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

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

2009-01-16

arxiv.org/abs/0901.2561v2

Mathematics

Geometric Topology

12 pages, 2 figures, some references and clarifications added

Scientific paper

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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