On a Roelcke-precompact Polish group that cannot act transitively on a complete metric space
| Autor: | Itaï Ben Yaacov |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Transitive relation
Group (mathematics) General Mathematics 010102 general mathematics Bohr compactification Mathematics::General Topology Permutation group Space (mathematics) 01 natural sciences Complete metric space Surjective function Combinatorics Morphism 0103 physical sciences 010307 mathematical physics 0101 mathematics Mathematics |
| Zdroj: | Israel Journal of Mathematics. 224:105-132 |
| ISSN: | 1565-8511 0021-2172 |
| DOI: | 10.1007/s11856-018-1638-8 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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