AI 中文总结
该研究将经典拓扑概念适配到粗几何框架，解决拓扑障碍，建立粗恰当作用等特征描述，肯定解决罗森达尔的两个开放问题，还通过粗狄利克雷域构造基本集、推广引理并应用于尤里松通用空间等距群。
AI 中文摘要
我们将经典的恰当不连续群作用的拓扑概念适配到粗几何框架。通过用一致粗等价替代局部紧致性和离散性的严格约束，系统地解决了卡波维奇所确定的拓扑障碍。我们建立了粗恰当作用和余有界性的全面大规模特征描述。作为主要应用，我们肯定地解决了罗森达尔提出的两个关于波兰群中共紧子群的内在粗有界性和等价性的开放问题（问题B.7和B.8）。此外，我们通过粗狄利克雷域构造大规模基本集，推广了施瓦尔茨 - 米尔诺引理，并将尤里松通用空间的等距群作为一个典型应用。
英文摘要
We adapt the classical topological notions of properly discontinuous group actions to the framework of coarse geometry. By substituting the rigid constraints of local compactness and discreteness with uniform coarse equivalences, we resolve the topological obstructions identified by Kapovich. We establish large-scale characterisations for coarsely proper actions and coboundedness. As a primary application, we provide an affirmative answer to an open problem posed by Rosendal (Problem B.7) regarding the intrinsic coarse equivalence of cocompact subgroups in Polish groups. Furthermore, we construct large-scale fundamental sets via coarse Dirichlet domains, generalise the Švarc-Milnor Lemma, and utilise the isometry group of the Urysohn universal space as an application.
