\rm CAT}(0)$ cube complex. We describe applications towards lattices and hyperbolic 3-manifold groups. In particular, by combining the theory of special cube complexes, the surface subgroup result of Kahn-Markovic, and Agol's criterion, we find that every subgroup separable closed hyperbolic 3-manifold is virtually fibered.