Hyperbolic groups acting improperly

Abstract

In this paper we study hyperbolic groups acting on CAT(0) cube complexes. The first main result (Theorem A) is a structural result about the Sageev construction, in which we relate quasi-convexity of hyperplane stabilizers with quasi-convexity of cell stabilizers. The second main result (Theorem D) generalizes both Agol's theorem on cubulated hyperbolic groups and Wise's Quasi-convex Hierarchy Theorem.Comment: 52pp. In v3, some unnecessary assumptions are dropped from some technical results, especially in Section 5 and Corollary 6.5. The main results are unchanged, but the improved technical results are expected to be useful in future work. Several other small improvements to the exposition have been mad