Isometrisable group actions

TL;DR

This paper introduces a topological property for group actions on separable metrizable spaces that characterizes when a compatible metric invariant under the group action exists, extending previous results to non-locally compact spaces.

Contribution

The paper generalizes a known result by establishing an equivalence between a topological property of group actions and the existence of G-invariant metrics on separable metrizable spaces.

Findings

Characterizes when a G-invariant compatible metric exists for group actions.

Extends Marjanovic's result from locally compact to general separable metrizable spaces.

Provides a new topological criterion for isometrizability of group actions.

Abstract

Given a separable metrisable space X, and a group G of homeomorphisms of X, we introduce a topological property of the action of G on X which is equivalent to the existence of a G-invariant compatible metric on X. This extends a result of Marjanovic obtained under the additional assumption that X is locally compact.