A Frechet law and an Erds-Philipp law for maximal cuspidal windings

In this paper we establish a Frechet law for maximal cuspidal windings of the geodesic flow on a Riemannian surface associated with an arbitrary finitely generated, essentially free Fuchsian group with parabolic elements. This result extends previous work by Galambos and Dolgopyat and is obtained by applying extreme value theory. Subsequently, we show that this law gives rise to an Erds-Philipp law and to various generalized Khintchine-type results for maximal cuspidal windings. These results strengthen previous results by Sullivan, Stratmann and Velani for Kleinian groups, and extend earlier work by Philipp on continued fractions, which was inspired by a conjecture of Erds.