Volume, entropy, and diameter in $\mathrm{SO}(p,q+1)$-higher Teichmüller spaces

Abstract

We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations $\rho :\Gamma \to \mathrm{SO}(p,q+1)$ of closed $p$-manifold groups. In particular: We provide a uniform lower bound of the product entropy times volume that depends only on the geometry of the abstract group $\Gamma$. We prove that the entropy is bounded from above by $p-1$ with equality if and only if $\rho$ is conjugate to a representation inside $\mathrm{S}(\mathrm{O}(p,1)\times \mathrm{O}(q))$, which answers affirmatively to a question of Glorieux and Monclair. Lastly, we prove finiteness and compactness results for groups admitting convex cocompact representations with bounded diameter.

arXiv comment29 pages. Comments are welcome! Main changes from v1: we improved the statement of Lemma 4.2; we rewrote the proof of Lemma 4.3 (we thank in particular Timothé Lemistre for sharing with us his strategy of proof); we updated the acknowledgments