PUB - Publikationen an der Universität Bielefeld

Investigating the impact of local compactness and connectedness in the theory of proper actions on locally compact and connected spaces, we introduce a new class of isometric actions on separable metric spaces called Cauchy-indivisible actions. The new class coincides with that of proper actions on locally compact metric spaces, without assuming connectivity, and, as examples show, may be different in general. In order to provide some basic theory for this new class of actions, we embed a Cauchy-indivisible action in a proper action of a semigroup in the completion of the underlying space. We show that, if this semigroup is a group, there are remarkable connections between Cauchy indivisibility and properness, while the original group has a Weil completion and vice versa. Further connections in this direction establish a relation between Borel sections for Cauchy-indivisible actions and fundamental sets for proper actions. Some open questions are added.