# Projective Anosov representations, convex cocompact actions, and rigidity

## TL;DR

This paper explores the actions of projective Anosov representations on convex domains, establishing conditions for convex cocompactness, and presents applications including a new characterization of Anosov representations and rigidity results related to boundary maps.

## Contribution

It demonstrates that many projective Anosov representations act convex cocompactly on convex domains and introduces new rigidity results involving entropy and boundary map regularity.

## Key findings

- Many projective Anosov representations act convex cocompactly on convex domains.
- A new characterization of Anosov representations via convex cocompact actions.
- Rigidity results show boundary maps are rarely smooth submanifolds.

## Abstract

In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any projective Anosov representation of that group acts convex cocompactly on some properly convex domain in real projective space. We also show that if a projective Anosov representation preserves a properly convex domain, then it acts convex cocompactly on some (possibly different) properly convex domain. We then give three applications. First, we show that Anosov representations into general semisimple Lie groups can be defined in terms of the existence of a convex cocompact action on a properly convex domain in some real projective space (which depends on the semisimple Lie group and parabolic subgroup). Next, we prove a rigidity result involving the Hilbert entropy of a projective Anosov representation. Finally, we prove a rigidity result which shows that the image of the boundary map associated to a projective Anosov representation is rarely a $C^2$ submanifold of projective space. This final rigidity result also applies to Hitchin representations.