Counting closed geodesics in strata.

We compute the asymptotic growth rate of the number N(C,R) of closed geodesics of length ≤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 γ of length at most R such that γ spends at least θ-fraction of its 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(S) of Riemann surfaces, and for 0≤θ≤1 we find an upper-bound for the number of geodesic paths of length ≤R in C which connect a point near x to a point near y and spend at least a θ-fraction of the time outside of a compact subset of C. [ABSTRACT FROM AUTHOR]