Proper actions and proper invariant metrics

Abstract

Abels H, Manoussos A, Noskov G. Proper actions and proper invariant metrics. Journal of the London Mathematical Society. 2011;83(3):619-636.We show that if a (locally compact) group $G$ acts properly on a locallycompact $\sigma$-compact space $X$ then there is a family of $G$-invariantproper continuous finite-valued pseudometrics which induces the topology of$X$. If $X$ is furthermore metrizable then $G$ acts properly on $X$ if and onlyif there exists a $G$-invariant proper compatible metric on $X$