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

TLDR: In this paper, the authors 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.

Abstract: 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.