Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-09T04:28:22.611Z Has data issue: false hasContentIssue false
  • English
  • Français

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, CA 91125, 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, 48149 Münster, Germany
JOSEPH ZIELINSKI
Affiliation:
Department of Mathematics, GAB 435, University of North Texas, Denton, TX 76201, USA (e-mail: Joseph.Zielinski@unt.edu, zielinski.math@gmail.com)
*
e-mail: apanagio@uni-muenster.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 1 , January 2022 , pp. 180 - 194
Check for updates
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

References

REFERENCES

Adams, S., Elliott, G. A. and Giordano, T.. Amenable actions of groups. Trans. Amer. Math. Soc. 344(2) (1994), 803–822.CrossRefGoogle Scholar
Becker, H. and Kechris, A. S.. The Descriptive Set Theory of Polish Group Actions. Cambridge University Press, New York, 1996.CrossRefGoogle Scholar
Feldman, J. and Moore, C. C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. Trans. Amer. Math. Soc. 234(2) (1977), 289–324.CrossRefGoogle Scholar
Feldman, J. and Ramsey, A.. Countable sections for free actions of groups. Adv. Math. 55 (1985), 224–227.CrossRefGoogle Scholar
Gao, S.. On automorphism groups of countable structures. J. Symb. Log. 63 (1998), 891–896.CrossRefGoogle Scholar
Gao, S.. Invariant Descriptive Set Theory. Chapman and Hall, Boca Raton, FL, 2008.CrossRefGoogle Scholar
Gao, S. and Kechris, A. S.. On the classification of Polish metric spaces up to isometry. Mem. Amer. Math. Soc. 161(766) (2003), 1–78.Google Scholar
Golodets, V. Y. and Sinel’shchikov, S. D.. Amenable ergodic group actions and images of cocycles. Dokl. Akad. Nauk 312 (1990), 1296–1299; Engl. transl. Soviet Math. Dokl. 41 (1990), 523–526.Google Scholar
Glasner, E.. Tsirelson, B. and Weiss, B.. The automorphism group of the Gaussian measure cannot act pointwise. Israel J. Math. 148 (2005), 305–329.CrossRefGoogle Scholar
Glasner, E. and Weiss, B.. Spatial and non-spatial actions of Polish groups. Ergod. Th. & Dynam. Sys. 25 (2005), 1521–1538.CrossRefGoogle Scholar
Hjorth, G.. Classification and orbit equivalence relations. Amer Math. Soc. 75 (2000), 195.Google Scholar
Hjorth, G.. A dichotomy for being essentially countable. Contemp. Math 380(7) (2005), 109–127.CrossRefGoogle Scholar
Kechris, A. S.. Countable sections for locally compact group actions. Ergod. Th. & Dynam. Sys. 12 (1992), 283–295.CrossRefGoogle Scholar
Kechris, A. S.. Global Aspects of Ergodic Group Actions. American Mathematical Society, Providence, RI, 2010.CrossRefGoogle Scholar
Kechris, A. S.. The theory of countable Borel equivalence relations. Preprint, 2020, www.math.caltech.edu/kechris/.Google Scholar
Kwiatkowska, A. and Solecki, S.. Spatial models of Boolean actions and groups of isometries. Ergod. Th. & Dynam. Sys. 31(2) (2011), 405–421.CrossRefGoogle Scholar
Lupini, M. and Panagiotopoulos, A.. Games orbits play and obstructions to Borel reducibility. Groups Geom. Dyn. 12(4) (2018), 1461–1483.CrossRefGoogle Scholar
Malicki, M.. Abelian pro-countable groups and orbit equivalence relations. Fund. Math. 233(1) (2016), 83–99.Google Scholar
McShane, E. J.. Extension of range of functions. Bull. Amer. Math. Soc. 40 (1934), 837–842.CrossRefGoogle Scholar
Solecki, S.. Actions of non-compact and non-locally compact groups. J. Symb. Log. 65(4) (2000), 1881–1894.CrossRefGoogle Scholar
Thompson, A.. A metamathematical condition equivalent to the existence of a complete left invariant metric. J. Symb. Log. 71(4) (2006), 1108–1124.CrossRefGoogle Scholar