# Decidability in Groups

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?

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You might like to look at the answers to this question, and the question it refers to: mathoverflow.net/questions/16532/… . – HJRW Mar 31 '11 at 9:06
Thanks. If a question is interesting, then it will come back. – Yiftach Barnea Mar 31 '11 at 9:44
There is active research going on about whether such problems are decidable if we assume the word problem has a solution. For instance, determining whether a group is a 3-manifold group is decidable, given a presentation and a solution to the word problem. – JeremyKun Oct 18 '11 at 2:20

The problem whether $G$ is perfect, that is $G=[G,G]$ is decidable because you need to abelianize all relations and solve a system of linear equations over $\mathbb Z$.