Actions of non-compact and non-locally compact polish groups

Sławomir Solecki

Research output: Contribution to journalArticlepeer-review

Abstract

We show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces.

Original languageEnglish (US)
Pages (from-to)1881-1894
Number of pages14
JournalJournal of Symbolic Logic
Volume65
Issue number4
DOIs
StatePublished - Dec 2000

Keywords

  • Continuous action
  • Orbit equivalence relation
  • Polish group

ASJC Scopus subject areas

  • Philosophy
  • Logic

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