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

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...

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Bibliographic Details
Published in:arXiv.org 2024-02
Main Authors: Mazzoli, Filippo, Viaggi, Gabriele
Format: Article
Language:English
Subjects:
Entropy
Lower bounds
Representations
Online Access:Get full text
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Summary: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.
ISSN:2331-8422