# Is there any way to check whether a group is residually solvable?

For a given group presentation of a group(finitely presented), I want to check whether it is residually solvable or not. Is there any good way to do it?

Actually, I'm curious whether the finitely presented group $G = < x_i, 1\leq i \leq m | w_i x_i w_i^{-1}x_{i+1}, 1\leq i \leq n-1 >$, where $w_i=x_j^{\pm 1}$ for some $j$.

• @HJ: There are misprints in the definition of $G$. Could you please correct them? Is it supposed to be an LOG presentation? – Mark Sapir Dec 8 '10 at 21:36
• HJ - as stated, this is just a free group. (Killing a conjugate of $x_i$ is the same as killing $x_i$.) I presume you're aware of Magnus's Theorem that free groups are residually nilpotent (in particular, residually solvable). – HJRW Dec 8 '10 at 21:42
• @Henry, it is a clear misprint. He probably meant an LOG presentation. – Mark Sapir Dec 8 '10 at 21:53
• @Mark, Yes, I corrected the presentation. There were missing $x_[i+1}$. – hopflink Dec 9 '10 at 20:29

 Input: f.p. group G
 
 Test if G is residually solvable. If it is not, output "non-trivial". If it is, find the abelianization of G. If the abelianization is trivial, output "trivial". 
• Otherwise output "non-trivial".