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

James Belk, Collin Bleak

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)
1 Downloads (Pure)

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 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 Minasyanin 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.
Original languageEnglish
Pages (from-to)3157-3172
Number of pages16
JournalTransactions of the American Mathematical Society
Volume369
Issue number5
Early online date27 Dec 2016
DOIs
Publication statusPublished - May 2017

Keywords

  • Undecidable torsion problem
  • Brin-Thompson groups
  • Rational group
  • Transducer
  • Reversible Turing machine

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