Return to Revisions

Sorry I came to this question late. $F_q$ is undecidable from a presentation for each fixed $3\leq q<\infty$. First note that for finitely presented groups $F_q$ is equivalent to the homological finiteness condition $FP_q$ so it is enough to show this condition undecidable. This was done for $q =3$ in Section 5 of Cremanns, Robert; Otto, Friedrich, For groups the property of having finite derivation type is equivalent to the homological finiteness condition FP3. J. Symbolic Comput. 22 (1996), no. 2, 155–177 https://www.sciencedirect.com/science/article/pii/S0747717196900462. The same construction works for any fixed finite $q\geq 3$. They use essentially the same construction as the proof that Markov properties are undecidable but different details.