ON A ROELCKE-PRECOMPACT POLISH GROUP THAT CANNOT ACT TRANSITIVELY ON A COMPLETE METRIC SPACE

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 for certain Polish groups, namely Aut* (mu) and Homeo(+)[0,1], such an action can never be transitive (unless the space acted upon is a singleton). We also point out that in all known examples, this pathology coincides with the pathology of Polish groups that are not closed permutation groups and yet have discrete uniform distance, asking whether there is a relation. We conclude with a general characterisation/classification of transitive continuous isometric actions of a Roelcke-precompact Polish group on a complete metric space. In particular, the morphism from a Roelcke-precompact Polish group to its Bohr compactification is surjective.