Hyperbolic groups acting improperly

TL;DR

This paper investigates hyperbolic groups acting on CAT(0) cube complexes, establishing new structural insights and generalizations of key theorems related to quasi-convexity and group actions.

Contribution

It provides a structural analysis of the Sageev construction and generalizes important theorems on cubulated hyperbolic groups and quasi-convex hierarchies.

Findings

Relation between quasi-convexity of hyperplane and cell stabilizers

Generalization of Agol's theorem on cubulated hyperbolic groups

Extension of Wise's Quasi-convex Hierarchy Theorem

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.