A remark on the word length in surface groups

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.

arXiv comment18 pages, 4 figures. Corollary 4.1 added in new version