Counting Curves in Hyperbolic Surfaces

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.