Simple closed curves, word length, and nilpotent quotients of free groups

Details Simple closed curves, word length, and nilpotent quotients of free groups Simple closed curves, word length, and nilpotent quotients of free groups

2010-09-02

arxiv.org/abs/1009.0332v1

Mathematics

Geometric Topology

Scientific paper

We consider the fundamental group $\pi$ of a surface of finite type equipped with the infinite generating set consisting of all simple closed curves. We show that every nilpotent quotient of $\pi$ has finite diameter with respect to the word metric given by this set. This is in contrast with a result of Danny Calegari that shows that $\pi$ has infinite diameter with respect to this set. Furthermore, we give a general criterion for a finitely generated group equipped with a generating set to have this property.

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