Counting curves in hyperbolic surfaces

Let Σ be a hyperbolic surface. We study the set of curves on Σ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary γ 0 . For example, in the particular case that Σ is a once-punctured torus, we prove that the cardinality of the set of curves of type γ 0 and of at most length L is asymptotic to L 2 times a constant.