Counting closed geodesics in strata

Abstract

We compute the asymptotic growth rate of the number N(C, R) of closed geodesics of length less than R in a connected component C of a stratum of quadratic differentials. We prove that for any 0 < θ< 1, the number of closed geodesics of length at most R that spend at least θ-fraction of time outside of a compact subset of C is exponentially smaller than N(C, R). The theorem follows from a lattice counting statement. For points x, y in the moduli space M of Riemann surfaces, and for 0 < θ< 1, we find an upper-bound for the number of geodesic paths of length less than R in C which connect a point near x to a point near y and spend a θ-fraction of the time outside of a compact subset of C.