The Weyl law for lengths

Question

For what I know, this must be a standard fact, but I can't spot it in the literature I have on hands. What is the asymptotic of the geodesic lengths spectrum for the modular surface $X(1)$? (That is, what's the analog of the Weyl's law for this case?)

Answer 1

There is a more general result (Margulis' thesis) which concerns closed orbits of the geodesic flow on Riemannian manifolds with pinched negative curvature and finite volume. In the case of hyperbolic surfaces, if $\pi$ is the counting function for closed prime geodesics it yields: $$ \pi(T) \sim \frac {e^T}T $$ and it follows that the asymptotic is the same for non-prime grodesics as well. Margulis' thesis has been published in a book (On Some Aspects of the Theory of Anosov Systems, Springer). I found this result in the second part (a survey of further results, which is maybe more accessible than the original text) on page 79 in the Remark after Theorem 1.1 (the theorem is stated for compact manifolds but the remark includes the case of finite-volume).

Margulis' proof uses dynamical properties of the geodesic flow; there is also a proof due to Huber which uses the Selberg trace formula, which you can find in Buser's book (Geometry and Spectrum of compact Riemann surfaces) in the case of compact surfaces (I did not check whether the proof he gives adapts immediately to the finite volume case).

EDIT: a paper of Dal'bo-Peigné presents a proof valid for all geometrically finite torsion-free Fuchsian groups. It is available here. The paper is written in a more general context but contains references to older work on surfaces.