Counting arcs on hyperbolic surfaces

TL;DR

This paper establishes the asymptotic growth rate of the number of arcs of bounded length on hyperbolic surfaces with boundary, revealing a polynomial growth related to the surface's topology.

Contribution

It provides the first precise asymptotic formulas for counting arcs on hyperbolic surfaces, including both boundary-to-boundary and cusp-to-cusp arcs.

Findings

Number of arcs grows as L^{6g-6+2(n+p)} for surfaces with genus g, n boundaries, p punctures.

Derived asymptotics for arcs between boundary components and cusps.

Constants depend on the surface's geometric and topological properties.

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.