User nick hildebrand - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:22:05Z http://mathoverflow.net/feeds/user/4682 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18423/proving-that-a-group-is-free Proving that a group is free Nick Hildebrand 2010-03-16T20:56:11Z 2010-05-11T15:56:03Z <p>I've got a group \$G\$ that I'm trying to prove is free. I already know that \$G\$ is torsion-free. Moreover, I can "almost" prove what I want : I can find a finite index subgroup \$G'\$ of \$G\$ that is definitely free.</p> <p>This leads me to the following question. Can anyone give me an example of a torsion-free group \$G\$ that is not free but contains a free subgroup of finite index? I've tried pretty hard to find groups like this, but i can't seem to avoid introducing torsion. Thanks!</p> http://mathoverflow.net/questions/20539/decidability-of-conjugacy-problem-for-finitely-generated-subgroups-of-free-groups Decidability of conjugacy problem for finitely generated subgroups of free groups Nick Hildebrand 2010-04-06T19:41:25Z 2010-04-16T16:38:14Z <p>The conjugacy problem for a free group \$F_n\$ on \$n\$ letters has an easy solution. Each element of \$F_n\$ is conjugate to a unique and easily computable "cyclically reduced element" (this means that if you arrange the word around a circle, then there are no cancellations), so two elements of \$F_n\$ are conjugate if and only if they have the same cyclically reduced conjugates.</p> <p>I've been trying unsuccessfully to generalize this to solve the following problem. Let \${x_1,\ldots,x_k}\$ and \${y_1,\ldots,y_{k'}}\$ be two finite sets of elements of \$F_n\$. Let \$G_x\$ and \$G_y\$ be the subgroups of \$F_n\$ generated by the \$x_i\$ and the \$y_i\$, respectively. Is there an algorithm to decide if \$G_x\$ and \$G_y\$ are conjugate? Does anyone know how to do this? Thank you very much!</p> http://mathoverflow.net/questions/20471/why-are-free-groups-residually-finite/20488#20488 Comment by Nick Hildebrand Nick Hildebrand 2010-04-06T19:15:47Z 2010-04-06T19:15:47Z Doesn't this proof generalize to give Malcev's theorem? The proof I know for finitely generated linear groups over fields goes along these same lines. I'm not sure about finitely generated linear groups over arbitrary commutative rings. http://mathoverflow.net/questions/18423/proving-that-a-group-is-free/18425#18425 Comment by Nick Hildebrand Nick Hildebrand 2010-03-16T22:31:04Z 2010-03-16T22:31:04Z Thank you very much! Given the people whose work you had to use to prove this, I don't feel so bad for not being able to do it myself =) http://mathoverflow.net/questions/18423/proving-that-a-group-is-free/18424#18424 Comment by Nick Hildebrand Nick Hildebrand 2010-03-16T21:07:51Z 2010-03-16T21:07:51Z Isn't Z+0 infinite index? The quotient is Z, which is an infinite group.