Siegel-Veech Constants for Cyclic Covers of Generic Translation Surfaces

TL;DR

This paper derives formulas for counting cylinders on cyclic covers of translation surfaces, revealing invariants and ratios independent of branch points, with implications for understanding geometric structures.

Contribution

It provides explicit formulas for Siegel-Veech constants on cyclic covers, linking topological and number-theoretic properties, and classifies connected components of related loci.

Findings

The ratio of Siegel-Veech constants for covers and base surfaces depends only on topological invariants.

The ratio for area^3 constants equals the reciprocal of the cover degree.

Formulas depend solely on topological invariants and number-theoretic properties.

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 degree of the cover. A key ingredient is a classification of the connected components of certain loci of cyclic branched covers.