Menger curve and Spherical CR uniformization of a closed hyperbolic 3-orbifold
| Autor: | Ma, Jiming, Xie, Baohua |
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| Rok vydání: | 2022 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $\rho$ of $G_{6,3}$ into $\mathbf{PU}(2,1)$. We show the 3-orbifold at infinity of $\rho(G_{6,3})$ 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}. Comment: 28 pages. arXiv admin note: text overlap with arXiv:1401.0308 by other authors |
| Databáze: | arXiv |
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