Properness, Cauchy-indivisibility and the Weil completion of a group of isometries

TL;DR

This paper introduces 'Cauchy-indivisible' actions on metric spaces, explores their properties, and links them to proper actions and Weil completions, expanding understanding of isometry groups in metric geometry.

Contribution

It defines a new class of metric actions, establishes their relation to proper actions, and connects them to Weil completions and fundamental sets in metric spaces.

Findings

Cauchy-indivisible actions coincide with proper actions on locally compact spaces.

A group with a Cauchy-indivisible action has a Weil completion.

Connections between Borel sections and fundamental sets are explored.

Abstract

In this paper we introduce a new class of metric actions on separable (not necessarily connected) metric spaces called "Cauchy-indivisible" actions. This new class coincides with that of proper actions on locally compact metric spaces and, as examples show, it may be different in general. The concept of "Cauchy-indivisibility" follows a more general research direction proposal in which we investigate the impact of basic notions in substantial results, like the impact of local compactness and connectivity in the theory of proper transformation groups. In order to provide some basic theory for this new class of actions we embed a "Cauchy-indivisible" action of a group of isometries of a separable metric space in a proper action of a semigroup in the completion of the underlying space. We show that, in case this subgroup is a group, the original group has a "Weil completion" and vice versa. Finally, in order to establish further connections between "Cauchy-indivisible" actions on separable metric spaces and proper actions on locally compact metric spaces we investigate the relation between "Borel sections" for "Cauchy-indivisible" actions and "fundamental sets" for proper actions. Some open questions are added.