Convex cocompactness in pseudo-Riemannian hyperbolic spaces

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 $$\mathrm {PO}(p,q)$$ by considering their action on the associated pseudo-Riemannian hyperbolic space $$\mathbb {H}^{p,q-1}$$ in place of the Riemannian symmetric space. Following work of Barbot and Merigot in anti-de Sitter geometry, we find an intimate connection between Anosov representations and a natural notion of convex cocompactness in this setting.