The double of a virtually compact special Gromov-hyperbolic group along a quasiconvex subgroup is virtually compact special, with a generalization to certain graphs of groups.
The virtual Haken conjecture
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abstract
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 "Residual finiteness, QCERF and fillings of hyperbolic groups".
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math.GR 1years
2026 1verdicts
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Virtual specialness of the double
The double of a virtually compact special Gromov-hyperbolic group along a quasiconvex subgroup is virtually compact special, with a generalization to certain graphs of groups.