A. Vdovina: Products of commutators, buildings and surfaces groups.

Abstract: Let $\FF$ be a free group, and $[\FF,\FF]$ be its commutator subgroup. We define the {\em genus} of a word $w \in [\FF,\FF]$ to be the least positive integer $g$ such that $w$ is a product of $g$ commutators in $\FF$. Every element of genus $g$ can be presented as a product of $g$ commutators by many ways. More interesting question, asked by E.Rips in 1996 at a geometric group theory conference in Anogia,Greece, is to find a word of genus g which has many genus g presentations up to an action of the modular group Mod(S_g).
We show, that, starting with sufficiently high genus, there is such a genus g word, and the number its of genus g presentations is factorial on g.