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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2210.09700 |
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| _version_ | 1866910319213805568 |
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| author | Song, Jijian Xu, Bin Ye, Yu |
| author_facet | Song, Jijian Xu, Bin Ye, Yu |
| contents | We are motivated by cone spherical metrics on compact Riemann surfaces of positive genus to solve a special case of the Hurwitz problem. Precisely speaking, letting $d,\,g$ and $\ell$ be three positive integers and $Λ$ be the following collection of $(\ell+2)$ partitions of a positive integer $d$: \[(a_1,\cdots, a_p),\,(b_1,\cdots, b_q),\,(m_1+1,1,\cdots,1),\cdots, (m_{\ell}+1,1,\cdots,1),\] where $(m_1,\cdots, m_{\ell})$ is a partition of $p+q-2+2g$, we prove that there exists a branched cover from some compact Riemann surface of genus $g$ to the Riemann sphere ${\Bbb P}^1$ with branch data $Λ$. An analogue for the genus-zero case was found by the first two authors ({\it Algebra Colloq.} {\bf 27} (2020), no. 2, 231-246), who were stimulated by such metrics on ${\Bbb P}^1$ and conjectured the veracity of the above statement there. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_09700 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A note on the Hurwitz problem and cone spherical metrics Song, Jijian Xu, Bin Ye, Yu Group Theory Geometric Topology 20B35, 30F30 We are motivated by cone spherical metrics on compact Riemann surfaces of positive genus to solve a special case of the Hurwitz problem. Precisely speaking, letting $d,\,g$ and $\ell$ be three positive integers and $Λ$ be the following collection of $(\ell+2)$ partitions of a positive integer $d$: \[(a_1,\cdots, a_p),\,(b_1,\cdots, b_q),\,(m_1+1,1,\cdots,1),\cdots, (m_{\ell}+1,1,\cdots,1),\] where $(m_1,\cdots, m_{\ell})$ is a partition of $p+q-2+2g$, we prove that there exists a branched cover from some compact Riemann surface of genus $g$ to the Riemann sphere ${\Bbb P}^1$ with branch data $Λ$. An analogue for the genus-zero case was found by the first two authors ({\it Algebra Colloq.} {\bf 27} (2020), no. 2, 231-246), who were stimulated by such metrics on ${\Bbb P}^1$ and conjectured the veracity of the above statement there. |
| title | A note on the Hurwitz problem and cone spherical metrics |
| topic | Group Theory Geometric Topology 20B35, 30F30 |
| url | https://arxiv.org/abs/2210.09700 |