# Convex cocompactness in pseudo-Riemannian hyperbolic spaces

## TL;DR

This paper explores the concept of convex cocompactness for representations of hyperbolic groups into pseudo-Riemannian hyperbolic spaces, establishing a link with Anosov representations in higher-rank Lie groups.

## Contribution

It introduces a new perspective on convex cocompactness in pseudo-Riemannian hyperbolic spaces and connects it with Anosov representations, extending classical notions from Riemannian to pseudo-Riemannian geometry.

## Key findings

- Established a connection between Anosov representations and convex cocompactness in pseudo-Riemannian hyperbolic spaces.
- Demonstrated that certain representations act properly and cocompactly on convex sets in these spaces.
- Extended the analogy between hyperbolic geometry and higher-rank Lie group representations.

## Abstract

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups PO(p,q) by considering their action on the associated pseudo-Riemannian hyperbolic space H^{p,q-1} in place of the Riemannian symmetric space. Following work of Barbot and M\'erigot in anti-de Sitter geometry, we find an intimate connection between Anosov representations and the natural notion of convex cocompactness in this setting.