## Abstract

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.

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 language | English |
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Pages (from-to) | 441-455 |

Journal | Transactions of the American Mathematical Society |

Volume | 372 |

Issue number | 1 |

Early online date | 20 Mar 2019 |

DOIs | |

Publication status | Published - 2 Jul 2020 |