We give the asymptotic growth of the number of arcs of bounded length between boundary components on hyperbolic surfaces with boundary. Specifically, if has genus , boundary components and punctures, then the number of orthogeodesic arcs in each pure mapping class group orbit of length at most is asymptotic to 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.
Cite this article
Nick Bell, Counting arcs on hyperbolic surfaces. Groups Geom. Dyn. 17 (2023), no. 2, pp. 459–478DOI 10.4171/GGD/705