Spaces of Elliptic Differentials
نویسنده
چکیده
We study modular fibers of elliptic differentials, i.e. investigate spaces of coverings (Y, τ) → (C/Z ⊕ Zi, dz). For genus 2 torus covers with fixed degree we show, that the modular fibers Fd(1, 1) are connected torus covers with Veech group SL2(Z). Using results of Eskin, Masur and Schmoll [EMS] we calculate χ(Fd(1, 1)) and the parity of the spin structure of the quadratic differential (Fd(1, 1)/(− id), qd). We state and apply formulæ for the asymptotic quadratic growth rates of various types of geodesic segments on (Y, τ) ∈ Fd(1, 1). The quadratic growth rates are expressed in terms of the SL2(Z) orbit closure of (Y, τ) in Fd(1, 1) and the flat geometry of Fd(1, 1). These are extended notes from a talk the author gave during the Activity on Algebraic and Topological Dynamics at the Max-Planck-Institute for Mathematics, Bonn summer 2004.
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