Siegel-Veech Constants for Cyclic Covers of Generic Translation Surfaces

Abstract

We compute the asymptotic number of cylinders, weighted by their area to any non-negative power, on any cyclic branched cover of any generic translation surface in any stratum. Our formulas depend only on topological invariants of the cover and number-theoretic properties of the degree: in particular, the ratio of the related Siegel-Veech constants for the locus of covers and for the base stratum component is independent of the number of branch values. One surprising corollary is that this ratio for $area^3$ Siegel-Veech constants is always equal to the reciprocal of the the degree of the cover. A key ingredient is a classification of the connected components of certain loci of cyclic branched covers.