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On Polish groups admitting non-essentially countable actions

Part of: Set theory Connections with other structures, applications Locally compact groups and their algebras

Published online by Cambridge University Press:  29 December 2020

ALEXANDER S. KECHRIS ,
MACIEJ MALICKI ,
ARISTOTELIS PANAGIOTOPOULOS  and
JOSEPH ZIELINSKI
ALEXANDER S. KECHRIS
Affiliation:
Department of Mathematics, Caltech, 1200 E. California Blvd, Pasadena, CA91125, USA (e-mail:kechris@caltech.edu)
MACIEJ MALICKI
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-656, Warsaw, Poland (e-mail:mamalicki@gmail.com)
ARISTOTELIS PANAGIOTOPOULOS*
Affiliation:
Institut für Mathematische Logik und Grundlagenforschung, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149Münster, Germany
JOSEPH ZIELINSKI
Affiliation:
Department of Mathematics, GAB 435, University of North Texas, Denton, TX76201, USA (e-mail:Joseph.Zielinski@unt.edu, zielinski.math@gmail.com)
*
e-mail:apanagio@uni-muenster.de
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Abstract

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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.

Keywords

Borel reductionessentially countablelocally compact groupnon-archimedean Polish groupsstormy action

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 54H20: Topological dynamics 22D05: General properties and structure of locally compact groups
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 1 , January 2022 , pp. 180 - 194
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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