$\sigma$-lacunary actions of Polish groups
HTML articles powered by AMS MathViewer
- by Jan Grebík PDF
- Proc. Amer. Math. Soc. 148 (2020), 3583-3589 Request permission
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.References
- Howard Becker and Alexander S. Kechris, The descriptive set theory of Polish group actions, London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, 1996. MR 1425877, DOI 10.1017/CBO9780511735264
- M. Cotton. Abelian group actions and hypersmooth eqivalence relations. PhD thesis 2019.
- Longyun Ding and Su Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math. 307 (2017), 312–343. MR 3590520, DOI 10.1016/j.aim.2016.11.019
- Su Gao and Steve Jackson, Countable abelian group actions and hyperfinite equivalence relations, Invent. Math. 201 (2015), no. 1, 309–383. MR 3359054, DOI 10.1007/s00222-015-0603-y
- Greg Hjorth and Alexander S. Kechris, Recent developments in the theory of Borel reducibility, Fund. Math. 170 (2001), no. 1-2, 21–52. Dedicated to the memory of Jerzy Łoś. MR 1881047, DOI 10.4064/fm170-1-2
- Vladimir Kanovei, Borel equivalence relations, University Lecture Series, vol. 44, American Mathematical Society, Providence, RI, 2008. Structure and classification. MR 2441635, DOI 10.1090/ulect/044
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- A. S. Kechris, Classical Descriptive Set Theory–Corrections. http://www.math.caltech.edu/~kechris/papers/CDST-corrections.pdf
- Alexander S. Kechris, Countable sections for locally compact group actions, Ergodic Theory Dynam. Systems 12 (1992), no. 2, 283–295. MR 1176624, DOI 10.1017/S0143385700006751
- A. S. Kechris, The theory of countable Borel equivalence relations, preprint.
- Alexander S. Kechris and Henry L. Macdonald, Borel equivalence relations and cardinal algebras, Fund. Math. 235 (2016), no. 2, 183–198. MR 3549382, DOI 10.4064/fm242-4-2016
- B. D. Miller. Lacunary sets for actions of tsi groups, preprint.
Additional Information
- 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
- 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
- © Copyright 2020 American Mathematical Society
- 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
- MathSciNet review: 4108862