Vol. 259, No. 2, 2012

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

Antonios Manoussos and Polychronis Strantzalos

Vol. 259 (2012), No. 2, 421–443

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.

proper action, Weil completion, Cauchy indivisibility, Borel section, fundamental set
Mathematical Subject Classification 2010
Primary: 37B05, 54H20
Secondary: 54H15
Received: 6 September 2011
Accepted: 8 June 2012
Published: 3 October 2012
Antonios Manoussos
Fakultät für Mathematik, SFB 701
Universität Bielefeld
Postfach 100131
D-33501 Bielefeld
Polychronis Strantzalos
Department of Mathematics
University of Athens
GR-157 84 Athens