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

We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations p: Gamma -> 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 I. We prove that the entropy is bounded from above by p-1 with equality if and only if p is conjugate to a representation inside S(O(p, 1) xO(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.