DOCUMENTA MATHEMATICA, Vol. 18 (2013), 1045-1087

Ian Agol

The Virtual Haken Conjecture

We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken question of Waldhausen and Thurston's virtual fibering question. An appendix to this paper by Agol, Groves, and Manning proves a generalization of the main result of \cite{AGM08}.

2010 Mathematics Subject Classification: 57M, 20F67

Keywords and Phrases: hyperbolic, group, Haken

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