What is wrong with the growth of simple closed geodesics on nonorientable surfaces?




A celebrated result of Mirzakhani states that, if (S,m) is a finite area orientable hyperbolic surface, then the number of simple closed geodesics of length less than L on (S,m) is asymptotically equivalent to a positive constant times LdimML(S), where ML(S) denotes the space of measured laminations on S. We observed on some explicit examples that this result does not hold for nonorientable hyperbolic surfaces. The aim of this article is to explain this surprising phenomenon.