Isometry groups of proper metric spaces

TL;DR

This paper characterizes when a group of homeomorphisms can be the full isometry group of a proper metric space, and explores the actions and properties of locally compact Polish groups as isometry groups.

Contribution

It provides necessary and sufficient conditions for groups to be full isometry groups of proper metric spaces and characterizes certain classes of locally compact Polish groups.

Findings

Every locally compact Polish group acts freely as a full isometry group on a suitable space.

Characterization of locally compact Polish groups acting effectively and almost transitively as isometry groups.

Density of proper metrics with trivial isometry group on spaces with more than two points.

Abstract

Given a locally compact Polish space X, a necessary and sufficient condition for a group G of homeomorphisms of X to be the full isometry group of (X,d) for some proper metric d on X is given. It is shown that every locally compact Polish group G acts freely on GxY as the full isometry group of GxY with respect to a certain proper metric on GxY, where Y is an arbitrary locally compact Polish space with (card(G),card(Y)) different from (1,2). Locally compact Polish groups which act effectively and almost transitively on complete metric spaces as full isometry groups are characterized. Locally compact Polish non-Abelian groups on which every left invariant metric is automatically right invariant are characterized and fully classified. It is demonstrated that for every locally compact Polish space X having more than two points the set of proper metrics d such that Iso(X,d) = {id} is dense in the space of all proper metrics on X.