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Summary

We introduce essentially countable and reducible-to-countable Borel equivalence relations and discuss situations where they appear in various contexts.

It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

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3 - Essentially Countable Relations

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16 - Open Problems

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On Polish groups admitting non-essentially countable actions

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3 results

3 - Essentially Countable Relations

Summary

16 - Open Problems

Summary

On Polish groups admitting non-essentially countable actions