Isometry groups among topological groups

TL;DR

This paper characterizes when topological groups are isomorphic to isometry groups of metric spaces, providing constructions for various classes of groups and linking them to geometric and Banach space isometries.

Contribution

It establishes conditions under which topological groups are isomorphic to isometry groups of metric spaces, including constructions for Polish and locally compact groups.

Findings

Topological groups are isomorphic to isometry groups iff they meet certain closure conditions.

Every Polish group can be realized as an isometry group of a specific metric space.

Existence of a Banach space with a group of isometries fixing a point.

Abstract

It is shown that a topological group G is topologically isomorphic to the isometry group of a (complete) metric space iff G coincides with its G-delta-closure in the Rajkov completion of G (resp. if G is Rajkov-complete). It is also shown that for every Polish (resp. compact Polish; locally compact Polish) group G there is a complete (resp. proper) metric d on X inducing the topology of X such that G is isomorphic to Iso(X,d) where X = l_2 (resp. X = Q; X = Q\{point} where Q is the Hilbert cube). It is demonstrated that there are a separable Banach space E and a nonzero vector e in E such that G is isomorphic to the group of all (linear) isometries of E which leave the point e fixed. Similar results are proved for an arbitrary complete topological group.