A remark on the word length in surface groups

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Let Σ be a surface of negative Euler characteristic and S a generating set for π1(Σ, 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 π1(Σ, p) is given by its intersection number with a well-chosen collection of curves and arcs on Σ. The same holds for
the word length of (a free homotopy class of) an immersed curve on Σ. 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.
Original languageEnglish
Pages (from-to)441-455
JournalTransactions of the American Mathematical Society
Issue number1
Early online date20 Mar 2019
Publication statusPublished - 2 Jul 2020


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