The 2-rank of finite groups acting on hyperelliptic 3-manifolds
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909114258423808 |
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| author | Sbarra, Max Leopold Frisch Mecchia, Mattia |
| author_facet | Sbarra, Max Leopold Frisch Mecchia, Mattia |
| contents | We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $S^3.$ Such involutions are called \textit{hyperelliptic} as the manifolds admitting such an action. We prove that the sectional 2-rank of a finite group acting on a 3-manifold and containing a hyperelliptic involution with fixed-point set with two components has sectional 2-rank at most four; this upper bound is sharp. The cases where the hyperelliptic involution has a fixed-point set with a number of components different from 2 have been already considered in literature. Our result completes the analysis and we obtain some general results where the number of the components of the fixed-point set is not fixed. In particular, we obtain that a finite group acting on a 3-manifold and containing a hyperelliptic involution has 2-rank at most four, and four is the best possible upper bound.
Finally, we restrict to the basic case of simple groups acting on hyperelliptic 3-manifolds: we use our result about the sectional 2-rank to prove that a simple group containing a hyperelliptic involution is isomorphic to $PSL(2,q)$ for some odd prime power $q$, or to one of four other small simple groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_13251 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The 2-rank of finite groups acting on hyperelliptic 3-manifolds Sbarra, Max Leopold Frisch Mecchia, Mattia Geometric Topology We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $S^3.$ Such involutions are called \textit{hyperelliptic} as the manifolds admitting such an action. We prove that the sectional 2-rank of a finite group acting on a 3-manifold and containing a hyperelliptic involution with fixed-point set with two components has sectional 2-rank at most four; this upper bound is sharp. The cases where the hyperelliptic involution has a fixed-point set with a number of components different from 2 have been already considered in literature. Our result completes the analysis and we obtain some general results where the number of the components of the fixed-point set is not fixed. In particular, we obtain that a finite group acting on a 3-manifold and containing a hyperelliptic involution has 2-rank at most four, and four is the best possible upper bound. Finally, we restrict to the basic case of simple groups acting on hyperelliptic 3-manifolds: we use our result about the sectional 2-rank to prove that a simple group containing a hyperelliptic involution is isomorphic to $PSL(2,q)$ for some odd prime power $q$, or to one of four other small simple groups. |
| title | The 2-rank of finite groups acting on hyperelliptic 3-manifolds |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2312.13251 |