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On the self-intersections of curves deep in the lower central series of a surface group

Justin Malestein, Andrew Putman
Arxiv ID: 0901.2561Last updated: 6/8/2020
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.

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