The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves and Jason Manning).
Ian AgolDaniel Groves University of California Berkeley Department of Mathematics 970 Evans Hall #3840 Statistics, and Computer Science Berkeley, CA, 94720-3840 University of Illinois at Chicago USA 322 Science and Engineering Of
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 citeAGM08.
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Ian Agol, The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves and Jason Manning).. Doc. Math. 18 (2013), pp. 1045–1087DOI 10.4171/DM/421