A remark on the word length in surface groups

TL;DR

This paper establishes a geometric interpretation of word length in surface groups via intersection numbers, linking algebraic properties to topological features, and analyzes the growth of immersed curves under mapping class group actions.

Contribution

It provides a new geometric formula for word length in surface groups and studies the asymptotic growth of immersed curves within mapping class group orbits.

Findings

Word length equals intersection number with certain curves and arcs.

Asymptotic growth rate of immersed curves of bounded word length.

Extension of the formula to free homotopy classes of immersed curves.

Abstract

Let $\Sigma$ be a surface of negative Euler characteristic and $S$ a generating set for $\pi_1(\Sigma,p)$ consisting of simple loops that are pairwise disjoint (except at $p$). We show that the word length with respect to $S$ of an element of $\pi_1(\Sigma,p)$ is given by its intersection number with a well-chosen collection of curves and arcs on $\Sigma$. The same holds for the word length of (a free homotopy class of) an immersed curve on $\Sigma$. As a consequence, we obtain the asymptotic growth of the number of immersed curves of bounded word length, as the length grows, in each mapping class group orbit.