  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.
