Oligomorphic groups, their automorphism groups, and the complexity of their isomorphism
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913746533744640 |
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| author | Paolini, Gianluca Nies, Andre |
| author_facet | Paolini, Gianluca Nies, Andre |
| contents | The paper follows two interconnected directions.
1. Let $G$ be a Roelcke precompact closed subgroup of the group $\Sym(ω)$ of permutations of the natural numbers. Then $\Inn(G)$ is closed in $\Aut(G)$, where $\Aut(G)$ carries the topology of pointwise convergence for its (faithful) action on the cosets of open subgroups. Under the stronger hypothesis that~$G$ is oligomorphic, $N_G/G$ is profinite, where $N_G$ denotes the normaliser of~$G$ in $\Sym(ω)$, and the topological group $\Out(G)= \Aut(G)/\Inn(G)$ is totally disconnected, locally compact.
2a. We provide a general method to show smoothness of the isomorphism relation for appropriate Borel classes of oligomorphic groups. We apply it to two such classes: the oligomorphic groups with no algebraicity, and the oligomorphic groups with finitely many {essential} subgroups up to conjugacy.
2b. Using this method we also show that if $G$ is in such a Borel class, then $\Aut(G)$ is topologically isomorphic to an oligomorphic group, and $\Out(G)$ is profinite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_02248 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Oligomorphic groups, their automorphism groups, and the complexity of their isomorphism Paolini, Gianluca Nies, Andre Logic Group Theory The paper follows two interconnected directions. 1. Let $G$ be a Roelcke precompact closed subgroup of the group $\Sym(ω)$ of permutations of the natural numbers. Then $\Inn(G)$ is closed in $\Aut(G)$, where $\Aut(G)$ carries the topology of pointwise convergence for its (faithful) action on the cosets of open subgroups. Under the stronger hypothesis that~$G$ is oligomorphic, $N_G/G$ is profinite, where $N_G$ denotes the normaliser of~$G$ in $\Sym(ω)$, and the topological group $\Out(G)= \Aut(G)/\Inn(G)$ is totally disconnected, locally compact. 2a. We provide a general method to show smoothness of the isomorphism relation for appropriate Borel classes of oligomorphic groups. We apply it to two such classes: the oligomorphic groups with no algebraicity, and the oligomorphic groups with finitely many {essential} subgroups up to conjugacy. 2b. Using this method we also show that if $G$ is in such a Borel class, then $\Aut(G)$ is topologically isomorphic to an oligomorphic group, and $\Out(G)$ is profinite. |
| title | Oligomorphic groups, their automorphism groups, and the complexity of their isomorphism |
| topic | Logic Group Theory |
| url | https://arxiv.org/abs/2410.02248 |