The virtual Haken conjecture

TL;DR

This paper proves that hyperbolic groups with cubulations are virtually special, leading to significant implications for 3-manifold topology, including the existence of finite-sheeted Haken covers and solutions to longstanding conjectures.

Contribution

It establishes that cubulated hyperbolic groups are virtually special, confirming the virtual Haken conjecture for hyperbolic 3-manifolds.

Findings

Hyperbolic groups with cubulations are virtually special.

Closed hyperbolic 3-manifolds have finite-sheeted Haken covers.

Quasi-convex subgroups are separable in these groups.

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".