On compact uniformly recurrent subgroups
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913344293699584 |
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| author | Caprace, Pierre-Emmanuel Goffer, Gil Lederle, Waltraud Tsankov, Todor |
| author_facet | Caprace, Pierre-Emmanuel Goffer, Gil Lederle, Waltraud Tsankov, Todor |
| contents | Let a group $Γ$ act on a paracompact, locally compact, Hausdorff space $M$ by homeomorphisms and let $2^M$ denote the set of closed subsets of $M$. We endow $2^M$ with the Chabauty topology, which is compact and admits a natural $Γ$-action by homeomorphisms. We show that for every minimal $Γ$-invariant closed subset $\mathcal Y$ of $2^M$ consisting of compact sets, the union $\bigcup \mathcal{Y}\subset M$ has compact closure.
As an application, we deduce that every compact uniformly recurrent subgroup of a locally compact group is contained in a compact normal subgroup. This generalizes a result of Ušakov on compact subgroups whose normalizer is compact. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_16297 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On compact uniformly recurrent subgroups Caprace, Pierre-Emmanuel Goffer, Gil Lederle, Waltraud Tsankov, Todor Group Theory Let a group $Γ$ act on a paracompact, locally compact, Hausdorff space $M$ by homeomorphisms and let $2^M$ denote the set of closed subsets of $M$. We endow $2^M$ with the Chabauty topology, which is compact and admits a natural $Γ$-action by homeomorphisms. We show that for every minimal $Γ$-invariant closed subset $\mathcal Y$ of $2^M$ consisting of compact sets, the union $\bigcup \mathcal{Y}\subset M$ has compact closure. As an application, we deduce that every compact uniformly recurrent subgroup of a locally compact group is contained in a compact normal subgroup. This generalizes a result of Ušakov on compact subgroups whose normalizer is compact. |
| title | On compact uniformly recurrent subgroups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2210.16297 |