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$ \sigma$-lacunary actions of Polish groups


Author: Jan Grebík
Journal: Proc. Amer. Math. Soc. 148 (2020), 3583-3589
MSC (2010): Primary 03E15, 28A05; Secondary 22A05, 54H05, 54H11
DOI: https://doi.org/10.1090/proc/14982
Published electronically: March 17, 2020
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Abstract: We show that every essentially countable orbit equivalence relation induced by a continuous action of a Polish group on a Polish space is $ \sigma $-lacunary. In combination with Gao and Jackson [Invent. Math. 201 (2015), pp. 309-383] we obtain a straightforward proof of the result from Ding and Gao [Adv. Math. 307 (2017), pp. 312-343] that every essentially countable equivalence relation that is induced by an action of an abelian nonarchimedean Polish group is Borel reducible to $ \mathbb{E}_0$, i.e., it is essentially hyperfinite.


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Jan Grebík
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom; Institute of Mathematics of the Czech Academy of Sciences, Žitná 609/25, 110 00 Praha 1-Nové Město, Czech Republic; and Department of Algebra, MFF UK, Sokolovská 83, 186 00 Praha 8, Czech Republic
Email: jan.grebik@warwick.ac.uk; grebikj@gmail.com

DOI: https://doi.org/10.1090/proc/14982
Received by editor(s): November 4, 2019
Received by editor(s) in revised form: December 11, 2019
Published electronically: March 17, 2020
Additional Notes: The author was supported by the GACR project 17-33849L and RVO: 67985840. The research was conducted during the author’s visit at Cornell University that was partially funded by the grant GAUK 900119 of Charles University.
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2020 American Mathematical Society