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

Journal of Symbolic Logic 65 (4):1881-1894 (2000)
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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.

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Countable borel equivalence relations.S. Jackson, A. S. Kechris & A. Louveau - 2002 - Journal of Mathematical Logic 2 (01):1-80.

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