Arcs on hyperbolic surfaces

: a view towards counting

Abstract

We give the asymptotic growth of the number of arcs of bounded length between
boundary components on hyperbolic surfaces with boundary, analogous to a result of Mirzakhani for curves. 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. In demonstrating these results, we develop tools to examine a well-known
association between arcs and curves. We demonstrate that this association
is not injective but is uniformly k-to-1 across arcs of the same type, and on the
pair of pants P it maps at most two two-ended arcs to the same curve. We also
derive arbitrarily large families of arcs whose lengths are equal under any hyperbolic metric.

Date of Award	21 Jun 2022
Original language	English
Awarding Institution	University of Bristol
Supervisor	Viveka Erlandsson (Supervisor)