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Some undecidability results for asynchronous transducers and the Brin-Thompson group $ 2V$


Authors: James Belk and Collin Bleak
Journal: Trans. Amer. Math. Soc. 369 (2017), 3157-3172
MSC (2010): Primary 20F10; Secondary 20B27, 20E32, 37B99
DOI: https://doi.org/10.1090/tran/6963
Published electronically: December 27, 2016
MathSciNet review: 3605967
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Abstract: Using a result of Kari and Ollinger, we prove that the torsion problem for elements of the Brin-Thompson group $ 2V$ is undecidable. As a result, we show that there does not exist an algorithm to determine whether an element of the rational group $ \mathcal {R}$ of Grigorchuk, Nekrashevich, and Sushchanskiĭ has finite order. A modification of the construction gives other undecidability results about the dynamics of the action of elements of $ 2V$ on Cantor space. Arzhantseva, Lafont, and Minasyan proved in 2012 that there exists a finitely presented group with solvable word problem and unsolvable torsion problem. To our knowledge, $ 2V$ furnishes the first concrete example of such a group and gives an example of a direct undecidability result in the extended family of R. Thompson type groups.


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Additional Information

James Belk
Affiliation: Mathematics Program, Bard College, P.O. Box 5000, Annandale, New York 12504
Email: belk@bard.edu

Collin Bleak
Affiliation: School of Mathematics and Statistics, Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland
Email: cb211@st-andrews.ac.uk

DOI: https://doi.org/10.1090/tran/6963
Keywords: Undecidable torsion problem, Brin-Thompson groups, rational group, transducer, reversible Turing machine
Received by editor(s): April 26, 2015
Published electronically: December 27, 2016
Article copyright: © Copyright 2016 American Mathematical Society