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Effectively closed subgroups of the infinite symmetric group


Authors: Noam Greenberg, Alexander Melnikov, Andre Nies and Daniel Turetsky
Journal: Proc. Amer. Math. Soc. 146 (2018), 5421-5435
MSC (2010): Primary 03D80, 20B35
DOI: https://doi.org/10.1090/proc/14055
Published electronically: September 10, 2018
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Abstract: We apply methods of computable structure theory to study effectively closed subgroups of $ S_\infty $. The main result of the paper says that there exists an effectively closed presentation of $ \mathbb{Z}_2$ which is not the automorphism group of any computable structure $ M$. In contrast, we show that every effectively closed discrete group is topologically isomorphic to $ \rm {Aut}(M)$ for some computable structure $ M$. We also prove that there exists an effectively closed compact (thus, profinite) subgroup of $ S_\infty $ that has no computable Polish presentation. In contrast, every profinite computable Polish group is topologically isomorphic to an effectively closed subgroup of $ S_\infty $. We also look at oligomorphic subgroups of $ S_\infty $; we construct a $ \Sigma ^1_1$ closed oligomorphic group in which the orbit equivalence relation is not uniformly HYP. Our proofs rely on methods of computable analysis, techniques of computable structure theory, elements of higher recursion theory, and the priority method.


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Additional Information

Noam Greenberg
Affiliation: Department of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington, 6140 New Zealand
Email: greenberg@msor.vuw.ac.nz

Alexander Melnikov
Affiliation: Massey University, Private Bag 102904 NSMC, Auckland, 0745 New Zealand
Email: alexander.g.melnikov@gmail.com

Andre Nies
Affiliation: Department of Computer Science, The University of Auckland, Private Bag 92019, Auckland, 1020 New Zealand
Email: andre@cs.auckland.ac.nz

Daniel Turetsky
Affiliation: Department of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington, 6140 New Zealand
Email: dan.turetsky@vuw.ac.nz

DOI: https://doi.org/10.1090/proc/14055
Received by editor(s): September 19, 2017
Received by editor(s) in revised form: November 6, 2017, and December 12, 2017
Published electronically: September 10, 2018
Additional Notes: The first author and third authors were partially supported by the Marsden Fund of New Zealand.
The second author was partially supported by the Marsden Fund of New Zealand and Massey University Research Fund.
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2018 American Mathematical Society

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