On Einstein Structures for $\mathrm{SO}_0(p,p+1)$-Surface Group Representations
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| Format: | Preprint |
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2025
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| _version_ | 1866917282491400192 |
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| author | Davalo, Colin Evans, Parker |
| author_facet | Davalo, Colin Evans, Parker |
| contents | Let $S$ be a closed surface of genus $g \geq 2$. We study the cocompact domain of discontinuity $Ω_ρ$ in the Einstein universe $\mathrm{Ein}^{p-1,p}$ defined by Guichard-Wienhard and Kapovich-Leeb-Porti for a class of $p$-Anosov representations $ρ:π_1S \rightarrow \mathrm{SO}_0(p,p+1)$ including Hitchin representations, for $p \geq 3$. The quotient $M_ρ = ρ(π_1S)\backslash Ω_ρ$ is abstractly known to be realizable as a fiber bundle over $S$, with unknown fiber of unique homotopy type $F_ρ$. We explicitly exhibit $M_ρ$ as a smooth $\mathfrak{F}_ρ$-fiber bundle over $S$, determining the diffeomorphism type of $M_ρ$ and the unique homotopy type $F_ρ$. Surprisingly, in many situations the fiber bundle $\mathfrak{F}_ρ \rightarrow M_ρ\rightarrow S$ is trivial. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_12779 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Einstein Structures for $\mathrm{SO}_0(p,p+1)$-Surface Group Representations Davalo, Colin Evans, Parker Geometric Topology Differential Geometry 57N16, 22E40 (Primary) Let $S$ be a closed surface of genus $g \geq 2$. We study the cocompact domain of discontinuity $Ω_ρ$ in the Einstein universe $\mathrm{Ein}^{p-1,p}$ defined by Guichard-Wienhard and Kapovich-Leeb-Porti for a class of $p$-Anosov representations $ρ:π_1S \rightarrow \mathrm{SO}_0(p,p+1)$ including Hitchin representations, for $p \geq 3$. The quotient $M_ρ = ρ(π_1S)\backslash Ω_ρ$ is abstractly known to be realizable as a fiber bundle over $S$, with unknown fiber of unique homotopy type $F_ρ$. We explicitly exhibit $M_ρ$ as a smooth $\mathfrak{F}_ρ$-fiber bundle over $S$, determining the diffeomorphism type of $M_ρ$ and the unique homotopy type $F_ρ$. Surprisingly, in many situations the fiber bundle $\mathfrak{F}_ρ \rightarrow M_ρ\rightarrow S$ is trivial. |
| title | On Einstein Structures for $\mathrm{SO}_0(p,p+1)$-Surface Group Representations |
| topic | Geometric Topology Differential Geometry 57N16, 22E40 (Primary) |
| url | https://arxiv.org/abs/2510.12779 |