Effectively closed subgroups of the infinite symmetric group
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- by Noam Greenberg, Alexander Melnikov, Andre Nies and Daniel Turetsky PDF
- Proc. Amer. Math. Soc. 146 (2018), 5421-5435 Request permission
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.References
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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
- MR Author ID: 757288
- ORCID: 0000-0003-2917-3848
- Email: greenberg@msor.vuw.ac.nz
- Alexander Melnikov
- Affiliation: Massey University, Private Bag 102904 NSMC, Auckland, 0745 New Zealand
- MR Author ID: 878230
- ORCID: 0000-0001-8781-7432
- Email: alexander.g.melnikov@gmail.com
- Andre Nies
- Affiliation: Department of Computer Science, The University of Auckland, Private Bag 92019, Auckland, 1020 New Zealand
- MR Author ID: 328692
- 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
- MR Author ID: 894314
- Email: dan.turetsky@vuw.ac.nz
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 5421-5435
- MSC (2010): Primary 03D80, 20B35
- DOI: https://doi.org/10.1090/proc/14055
- MathSciNet review: 3866879