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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2201.04765 |
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| _version_ | 1866914821046272000 |
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| author | Ma, Jiming Xie, Baohua |
| author_facet | Ma, Jiming Xie, Baohua |
| contents | Let $$G_{6,3}=\langle a_0, \cdots, a_5| a_{i}^{3}=id, a_{i} a_{i+1}= a_{i+1} a_{i}, i \in \mathbb{Z}/6\mathbb{Z}\rangle$$ be a hyperbolic group with boundary the Menger curve. J. Granier \cite{Granier} constructed a discrete, convex cocompact and faithful representation $ρ$ of $G_{6,3}$ into $\mathbf{PU}(2,1)$. We show the 3-orbifold at infinity of $ρ(G_{6,3})$ is a closed hyperbolic 3-orbifold, with underlying space the 3-sphere and singular locus the $\mathbb{Z}_3$-coned chain-link $C(6,-2)$. This answers the second part of Misha Kapovich's Conjecture 10.6\cite{Kapovich}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_04765 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Menger curve and Spherical CR uniformization of a closed hyperbolic 3-orbifold Ma, Jiming Xie, Baohua Geometric Topology Let $$G_{6,3}=\langle a_0, \cdots, a_5| a_{i}^{3}=id, a_{i} a_{i+1}= a_{i+1} a_{i}, i \in \mathbb{Z}/6\mathbb{Z}\rangle$$ be a hyperbolic group with boundary the Menger curve. J. Granier \cite{Granier} constructed a discrete, convex cocompact and faithful representation $ρ$ of $G_{6,3}$ into $\mathbf{PU}(2,1)$. We show the 3-orbifold at infinity of $ρ(G_{6,3})$ is a closed hyperbolic 3-orbifold, with underlying space the 3-sphere and singular locus the $\mathbb{Z}_3$-coned chain-link $C(6,-2)$. This answers the second part of Misha Kapovich's Conjecture 10.6\cite{Kapovich}. |
| title | Menger curve and Spherical CR uniformization of a closed hyperbolic 3-orbifold |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2201.04765 |