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**The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves and Jason Manning).**
*(English)*
Zbl 1286.57019

This paper proves an important result in geometric group theory with strong consequences for three-manifolds: cubulated hyperbolic groups are virtually special. It implies the virtual fibration theorem, answering a question of Thurston in 1982: every closed hyperbolic three-manifold has a finite sheeted covering that fibers over the circle. In particular it establishes the Haken conjecture, stated by Waldhausen in 1968, because this covering is also Haken. In addition, Agol’s theorem also yields that the fundamental group of a closed hyperbolic three-manifold is large (a finite index subgroup surjects onto a non cyclic free group) and LERF (locally extended residually finite). The implication of virtual fibration relies on work of Kahn-Markovich (on the existence of almost-geodesics immersed surfaces) and on work on cube complexes of non-positive curvature by Bergeron, Haglund, and specially Wise.

A group is called cubulated if it acts properly and cocompactly on a cube complex of non-positive curvature. Sageev’s construction provides such a complex to a configuration of immersed surfaces. The wall complex is another cube complex obtained by taking the coordinate hyperplanes on each cube. Haglund and Wise introduced the key notion of special cube complex, in terms of walls, and showed that special cube complexes are the ones admitting a local isometry to the standard cube complex of a right-angled Artin group. Moreover Wise proved that, for torsion-free hyperbolic groups, being the fundamental group of a compact special complex is equivalent to having a quasi-convex virtual hierarchy and established many properties for such groups, for instance linearity and largeness. Thus, besides the relevant applications to three-manifolds, Agol’s theorem implies that hyperbolic cubulated groups are large, linear, and have the property that quasi-convex subgroups are separable. In order to prove that a cube complex with hyperbolic fundamental group \(X\) is virtually special, Agol finds a covering \(\tilde X\to X\), possibly non-compact but with compact walls. This uses a weak separation theorem proved in an appendix by Agol, Groves and Manning, that generalizes a previous result of the same authors in [Geom. Topol. 13, No. 2, 1043–1073 (2009; Zbl 1229.20037)]. The walls have a hierarchy associated to a (finite) coloring of the incidence graph of the wall complex of \(\tilde X\). The result of cutting \(\tilde X\) along walls is a collection of cubical polyhedra, with a coloring on their faces that keeps track of how the walls are cut. Then the polyhedra must be glued again so that they form a finite covering of \(X\). For that purpose, a virtual gluing theorem and a theorem on probability measures on the space of colorings of a graph are proven.

A group is called cubulated if it acts properly and cocompactly on a cube complex of non-positive curvature. Sageev’s construction provides such a complex to a configuration of immersed surfaces. The wall complex is another cube complex obtained by taking the coordinate hyperplanes on each cube. Haglund and Wise introduced the key notion of special cube complex, in terms of walls, and showed that special cube complexes are the ones admitting a local isometry to the standard cube complex of a right-angled Artin group. Moreover Wise proved that, for torsion-free hyperbolic groups, being the fundamental group of a compact special complex is equivalent to having a quasi-convex virtual hierarchy and established many properties for such groups, for instance linearity and largeness. Thus, besides the relevant applications to three-manifolds, Agol’s theorem implies that hyperbolic cubulated groups are large, linear, and have the property that quasi-convex subgroups are separable. In order to prove that a cube complex with hyperbolic fundamental group \(X\) is virtually special, Agol finds a covering \(\tilde X\to X\), possibly non-compact but with compact walls. This uses a weak separation theorem proved in an appendix by Agol, Groves and Manning, that generalizes a previous result of the same authors in [Geom. Topol. 13, No. 2, 1043–1073 (2009; Zbl 1229.20037)]. The walls have a hierarchy associated to a (finite) coloring of the incidence graph of the wall complex of \(\tilde X\). The result of cutting \(\tilde X\) along walls is a collection of cubical polyhedra, with a coloring on their faces that keeps track of how the walls are cut. Then the polyhedra must be glued again so that they form a finite covering of \(X\). For that purpose, a virtual gluing theorem and a theorem on probability measures on the space of colorings of a graph are proven.

Reviewer: Joan Porti (Bellaterra)

### MSC:

57M99 | General low-dimensional topology |

57M50 | General geometric structures on low-dimensional manifolds |

20F67 | Hyperbolic groups and nonpositively curved groups |