Is there any way to check whether a group is residually solvable? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:19:10Z http://mathoverflow.net/feeds/question/48627 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48627/is-there-any-way-to-check-whether-a-group-is-residually-solvable Is there any way to check whether a group is residually solvable? HJ 2010-12-08T11:29:49Z 2010-12-09T20:28:03Z <p>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?</p> <p>Actually, I'm curious whether the finitely presented group $G = &lt; x_i, 1\leq i \leq m | w_i x_i w_i^{-1}x_{i+1}, 1\leq i \leq n-1 >$, where $w_i=x_j^{\pm 1}$ for some $j$.</p> http://mathoverflow.net/questions/48627/is-there-any-way-to-check-whether-a-group-is-residually-solvable/48637#48637 Answer by ndkrempel for Is there any way to check whether a group is residually solvable? ndkrempel 2010-12-08T13:09:59Z 2010-12-08T13:09:59Z <p>This problem is undecidable. If it were not, you could use it to construct an algorithm for testing if a given f.p. group is trivial or not, which is well known to be undecidable:</p> <p><code> Input: f.p. group G</p> <ul> <li>Test if G is residually solvable.</li> <li>If it is not, output "non-trivial".</li> <li>If it is, find the abelianization of G.</li> <li>If the abelianization is trivial, output "trivial".</li> <li>Otherwise output "non-trivial". </code></li> </ul> http://mathoverflow.net/questions/48627/is-there-any-way-to-check-whether-a-group-is-residually-solvable/48676#48676 Answer by Mark Sapir for Is there any way to check whether a group is residually solvable? Mark Sapir 2010-12-08T21:27:57Z 2010-12-08T21:27:57Z <p>It is not clear whether the set of finite residually solvable group presentations is even recursively enumerable. Unlike for the word or triviality problem where there exists an algorithm which says "yes" iff the answer is "yes", I do not think there exists such an algorithm in this case. But I do not think anybody proved that the algorithm does not exist. Same for residually finite groups. </p>