Journal of Symbolic Logic

Actions of Non-Compact and Non-Locally Compact Polish Groups

Slawomir Solecki

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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.

Article information

J. Symbolic Logic, Volume 65, Issue 4 (2000), 1881-1894.

First available in Project Euclid: 6 July 2007

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Primary: 03E15: Descriptive set theory [See also 28A05, 54H05]
Secondary: 54H15: Transformation groups and semigroups [See also 20M20, 22-XX, 57Sxx] 22D05: General properties and structure of locally compact groups

Polish Group Continuous Action Orbit Equivalence Relation


Solecki, Slawomir. Actions of Non-Compact and Non-Locally Compact Polish Groups. J. Symbolic Logic 65 (2000), no. 4, 1881--1894.

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