Counting Curves in Hyperbolic Surfaces

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.

49 pages, 11 (mostly hand-drawn) figures

References in corpus (2)

• Bounds on the number of non-simple closed geodesics on a surface
• Lower bound on the number of non-simple closed geodesics on surfaces

Cited by in corpus (1)

• Non-Abelian number theory and the structure of curves on surfaces