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

TL;DR

This paper investigates the properties of the fundamental group of a surface with respect to all simple closed curves as generators, showing that all its nilpotent quotients have finite diameter, contrasting with the infinite diameter of the group itself.

Contribution

It establishes that nilpotent quotients of such groups have finite diameter under the word metric, and provides a general criterion for finitely generated groups to have this property.

Findings

Nilpotent quotients have finite diameter with respect to simple closed curves.

The fundamental group of a surface has infinite diameter, contrasting with its nilpotent quotients.

A general criterion for groups to have finite diameter under certain generating sets.

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.