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
Number of pages15
JournalTransactions of the American Mathematical Society
Early online date20 Mar 2019
Publication statusE-pub ahead of print - 20 Mar 2019


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