most recent 30 from http://mathoverflow.net 2013-05-19T23:26:55Z http://mathoverflow.net/feeds/question/102744 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102744/commutator-length-in-free-groups Jason Mraz 2012-07-20T15:00:46Z 2012-07-20T15:00:46Z

<p>An interesting question: Given a group $G$ , we define the commutator-length of one of its elements $g$ (it's a known notion) to be the minimal number of commutators in the group such that their product equals $g$ (i.e.- $ cl(g) = min (n \qquad | \qquad g= [a_1 , b_1 ] [a_2,b_2 ] ...[a_n,b_n] )$ . </p> <p>Now, given a free-group $F= &lt; x_1 ,..., x_n > $ . Is it known that $cl (x_i x_j ) >1 $ ? (i.e.- a product of two generators can't be a commutator of two other elements) </p> <p>Thanks in advance </p>