On Roeckle-precompact Polish group which cannot act transitively on a complete metric space

TL;DR

This paper investigates conditions under which certain Polish groups can act transitively on complete metric spaces, showing that some groups cannot, and characterizing when such actions are transitive.

Contribution

It provides a characterization of transitive isometric actions of Roeckle-precompact Polish groups on complete metric spaces and relates this to properties of their Bohr compactification.

Findings

Certain Polish groups like Aut*(μ) and Homeo+[0,1] cannot act transitively unless on a singleton.

The morphism from Roeckle-precompact Polish groups to their Bohr compactification is surjective.

A general criterion for transitivity of isometric actions of Roeckle-precompact Polish groups.

Abstract

We study when a continuous isometric action of a Polish group on a complete metric space is, or can be, transitive. Our main results consist of showing that certain Polish groups, namely $\mathrm{Aut}^*(\mu)$ and $\mathrm{Homeo}^+[0,1]$, such an action can never be transitive (unless the space acted upon is a singleton). We also point out "circumstantial evidence" that this pathology could be related to that of Polish groups which are not closed permutation groups and yet have discrete uniform distance, and give a general characterisation of continuous isometric action of a Roeckle-precompact Polish group on a complete metric space is transitive. It follows that the morphism from a Roeckle-precompact Polish group to its Bohr compactification is surjective.