Hyperbolic groups acting improperly (Q6066647)

Cube complexes in group theory arise via the construction of \textit{M. Sageev} in [Proc. Lond. Math. Soc., III. Ser. 71, No. 3, 585--617 (1995; Zbl 0861.20041)] which takes as input a group \(G\) and a collection of codimension-one subgroups of \(G\) and produces a \(\mathsf{CAT}(0)\) cube complex \(X\) equipped with an isometric \(G\)-action on \(X\) with no global fixed point. The first main result proved in this paper establishes some fundamental properties about the Sageev construction. It is Theorem A: Let \(G\) be a hyperbolic group. The following conditions on a cocompact \(G\)-action on a \(\mathsf{CAT}(0)\) cube complex are equivalent: (a) all hyperplane stabilizers are quasiconvex; (b) all vertex stabilizers are quasiconvex; (c) all cell stabilizers are quasiconvex. A group \(G\) is virtually special if there is a finite-index subgroup \(G_{0} \leq G\) and a \(\mathsf{CAT}(0)\) cube complex \(X\) such that \(G_{0}\) acts freely and cubically on \(X\) and \(G_{0} \setminus X\) is a compact special cube complex. A key result due to \textit{I. Agol} [Doc. Math. 18, 1045--1087 (2013; Zbl 1286.57019)] establishes that if a hyperbolic group which acts properly and cocompactly on a \(\mathsf{CAT}(0)\) cube complex is virtually special. The second main result, which provides a simultaneous generalization of Agol's result and Wise's quasiconvex hierarchy theorem, is Theorem D: Suppose that \(G\) is a hyperbolic group acting cocompactly on a \(\mathsf{CAT}(0)\) cube complex \(X\) with quasiconvex and virtually special cell stabilizers. Then \(G\) is virtually special. Theorem D (together with Theorem A) simplifies the proof of the virtual Haken and virtual fibering theorems for hyperbolic 3-manifolds.