Spaces of elliptic differentials
Martin Schmoll
Abstract
We study modular fibers of elliptic differentials, which are roughly spaces of torus-coverings over a fixed base torus. For genus 2 torus covers with fixed degree we show, that the modular fibers Fd(1,1) are itself connected torus covers with Veech group SL2(Z). Using results of Eskin, Masur and Schmoll we calculate the Euler Characteristic and the parity of the spin structure of the quadratic differential (Fd(1,1)/(-id),qd). We state and apply formulas for the asymptotic quadratic growth rates of various types of geodesic segments on a surface S "contained" in Fd(1,1). The quadratic growth rates are expressed in terms of the SL2(Z) orbit closure of S ∈ Fd(1,1) and the flat geometry of Fd(1,1).
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