Liftable mapping class groups of regular abelian covers
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913604565991424 |
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| author | Dhanwani, Neeraj K. Kapari, Pankaj Rajeevsarathy, Kashyap Tomar, Ravi |
| author_facet | Dhanwani, Neeraj K. Kapari, Pankaj Rajeevsarathy, Kashyap Tomar, Ravi |
| contents | Let $S_g$ be the closed oriented surface of genus $g \geq 0$, and let $\mathrm{Mod}(S_g)$ be the mapping class group of $S_g$. For $g\geq 2$, we develop an algorithm to obtain a finite generating set for the liftable mapping class group $\mathrm{LMod}_p(S_g)$ of a regular abelian cover $p$ of $S_g$. A key ingredient of our method is a result that provides a generating set of a group $G$ acting on a connected graph $X$ such that the quotient graph $X/G$ is finite. As an application of our algorithm, when $k$ is prime, we provide a finite generating set for $\mathrm{LMod}_{p_k}(S_2)$ for cyclic cover $p_k:S_{k+1}\to S_2$. Using the Birman-Hilden theory, when $k=2,3$ and $g=2$, we also obtain a finite generating set for the normalizer of the Deck transformation group of $p_k$ in $\mathrm{Mod}(S_{k+1})$. We conclude the paper with an application of our algorithm that gives a finite generating set for $\mathrm{LMod}_p(S_2)$, where $p:S_5\to S_2$ is a cover with deck transformation group isomorphic to $\mathbb{Z}_2\oplus \mathbb{Z}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07319 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Liftable mapping class groups of regular abelian covers Dhanwani, Neeraj K. Kapari, Pankaj Rajeevsarathy, Kashyap Tomar, Ravi Geometric Topology 57K20, 57M60 Let $S_g$ be the closed oriented surface of genus $g \geq 0$, and let $\mathrm{Mod}(S_g)$ be the mapping class group of $S_g$. For $g\geq 2$, we develop an algorithm to obtain a finite generating set for the liftable mapping class group $\mathrm{LMod}_p(S_g)$ of a regular abelian cover $p$ of $S_g$. A key ingredient of our method is a result that provides a generating set of a group $G$ acting on a connected graph $X$ such that the quotient graph $X/G$ is finite. As an application of our algorithm, when $k$ is prime, we provide a finite generating set for $\mathrm{LMod}_{p_k}(S_2)$ for cyclic cover $p_k:S_{k+1}\to S_2$. Using the Birman-Hilden theory, when $k=2,3$ and $g=2$, we also obtain a finite generating set for the normalizer of the Deck transformation group of $p_k$ in $\mathrm{Mod}(S_{k+1})$. We conclude the paper with an application of our algorithm that gives a finite generating set for $\mathrm{LMod}_p(S_2)$, where $p:S_5\to S_2$ is a cover with deck transformation group isomorphic to $\mathbb{Z}_2\oplus \mathbb{Z}_2$. |
| title | Liftable mapping class groups of regular abelian covers |
| topic | Geometric Topology 57K20, 57M60 |
| url | https://arxiv.org/abs/2412.07319 |