Volume, entropy, and diameter in ${\rm SO}(p,q+1)$-higher Teichmüller spaces
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916137558605824 |
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| author | Mazzoli, Filippo Viaggi, Gabriele |
| author_facet | Mazzoli, Filippo Viaggi, Gabriele |
| contents | We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations $ρ: Γ\to \mathrm{SO}(p,q+1)$ 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 $Γ$. 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 ${\rm S}({\rm O}(p,1)\times{\rm O}(q))$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_17137 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Volume, entropy, and diameter in ${\rm SO}(p,q+1)$-higher Teichmüller spaces Mazzoli, Filippo Viaggi, Gabriele Differential Geometry Geometric Topology 53C50, 57N16, 22E40 We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations $ρ: Γ\to \mathrm{SO}(p,q+1)$ 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 $Γ$. 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 ${\rm S}({\rm O}(p,1)\times{\rm O}(q))$, 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. |
| title | Volume, entropy, and diameter in ${\rm SO}(p,q+1)$-higher Teichmüller spaces |
| topic | Differential Geometry Geometric Topology 53C50, 57N16, 22E40 |
| url | https://arxiv.org/abs/2312.17137 |