TITLE: Is any interesting question about a group G decidable from a presentation of G?
QUESTION [22 upvotes]: We say that a group G is in the class Fq if there is a CW-complex which is a BG (that is, which has fundamental group G and contractible universal cover) and which has finite q-skeleton.  Thus F0 contains all groups, F1 contains exactly the finitely generated groups, F2 the finitely presented groups, and so forth.
My question:  For a fixed q ≥ 3, is it possible to decide, from a finite presentation of a group G, whether G is in Fq or not?  I would assume not, but am not having much luck proving it.
One approach would be to prove that, if G is a group in Fq and H is a finitely presented subgroup, then H ∈ Fq as well.  This would make being in Fq a Markov property, or at least close enough to make it undecidable.
Henry Wilton's comment below makes it clear that being Fq is not even quasi-Markov, so the above idea won't work.  I still suspect that "G ∈ Fq" is not decidable, but now my intuition is from Rice's theorem:

If $\mathcal{B}$ is a nonempty set of computable functions with nonempty complement, then no algorithm accepts an input n and decides whether φn is an element of $\mathcal{B}$.

It seems likely to me that something similar is true of finite presentations and the groups they define.
John Stillwell notes below that this can't be true for a number of questions involving the abelianization of G.  This wouldn't affect the Rips construction/1-2-3 theorem discussion below if the homology-sphere idea works, since those groups are all perfect.
Any thoughts?

REPLY [11 votes]: 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.