A remark on the word length in surface groups

Erlandsson, Viveka

doi:10.1090/tran/7561

A remark on the word length in surface groups
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Trans. Amer. Math. Soc. 372 (2019), 441-455 Request permission

Abstract:

Let $\Sigma$ be a surface of negative Euler characteristic and let $S$ be 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.

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