Volume, Entropy, and Diameter in SO(p,q+1)-Higher Teichmüller Spaces

par Filippo Mazzoli (MPIM Leipzig)

lundi 18 mars 2024, 14:00 → 15:15 Europe/Paris

Amphithéâtre Léon Motchane (IHES)

The notion of 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 holonomies of convex cocompact hyperbolic manifolds or 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 H^p,q-convex cocompact representation of a group Γ of cohomological dimension p admits a unique invariant maximal spacelike p-dimensional manifold inside the pseudo-Riemannian hyperbolic space H^p,q, and that the space of H^p,q-convex cocompact representations of Γ forms a union of connected components in the associated SO(p,q+1)-character variety.

In this talk, I will describe some 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 a question of Glorieux-Monclair), a comparison between entropy and volume, and several compactness and finiteness criteria in this framework.