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

Abstract

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.