A boundary criterion for cubulation

@article{Bergeron2009ABC,
  title={A boundary criterion for cubulation},
  author={Nicolas Bergeron and Daniel T. Wise},
  journal={American Journal of Mathematics},
  year={2009},
  volume={134},
  pages={843 - 859}
}
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. 
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Agol's theorem on hyperbolic cubulations
  • Sam Shepherd
  • Mathematics
    Rocky Mountain Journal of Mathematics
  • 2021
The following theorem was proven by Agol: let $G$ be a hyperbolic group acting properly and cocompactly on a CAT(0) cube complex $X$, then $G$ has a finite index subgroup $G'$ that acts freely on $X$
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