Anosov representations, strongly convex cocompact groups and weak eigenvalue gaps

TL;DR

This paper characterizes Anosov representations of hyperbolic groups into real semisimple Lie groups using boundary maps, eigenvalue gap properties, and applies these results to strongly convex cocompact groups in projective linear groups.

Contribution

It introduces new characterizations of Anosov representations via boundary maps and eigenvalue gaps, and relates these to strongly convex cocompact subgroups.

Findings

Characterization of Anosov representations through boundary maps and gap properties

Conditions for groups with weak eigenvalue gaps to be Anosov

A new criterion for strongly convex cocompact subgroups in projective linear groups

Abstract

We provide characterizations of Anosov representations of word hyperbolic groups into real semisimple Lie groups in terms of the existence of equivariant limit maps on the Gromov boundary, the Cartan property and the uniform gap summation property introduced by Guichard-Gu\'eritaud-Kassel-Wienhard. We also study representations of finitely generated groups satisfying weak uniform gaps in eigenvalues and establish conditions to be Anosov. As an application, we also obtain a characterization of strongly convex cocompact subgroups of the projective linear group $\mathsf{PGL}_d(\mathbb{R})$.