This is not my area of research, but I am curious. Let $G=\left< X|R \right>$ be a finitely presented group, where $X$ and $R$ are finite. There are many questions which are undecidable for all such $G$, for example whether $G$ is trivial or whether a particular word is trivial in $G$. Is there any non-trivial question (by trivial I mean that the answer is always yes or always no) which is decidable? For instance, is there a class $S$ (non-empty and not equal to all the finitely presented groups) such you can always decide whether $G$ is in this class? 



Decidability in groups
The problem whether $G$ is perfect, that is $G=[G,G]$ is decidable because you need to abelianize all relations (replace the operation by "+" in every relation) and solve a system of linear equations over $\mathbb Z$. For example, if the defining relations are $xy^{-1}xxy^5x^{-8}=1, x^{-3}y^{-2}xyx^5 = 1 $, then the Abelianization gives $-5x+4y=0, 3x-y=0$. Now you need to check if these two relations "kill" $\mathbb{Z^2}$. That means for some integers $a,b,c,d$ we should have $x = a(-5x+4y)+b(3x-y), 
 y = c(-5x+4y)+d(3x-y) $. This gives 4 integer equations with four unknowns: $1=-5a+3b, 0=4a-b, 1=4c-d, 0=-5c+3d 
 $. This systems does not have an integer solution (it implies $7a=1 $), so $G\ne [G,G]$. 


