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

We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations $\rho :\mathrm{\Gamma }\to \mathrm{S}\mathrm{O}(p,q+1)$ 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 $\mathrm{\Gamma }$. 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.