Counting closed geodesics in strata

TL;DR

This paper analyzes the asymptotic growth of closed geodesics in strata of quadratic differentials, showing that geodesics spending significant time outside compact sets are exponentially rarer, with implications for moduli space geometry.

Contribution

It provides the first asymptotic count of closed geodesics in strata and establishes exponential decay for geodesics spending substantial time outside compact regions.

Findings

Asymptotic growth rate of closed geodesics in strata computed.

Geodesics spending >  of time outside compact sets are exponentially rare.

Upper bounds for geodesic paths connecting points in moduli space established.

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 < \theta < 1, the number of closed geodesics of length at most R that spend at least \theta-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 < \theta < 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 \theta-fraction of the time outside of a compact subset of C.