A remark on the word length in surface groups

2016/08/26 by Erlandsson, Viveka · 1 citation
#30F60 #32G15 #57M50 #57M60 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1608.07436

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.

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