Given a group $G$ of homeomorphism of a first-countable Hausdorff space $\mathcal{X}$, we prove that if the action of $G$ on $\mathcal{X}$ is minimal and has rigid stabilisers that act locally minimally, then the neighbourhood stabilisers of any two points in $\mathcal{X}$ are conjugated by a homeomorphism of $\mathcal{X}$. This allows us to study stabilisers of points in many classes of groups, such as topological full groups of Cantor minimal systems, Thompson groups, branch groups, and groups acting on trees with almost prescribed local actions.

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微支撑动作群点稳定器的研究

给定第一可数豪斯多夫空间 $\mathcal{X}$ 的同胚群 $G$，我们证明如果 $G$ 对 $\mathcal{X}$ 的作用是最小的并且具有局部作用的刚性稳定器至少，$\mathcal{X}$ 中任意两点的邻域稳定器由 $\mathcal{X}$ 的同胚共轭。这使我们能够研究许多类群中点的稳定器，例如康托最小系统的拓扑全群、汤普森群、分支群，以及作用在树上的群几乎是规定的局部动作。