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

TL;DR

This paper proves that key problems about the order of elements in the Brin-Thompson group 2V and related groups are undecidable, highlighting fundamental limits of algorithmic analysis in these algebraic structures.

Contribution

It establishes the first concrete example of a finitely presented group with solvable word problem but undecidable torsion problem, extending undecidability results to Thompson-type groups.

Findings

Undecidability of torsion problem in 2V

No algorithm to determine finite order in rational group R

New undecidability results about dynamics on Cantor Space

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 Sushchanskii 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 prove 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.