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arXiv · 2312.17137

Volume, entropy, and diameter in ${\rm SO}(p,q+1)$-higher Teichm\"uller spaces

Filippo Mazzoli·Gabriele Viaggi
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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 ${\rm S}({\rm O}(p,1)\times{\rm O}(q))$, 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.

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Filippo Mazzoli, Gabriele Viaggi. 2023-12-28. Volume, entropy, and diameter in ${\rm SO}(p,q+1)$-higher Teichm\"uller spaces. https://arxiv.org/abs/2312.17137

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