A characterization of proper actions with bornology and coarse geometry
Hiroaki Nagaya
TL;DR
This work develops a bornology-driven perspective on proper group actions, extending classical metric and uniform approaches (Palais, Yoshino) to coarse geometry. It defines bornological proper actions (B-proper) for actions of a bornological group on a bornological space and characterizes them via orbit data and induced maps, establishing equivalences with the topological picture under suitable bornologies. A central contribution is the construction of coarse structures $\mathcal{E}$ with $\mathcal{B}_\mathcal{E}=\mathcal{B}_X$ in which the action is equi-controlled, including a minimal coarse structure $\mathcal{E}(L,\mathcal{B}_X)$ that realizes bornological properness. The results unify and generalize properness notions across topology, uniform structures, and coarse geometry, providing a robust framework for analyzing group actions on large-scale spaces. Practical impact lies in a systematic approach to realizing proper actions without fixed metrics, enabling applications in $(G,X)$-manifolds and coarse geometric analysis.
Abstract
In 1961, Palais showed that every smooth proper Lie group action on a smooth manifold admits a compatible Riemannian metric on the manifold such that the action becomes isometric. In 2006, Yoshino studied a continuous proper action of a locally compact Hausdorff group on a locally compact Hausdorff space, and showed that the space carries a compatible uniform structure making the action equi continuous in an appropriate setting. In this paper, we focus on bornological proper actions on bornological spaces and prove that the space admits a compatible coarse structure such that the action becomes equi controlled.