Convex co-compact actions of relatively hyperbolic groups

TL;DR

This paper characterizes when discrete groups acting convex co-compactly on convex domains in real projective space are relatively hyperbolic, linking geometric properties of the domain to algebraic hyperbolicity.

Contribution

It provides necessary and sufficient geometric conditions for relative hyperbolicity of such groups, answering a previously open question.

Findings

Established criteria for relative hyperbolicity based on convex domain geometry

Connected convex projective actions to known hyperbolic group properties

Extended analogous results from CAT(0) spaces to projective geometry

Abstract

In this paper we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish necessary and sufficient conditions for the group to be relatively hyperbolic in terms of the geometry of the convex domain. This answers a question of Danciger-Gu\'eritaud-Kassel and is analogous to a result of Hruska-Kleiner for ${\rm CAT}(0)$ spaces.