On Roeckle-precompact Polish group which cannot act transitively on a complete metric space
Logic
2016-09-20 v2
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 and , 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.
Keywords
Cite
@article{arxiv.1510.00238,
title = {On Roeckle-precompact Polish group which cannot act transitively on a complete metric space},
author = {Itaï Ben Yaacov},
journal= {arXiv preprint arXiv:1510.00238},
year = {2016}
}