Relatively geometric actions on CAT(0) cube complexes

TL;DR

This paper develops the theory of relatively geometric actions of relatively hyperbolic groups on CAT(0) cube complexes, establishing key properties and analogs of important theorems in geometric group theory.

Contribution

It introduces and proves foundational results for relatively geometric actions, including convexity of quasi-convex subgroups and analogs of major theorems.

Findings

Full relatively quasi-convex subgroups are convex compact

An analog of Agol's Theorem is established

A version of Haglund--Wise's Canonical Completion and Retraction is proved

Abstract

We develop the foundations of the theory of relatively geometric actions of relatively hyperbolic groups on CAT(0) cube complexes, a notion introduced in our previous work [5]. In the relatively geometric setting we prove: full relatively quasi-convex subgroups are convex compact; an analog of Agol's Theorem; and a version of Haglund--Wise's Canonical Completion and Retraction.