On closure operations in the space of subgroups and applications
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913886236573696 |
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| author | Francoeur, Dominik Boudec, Adrien Le |
| author_facet | Francoeur, Dominik Boudec, Adrien Le |
| contents | We establish some interactions between uniformly recurrent subgroups (URSs) of a group $G$ and cosets topologies $τ_\mathcal{N}$ on $G$ associated to a family $\mathcal{N}$ of normal subgroups of $G$. We show that when $\mathcal{N}$ consists of finite index subgroups of $G$, there is a natural closure operation $\mathcal{H} \mapsto \mathrm{cl}_\mathcal{N}(\mathcal{H})$ that associates to a URS $\mathcal{H}$ another URS $\mathrm{cl}_\mathcal{N}(\mathcal{H})$, called the $τ_\mathcal{N}$-closure of $\mathcal{H}$. We give a characterization of the URSs $\mathcal{H}$ that are $τ_\mathcal{N}$-closed in terms of stabilizer URSs. This has consequences on arbitrary URSs when $G$ belongs to the class of groups for which every faithful minimal profinite action is topologically free. We also consider the largest amenable URS $\mathcal{A}_G$, and prove that for certain coset topologies on $G$, almost all subgroups $H \in \mathcal{A}_G$ have the same closure. For groups in which amenability is detected by a set of laws (a property that is variant of the Tits alternative), we deduce a criterion for $\mathcal{A}_G$ to be a singleton based on residual properties of $G$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_10222 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On closure operations in the space of subgroups and applications Francoeur, Dominik Boudec, Adrien Le Group Theory Operator Algebras We establish some interactions between uniformly recurrent subgroups (URSs) of a group $G$ and cosets topologies $τ_\mathcal{N}$ on $G$ associated to a family $\mathcal{N}$ of normal subgroups of $G$. We show that when $\mathcal{N}$ consists of finite index subgroups of $G$, there is a natural closure operation $\mathcal{H} \mapsto \mathrm{cl}_\mathcal{N}(\mathcal{H})$ that associates to a URS $\mathcal{H}$ another URS $\mathrm{cl}_\mathcal{N}(\mathcal{H})$, called the $τ_\mathcal{N}$-closure of $\mathcal{H}$. We give a characterization of the URSs $\mathcal{H}$ that are $τ_\mathcal{N}$-closed in terms of stabilizer URSs. This has consequences on arbitrary URSs when $G$ belongs to the class of groups for which every faithful minimal profinite action is topologically free. We also consider the largest amenable URS $\mathcal{A}_G$, and prove that for certain coset topologies on $G$, almost all subgroups $H \in \mathcal{A}_G$ have the same closure. For groups in which amenability is detected by a set of laws (a property that is variant of the Tits alternative), we deduce a criterion for $\mathcal{A}_G$ to be a singleton based on residual properties of $G$. |
| title | On closure operations in the space of subgroups and applications |
| topic | Group Theory Operator Algebras |
| url | https://arxiv.org/abs/2407.10222 |