Liftable mapping class groups of certain branched covers of torus
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909812271349760 |
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| author | Kapari, Pankaj |
| author_facet | Kapari, Pankaj |
| contents | Let $S_{g,n}$ be a closed oriented hyperbolic surface of genus $g$ with $n$ marked points, with the understanding that $S_{g,0}=S_g$. Let $\mathrm{Mod}(S_{h,n})$ be the mapping class group of $S_{h,n}$ and $\mathrm{LMod}_p(S_{h,n})$ be the liftable mapping class group associated to a cover $p:S_g\to S_{h,n}$. For the cover $p_k:S_k\to S_{1,2}$, Ghaswala, in his PhD thesis, derived a finite presentation for $\mathrm{LMod}_{p_k}(S_{1,2})$ when $k=2,3,4$ and a finite generating set when $k=5,6$ using the Reidemeister-Schreier rewriting process. In this paper, we derive a finite generating set for $\mathrm{LMod}_{p_k}(S_{1,2})$ for all $k\geq 2$. In the process, we also prove that the kernel of the homology representation $Ψ:\mathrm{Mod}(S_{1,2})\to \mathrm{GL}_3(\Z)$ is normally generated by a Dehn twist about a separating simple closed curve, and it is free with a countable basis. We also provide an explicit countable basis for $\kerΨ$ consisting of separating Dehn twists. As an application of Birman-Hilden theory, we provide a finite generating set for the normalizer of the Deck group of $p_k$ in $\mathrm{Mod}(S_k)$ when $k=2,3$. We conclude the paper by proving that $\mathrm{LMod}_{p_k}(S_{1,2})$ is maximal in $\mathrm{Mod}(S_{1,2})$ if and only if $k$ is prime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11788 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liftable mapping class groups of certain branched covers of torus Kapari, Pankaj Geometric Topology 57K20, 57M60 Let $S_{g,n}$ be a closed oriented hyperbolic surface of genus $g$ with $n$ marked points, with the understanding that $S_{g,0}=S_g$. Let $\mathrm{Mod}(S_{h,n})$ be the mapping class group of $S_{h,n}$ and $\mathrm{LMod}_p(S_{h,n})$ be the liftable mapping class group associated to a cover $p:S_g\to S_{h,n}$. For the cover $p_k:S_k\to S_{1,2}$, Ghaswala, in his PhD thesis, derived a finite presentation for $\mathrm{LMod}_{p_k}(S_{1,2})$ when $k=2,3,4$ and a finite generating set when $k=5,6$ using the Reidemeister-Schreier rewriting process. In this paper, we derive a finite generating set for $\mathrm{LMod}_{p_k}(S_{1,2})$ for all $k\geq 2$. In the process, we also prove that the kernel of the homology representation $Ψ:\mathrm{Mod}(S_{1,2})\to \mathrm{GL}_3(\Z)$ is normally generated by a Dehn twist about a separating simple closed curve, and it is free with a countable basis. We also provide an explicit countable basis for $\kerΨ$ consisting of separating Dehn twists. As an application of Birman-Hilden theory, we provide a finite generating set for the normalizer of the Deck group of $p_k$ in $\mathrm{Mod}(S_k)$ when $k=2,3$. We conclude the paper by proving that $\mathrm{LMod}_{p_k}(S_{1,2})$ is maximal in $\mathrm{Mod}(S_{1,2})$ if and only if $k$ is prime. |
| title | Liftable mapping class groups of certain branched covers of torus |
| topic | Geometric Topology 57K20, 57M60 |
| url | https://arxiv.org/abs/2509.11788 |