Counting Curves in Hyperbolic Surfaces

TL;DR

This paper investigates the asymptotic growth of the number of curves of a fixed type on hyperbolic surfaces, demonstrating quadratic growth in length for the case of a once-punctured torus.

Contribution

It provides the first asymptotic count of curves of a given type on hyperbolic surfaces, specifically establishing quadratic growth for the once-punctured torus case.

Findings

Number of curves of a fixed type grows quadratically with length on a once-punctured torus.

Asymptotic formula involves a specific constant depending on the curve type.

Method extends to general hyperbolic surfaces for counting curves of a given type.

Abstract

Let $\Sigma$ be a hyperbolic surface. We study the set of curves on $\Sigma$ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary $\gamma_0$. For example, in the particular case that $\Sigma$ is a once-punctured torus, we prove that the cardinality of the set of curves of type $\gamma_0$ and of at most length $L$ is asymptotic to $L^2$ times a constant.