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

Author： Malestein Justin  

Publisher： Springer Publishing Company

ISSN： 0046-5755

Source： Geometriae Dedicata, Vol.149, Iss.1, 2010-12, pp. : 73-84

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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.

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