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arXiv · 1510.00238

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

Itaï Ben Yaacov
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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 $\mathrm{Aut}^*(μ)$ 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.

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BibTeXRIS

Itaï Ben Yaacov. 2016-09-19. On Roeckle-precompact Polish group which cannot act transitively on a complete metric space. https://arxiv.org/abs/1510.00238

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