Moduli spaces of branched covers of Veech surfaces I: d-symmetric differentials

Details Moduli spaces of branched covers of Veech surfaces I: d-symmetric differentials Moduli spaces of branched covers of Veech surfaces I: d-symmetric differentials

2006-02-17

arxiv.org/abs/math/0602396v1

Mathematics

Geometric Topology

40 pages, 1 figure, submitted

Scientific paper

We give a description of asymptotic quadratic growth rates for geodesic segments on covers of Veech surfaces in terms of the modular fiber parameterizing coverings of a fixed Veech surface. To make the paper self contained we derive the necessary asymptotic formulas from the Gutkin-Judge formula. As an application of the method we define and analyze d-symmetric elliptic differentials and their modular fibers F^{sym}_d. For given genus g, g-symmetric elliptic differentials (with fixed base lattice) provide a 2-dimensional family of translation surfaces. We calculate several asymptotic constants, to establish their dependence on the translation geometry of F^{sym}_d and their sensitivity as SL(2,Z)-orbit invariants.

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