Some undecidability results for asynchronous transducers and the Brin-Thompson group 2𝑉
Using a result of Kari and Ollinger, we prove that the torsion problem for elements of the Brin-
Thompson group $2 V $ 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 $2 V $ on Cantor space. Arzhantseva, Lafont, and Minasyan proved in 2012 that there�…
Thompson group $2 V $ 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 $2 V $ on Cantor space. Arzhantseva, Lafont, and Minasyan proved in 2012 that there�…
Abstract
Using a result of Kari and Ollinger, we prove that the torsion problem for elements of the Brin-Thompson group is undecidable. As a result, we show that there does not exist an algorithm to determine whether an element of the rational group 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 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, 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. References
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