Coarsely Proper Actions of Topological Groups

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 systematically resolve the topological obstructions identified by Kapovich. We establish comprehensive large-scale characterisations for coarsely proper actions and coboundedness. As a primary application, we provide an affirmative resolution to two open problems posed by Rosendal (Problems B.7 and B.8) regarding the intrinsic coarse boundedness and 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 a canonical application.