Let $G=F\_2$ be the free group of rank $2$. Is there a constant $c>0$ such that the word length $|[u,v]|$ of every commutator $[u,v]=uvu^{-1}v^{-1}$ where $u,v\in G$, $|u|,|v|>0$ is at least $c(|u|+|v|)$ unless $[u,v]=[u\_1,v\_1]$ for some $u\_1,v\_1$ with $|u\_1|+|v\_1|<|u|+|v|$?



Length of commutators in the free group
Victor Guba sent me a proof. The proof is based on the old result by Wicks, Wicks, N. J. Commutators in free products. J. London Math. Soc. 37 (1962), 433–444., which describes all words which are commutators in a free group (Lemma 5 in the paper). By that result, a word is a commutator iff it is a conjugate of a reduced word of the form $ABCA^{-1}B^{-1}C^{-1}$. This immediately implies that one can take $c=1/4$ (possibly bigger).


