Liftability of periodic mapping classes under alternating covers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912566431711232 |
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| author | Sanghi, Apeksha Rajeevsarathy, Kashyap Dey, Rajesh |
| author_facet | Sanghi, Apeksha Rajeevsarathy, Kashyap Dey, Rajesh |
| contents | Let $S_g$ be the closed orientable surface of genus $g \geq 2$, and let $\mathrm{Mod}(S_g)$ be the mapping class group of $S_g$. Let $A_n$ denote the alternating group on $n$ letters. We derive necessary and sufficient conditions under which a periodic mapping class has a conjugate that lifts under the branched cover $S_g \to S_g/A_n$ induced by an action of $A_n$ on $S_g$. This provides a classification of the subgroups of $\mathrm{Mod}(S_g)$ that are isomorphic to $A_n \rtimes \mathbb{Z}_m$, up to a certain equivalence that we call weak conjugacy. As an application, we show that for $n \geq 7$, such a subgroup of $\mathrm{Mod}(S_g)$ cannot have an irreducible periodic mapping class. Furthermore, we show that for $n \geq 5$ and $n \neq 6$, if the order of such a subgroup is greater than $5g-5$, then $m \leq 26$. Moreover, for $g \geq 2$ and $n \geq 5$, we establish that there exists no subgroup of $\mathrm{Mod}(S_g)$ that is isomorphic to $A_n \rtimes \mathbb{Z}$, where the $\mathbb{Z}$-component is generated by a power of a Dehn twist. Finally, we provide a complete classification of the weak conjugacy classes of such subgroups in $\mathrm{Mod}(S_{10})$ and $\mathrm{Mod}(S_{11})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_02114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liftability of periodic mapping classes under alternating covers Sanghi, Apeksha Rajeevsarathy, Kashyap Dey, Rajesh Geometric Topology Primary 57K20, Secondary 57M60 Let $S_g$ be the closed orientable surface of genus $g \geq 2$, and let $\mathrm{Mod}(S_g)$ be the mapping class group of $S_g$. Let $A_n$ denote the alternating group on $n$ letters. We derive necessary and sufficient conditions under which a periodic mapping class has a conjugate that lifts under the branched cover $S_g \to S_g/A_n$ induced by an action of $A_n$ on $S_g$. This provides a classification of the subgroups of $\mathrm{Mod}(S_g)$ that are isomorphic to $A_n \rtimes \mathbb{Z}_m$, up to a certain equivalence that we call weak conjugacy. As an application, we show that for $n \geq 7$, such a subgroup of $\mathrm{Mod}(S_g)$ cannot have an irreducible periodic mapping class. Furthermore, we show that for $n \geq 5$ and $n \neq 6$, if the order of such a subgroup is greater than $5g-5$, then $m \leq 26$. Moreover, for $g \geq 2$ and $n \geq 5$, we establish that there exists no subgroup of $\mathrm{Mod}(S_g)$ that is isomorphic to $A_n \rtimes \mathbb{Z}$, where the $\mathbb{Z}$-component is generated by a power of a Dehn twist. Finally, we provide a complete classification of the weak conjugacy classes of such subgroups in $\mathrm{Mod}(S_{10})$ and $\mathrm{Mod}(S_{11})$. |
| title | Liftability of periodic mapping classes under alternating covers |
| topic | Geometric Topology Primary 57K20, Secondary 57M60 |
| url | https://arxiv.org/abs/2509.02114 |