  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.
