Abstract
AbstractIt 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.
Highlights
Motivated by foundational questions about the intrinsic complexity of various mathematical classification problems, one of the prominent ongoing projects in descriptive set theoryDownloaded from https://www.cambridge.org/core. 02 Nov 2021 at 16:55:25, subject to the Cambridge Core terms of use.seeks to organize the collection of all definable equivalence relations with respect to Borel reductions.Definition 1.1
We say that E on X is essentially countable if E ≤B F, for some countable Borel equivalence relation F
To the compact case, it is a theorem of Kechris [Ke92] that every orbit equivalence relation induced by any locally compact Polish group G is essentially countable
Summary
Motivated by foundational questions about the intrinsic complexity of various mathematical classification problems, one of the prominent ongoing projects in descriptive set theoryseeks to organize the collection of all definable equivalence relations with respect to Borel reductions.Definition 1.1. To the compact case, it is a theorem of Kechris [Ke92] that every orbit equivalence relation induced by any locally compact Polish group G is essentially countable.
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