Volume, entropy, and diameter in SO(p,q+1)-higher Teichmüller spaces.

The notion of $\mathbb{H}^{p,q}$-convex cocompact representations was introduced by Danciger, Guéritaud, and Kassel and provides a unifying framework for several interesting classes of discrete subgroups of the orthogonal groups SO(p,q+1), such as convex cocompact hyperbolic manifolds and maximal globally hyperbolic anti-de Sitter spacetimes of negative Euler characteristic. By recent works of Seppi-Smith-Toulisse and Beyrer-Kassel, we now know that any such representation admits a unique invariant maximal spacelike p-dimensional manifold inside the pseudo-Riemannian hyperbolic space $\mathbb{H}^{p,q}$, and that the space of $\mathbb{H}^{p,q}$-convex cocompact representations of a group $\Gamma$ consists of a union of connected components of the associated SO(p,q+1)-character variety. In this talk, I will describe a recent joint work with Gabriele Viaggi in which we provide various applications for the existence of invariant maximal spacelike submanifolds. These include a rigidity result for the pseudo-Riemannian critical exponent (which answers affirmatively to a question of Glorieux-Monclair), a comparison between entropy and volume, and several compactness and finiteness criteria in this framework.