Counting arcs on hyperbolic surfaces

Nick Bell

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arXiv · 2011.13969

Counting arcs on hyperbolic surfaces

Nick Bell

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Abstract

We give the asymptotic growth of the number of (multi-)arcs of bounded length between boundary components on complete finite-area hyperbolic surfaces with boundary. Specifically, if $S$ has genus $g$, $n$ boundary components and $p$ punctures, then the number of orthogeodesic arcs in each pure mapping class group orbit of length at most $L$ is asymptotic to $L^{6g-6+2(n+p)}$ times a constant. We prove an analogous result for arcs between cusps, where we define the length of such an arc to be the length of the sub-arc obtained by removing certain cuspidal regions from the surface.

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Nick Bell. 2020-11-27. Counting arcs on hyperbolic surfaces. https://arxiv.org/abs/2011.13969

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