子群空间中的闭合运算及其应用

Dominik Francoeur, Adrien Le Boudec
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引用次数: 0

摘要

我们在一个群 $G$ 的均匀重复子群(URSs)与 $G$ 上与 $G$ 的正常子群的一个族 $\mathcal{N}$ 相关的 cosets 拓扑 $\tau_\mathcal{N}$ 之间建立了一些相互作用。我们证明,当$\mathcal{N}$由$G$的有限索引子群组成时,存在一个自然封闭操作$\mathcal{H}。\mapsto(映射到)$mathrm{cl}_\mathcal{N}(\mathcal{H})$,它与一个 URS $\mathcal{H}$ 关联到另一个 URS$\mathrm{cl}_\mathcal{N}(\mathcal{H})$,称为 $\tau_\mathcal{N}$-closureof $\mathcal{H}$ 。我们从稳定器 URS 的角度给出了$\mathcal{H}$封闭的 URS 的特征。当 $G$ 属于每一个忠实最小无限作用都是拓扑自由的群时,这对任意 URS 有影响。我们还考虑了最大可驯化 URS $\mathcal{A}_G$,并证明对于 $G$ 上的某些余集拓扑,几乎所有在 \mathcal{A}_G$ 中的子群 $H \ 都有相同的闭合。我们根据 $G$ 的残差性质,推导出了 $\mathcal{A}_G$ 是单子的标准。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On closure operations in the space of subgroups and applications
We establish some interactions between uniformly recurrent subgroups (URSs) of a group $G$ and cosets topologies $\tau_\mathcal{N}$ 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 $\tau_\mathcal{N}$-closure of $\mathcal{H}$. We give a characterization of the URSs $\mathcal{H}$ that are $\tau_\mathcal{N}$-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, we deduce a criterion for $\mathcal{A}_G$ to be a singleton based on residual properties of $G$.
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