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

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