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

TL;DR

This paper provides estimates on the minimal self-intersections of elements deep in the lower central series of surface groups, offering new insights into their algebraic and topological properties.

Contribution

It introduces new bounds on self-intersections of elements in the lower central series and offers a novel topological proof of residual nilpotency for free and surface groups.

Findings

Nontrivial elements in the kth lower central series have at least k self-intersections.

Nontrivial elements in the kth lower central series of free groups have word length at least k.

New topological proof of residual nilpotency for free and surface groups.

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.