Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-26T15:55:50.162Z Has data issue: false hasContentIssue false
  • English
  • Français

THE COMPLEXITY OF TOPOLOGICAL GROUP ISOMORPHISM

Published online by Cambridge University Press:  23 October 2018

ALEXANDER S. KECHRIS ,
ANDRÉ NIES  and
KATRIN TENT
ALEXANDER S. KECHRIS
Affiliation:
DEPARTMENT OF MATHEMATICS CALTECH, PASADENA, CA91125, USAE-mail: kechris@caltech.edu
ANDRÉ NIES
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE THE UNIVERSITY OF AUCKLAND PRIVATE BAG92019AUCKLAND, NEW ZEALANDE-mail: andre@cs.auckland.ac.nz
KATRIN TENT
Affiliation:
MATHEMATISCHES INSTITUT UNIVERSITÄT MÜNSTER EINSTEINSTRASSE 62 48149 MÜNSTER, GERMANYE-mail: tent@math.uni-muenster.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the complexity of the topological isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Borel spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound.

Keywords

03E1520B35topological group isomorphismnon-Archimedean grouplocally compact groupoligomorphic groupBorel reducibility
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 3 , September 2018 , pp. 1190 - 1203
Copyright
Copyright © The Association for Symbolic Logic 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ahlbrandt, G. and Ziegler, M., Quasi finitely axiomatizable totally categorical theories. Annals of Pure and Applied Logic, vol. 30 (1986), no. 1, pp. 63–82.CrossRefGoogle Scholar
Becker, H. and Kechris, A., The Descriptive Set Theory of Polish Group Actions, vol. 232, Cambridge University Press, Cambridge, 1996.CrossRefGoogle Scholar
Fried, M. and Jarden, M., Field Arithmetic, vol. 11, Springer Science & Business Media, New York, 2006.Google Scholar
Gao, S., Invariant Descriptive Set Theory, Pure and Applied Mathematics, vol. 293, CRC Press, Boca Raton, FL, 2009.Google Scholar
Hodges, W., Model Theory, Encyclopedia of Mathematics, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
Kechris, A. S., Classical Descriptive Set Theory, vol. 156, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
Mekler, A., Stability of nilpotent groups of class 2 and prime exponent, this JOURNAL, vol. 46 (1981), no. 4, pp. 781–788.Google Scholar
Nies, A., The complexity of isomorphism between countably based profinite groups, preprint, 2016, arXiv:1604.00609.Google Scholar
Ribes, L. and Zalesskii, P., Profinite Groups, Springer, New York, 2000.CrossRefGoogle Scholar
Rosendal, C. and Zielinski, J., Compact metrizable structures and classification problems, this JOURNAL, vol. 83 (2018), no. 1, pp. 165–186.Google Scholar
Tsankov, T., Unitary representations of oligomorphic groups. Geometric and Functional Analysis, vol. 22 (2012), no. 2, pp. 528–555.CrossRefGoogle Scholar