Coarsely Proper Actions of Topological Groups

Abstract

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 \v{S}varc-Milnor Lemma, and utilise the isometry group of the Urysohn universal space as an application.