On the self-intersections of curves deep in the lower central series of a surface group (with J. Malestein) - [pdf] [ps]
Geom. Dedicata. 149 (2010), no. 1, 73–84.

Abstract :
We give various estimates of the minimal number of self-intersections of a nontrivial element of the kth term of the lower central series and derived series of the fundamental group of a surface. As an application, we obtain a new topological proof of the fact that free groups and fundamental groups of closed surfaces are residually nilpotent. Along the way, we prove that a nontrivial element of the kth term of the lower central series of a nonabelian free group has to have word length at least $k$ in a free generating set.