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

Antonios Manoussos and Polychronis Strantzalos

Vol. 259 (2012), No. 2, 421–443

DOI: 10.2140/pjm.2012.259.421

Abstract

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.

Keywords

proper action, Weil completion, Cauchy indivisibility, Borel section, fundamental set

Mathematical Subject Classification 2010

Primary: 37B05, 54H20

Secondary: 54H15

Milestones

Received: 6 September 2011

Accepted: 8 June 2012

Published: 3 October 2012