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 language | English (US) |
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Pages (from-to) | 1881-1894 |
Number of pages | 14 |
Journal | Journal of Symbolic Logic |
Volume | 65 |
Issue number | 4 |
DOIs | |
State | Published - Dec 2000 |
Keywords
- Continuous action
- Orbit equivalence relation
- Polish group
ASJC Scopus subject areas
- Philosophy
- Logic