Separation and Relative Quasi-convexity Criteria for Relatively Geometric Actions

TL;DR

This paper extends Bowditch's characterization of relative hyperbolicity to generalized fine actions on hyperbolic graphs, providing new criteria for relative quasiconvexity and insights into boundary separation in relatively geometric actions.

Contribution

It introduces generalized fine actions allowing peripheral subgroups to stabilize finite sub-graphs and establishes a new criterion for relative quasiconvexity in this setting.

Findings

Generalized fine actions include actions on CAT(0) cube complexes.

A subgroup cocompactly stabilizing a quasi-convex sub-graph is relatively quasiconvex.

Bowditch boundary points are separated by hyperplane stabilizers in these actions.

Abstract

Bowditch characterized relative hyperbolicity in terms of group actions on fine hyperbolic graphs with finitely many edge orbits and finite edge stabilizers. In this paper, we define generalized fine actions on hyperbolic graphs, in which the peripheral subgroups are allowed to stabilize finite sub-graphs rather than stabilizing a point. Generalized fine actions are useful for studying groups that act relatively geometrically on a CAT(0) cube complex, which were recently defined by the first two authors. Specifically, we show that a group acting relatively geometrically on a CAT(0) cube complex admits a generalized fine action on the one-skeleton of the cube complex. For generalized fine actions, we prove a criterion for relative quasiconvexity as subgroups that cocompactly stabilize quasi-convex sub-graphs, generalizing a result of Martinez-Pedroza and Wise in the setting of fine hyperbolic graphs. As an application, we obtain a characterization of boundary separation in generalized fine graphs and use it to prove that Bowditch boundary points in relatively geometric actions are always separated by a hyperplane stabilizer.