Finite subgroups of the profinite completion of good groups
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910671918071808 |
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| author | Boggi, Marco Zalesskii, Pavel |
| author_facet | Boggi, Marco Zalesskii, Pavel |
| contents | Let $G$ be a residually finite, good group of finite virtual cohomological dimension. We prove that the natural monomorphism $G\hookrightarrow\hat{G}$ induces a bijective correspondence between conjugacy classes of finite $p$-subgroups of $G$ and those of its profinite completion $\hat{G}$. Moreover, we prove that the centralizers and normalizers in $\hat{G}$ of finite $p$-subgroups of $G$ are the closures of the respective centralizers and normalizers in $G$. With somewhat more restrictive hypotheses, we prove the same results for finite solvable subgroups of $G$. In the last section, we give a few applications of this theorem to hyperelliptic mapping class groups and virtually compact special toral relatively hyperbolic groups (these include fundamental groups of $3$-orbifolds and of uniform standard arithmetic hyperbolic orbifolds). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08639 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite subgroups of the profinite completion of good groups Boggi, Marco Zalesskii, Pavel Group Theory Geometric Topology Let $G$ be a residually finite, good group of finite virtual cohomological dimension. We prove that the natural monomorphism $G\hookrightarrow\hat{G}$ induces a bijective correspondence between conjugacy classes of finite $p$-subgroups of $G$ and those of its profinite completion $\hat{G}$. Moreover, we prove that the centralizers and normalizers in $\hat{G}$ of finite $p$-subgroups of $G$ are the closures of the respective centralizers and normalizers in $G$. With somewhat more restrictive hypotheses, we prove the same results for finite solvable subgroups of $G$. In the last section, we give a few applications of this theorem to hyperelliptic mapping class groups and virtually compact special toral relatively hyperbolic groups (these include fundamental groups of $3$-orbifolds and of uniform standard arithmetic hyperbolic orbifolds). |
| title | Finite subgroups of the profinite completion of good groups |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2406.08639 |