Morse properties in convex projective geometry

TL;DR

This paper investigates hyperbolic directions in convex projective geometry by examining metric, boundary, and automorphism perspectives, generalizing known results beyond Gromov hyperbolic spaces.

Contribution

It introduces a unified framework connecting Morse and regular quasi-geodesics across multiple geometric contexts in convex projective domains.

Findings

Relationship between different Morse definitions established

Generalization of Benoist and Guichard's results

Insights into the structure of hyperbolic directions in convex domains

Abstract

We study properties of "hyperbolic directions" in groups acting cocompactly on properly convex domains in real projective space, from three different perspectives simultaneously: the (coarse) metric geometry of the Hilbert metric, the projective geometry of the boundary of the domain, and the singular value gaps of projective automorphisms. We describe the relationship between different definitions of "Morse" and "regular" quasi-geodesics arising in these three different contexts. This generalizes several results of Benoist and Guichard to the non-Gromov hyperbolic setting.