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THE COMPLEXITY OF TOPOLOGICAL GROUP ISOMORPHISM

Published online by Cambridge University Press:  23 October 2018

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

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.

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Copyright
Copyright © The Association for Symbolic Logic 2018 

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