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

Published online by Cambridge University Press:  12 March 2014

Sławomir Solecki
Sławomir Solecki*
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA, E-mail:ssolecki@indiana.edu
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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.

Keywords

Polish groupcontinuous actionorbit equivalence relation
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 4 , December 2000 , pp. 1881 - 1894
Copyright
Copyright © Association for Symbolic Logic 2000

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References

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