Dynamical properties of convex cocompact actions in projective space

TL;DR

This paper characterizes convex cocompact group actions in projective space through dynamical expansion properties and provides criteria for such actions in relatively hyperbolic groups, linking geometric and dynamical perspectives.

Contribution

It offers a new dynamical characterization of convex cocompactness in projective space and establishes conditions for relative convex cocompactness involving boundary homeomorphisms.

Findings

Convex cocompactness is equivalent to an expansion property in Grassmannians.

Provides a necessary and sufficient condition for convex cocompactness in relatively hyperbolic groups.

Shows the existence of an equivariant boundary homeomorphism characterizes relative convex cocompactness.

Abstract

We give a dynamical characterization of convex cocompact group actions on properly convex domains in projective space in the sense of Danciger-Gueritaud-Kassel: we show that convex cocompactness in $\mathbb{R} \mathrm{P}^d$ is equivalent to an expansion property of the group about its limit set, occuring in different Grassmannians. As an application, we give a sufficient and necessary condition for convex cocompactness for groups which are hyperbolic relative to a collection of convex cocompact subgroups. We show that convex cocompactness in this situation is equivalent to the existence of an equivariant homeomorphism from the Bowditch boundary to the quotient of the limit set of the group by the limit sets of its peripheral subgroups.