$\alpha$-Siegel-Veech constants for branched cyclic covers of tori

Martin Schmoll – Clemson University

$\alpha$-Siegel-Veech constants are roughly speaking the quadratic asymptotic constants for weighted counts of the number of geodesics below a given length on flat Riemann surfaces. Here the geodesics length is the parameter with respect to which the asymptotic is taken. Each geodesic loop is weighted by the area of the strip carved out by loops isotopic to the given one, exponentiated with the parameter $\alpha$. This uses the fact, that on the flat surfaces considered closed loops appear in families defining euclidean cylinders. For $\alpha=0$ one has the standard quadratic growth rate of cylinders and for $\alpha=1$ one has the so called area-Siegel-Veech constant related to the sum of Lyapunov exponents of the Kontsevich-Zorich cocycle. The asymptotic constants are normalized, so that to the quadratic growth rates of a unit area flat torus have $\alpha$-Siegel-Veech constants 1.

For (generic) branched cyclic covers of tori with given branching we present a general formula for $\alpha$-Siegel-Veech constants. The formula has arithmetic properties depending only on the degree of the cover and a number we call monodromy factor of the cover. In fact, the formula only depends on a unique decomposition of the covering degree determined by the monodromy factors’ primes and does not depend on any other branching data. Most surprising the area Siegel-Veech constant is 1, if the monodromy factors’ prime factorization contains the same primes as the factorization of the covers degree. This phenomenon was observed through computer experiments and generally conjectured by David Aulicino, who is the coauthor of the research presented.