In this talk I will describe results that extend the the work of Jaerisch, Kesseböhmer and Stratmann on univariate EVLs for cuspidal windings of geodesics on H^2/\Gamma where \Gamma is a certain kind of Fuchsian group. For this kind of \Gamma, H^2/\Gamma may have multiple cusps and many natural questions as to the behavior of geodesics in distinct cusps arise. We answer several such questions using multivariate extreme value theory.
The question of EVLs for maximal cuspidal excursions of geodesics was brought to life by Pollicott in 2006 when he demonstrated that a result of Galambos from 1972 concerning an EVL for entries of continued fractions expansions, could be translated into a statement about cuspidal excursions of geodesics on H^2/PSL(2,Z). This translation uses the coding theory for geodesics developed by Series and others.