The volumes of strata of Abelian or quadratic differentials play an important role in the study of dynamics on flat surfaces, related to dynamics in polygonal billiards. This article reviews all known ways to compute volumes in the quadratic case and
provides explicit values of volumes of the strata of meromorphic quadratic differentials with at most simple poles in all low dimensions.
We present an explicit formula relating volumes of strata of meromorphicquadratic differentials with at most simple poles on Riemann surfacesand counting functions of the number of flat cylinders filled by closedgeodesics in associated flat metric wi
th singularities. This generalizes the resultof Athreya, Eskin and Zorich in genus 0 to higher genera.
An Abelian differential gives rise to a flat structure (translation surface) on the underlying Riemann surface. In some directions the directional flow on the flat surface may contain a periodic region that is made up of maximal cylinders filled by p
arallel geodesics of the same length. The growth rate of the number of such regions counted with weights, as a function of the length, is quadratic with a coefficient, called Siegel-Veech constant, that is shared by almost all translation surfaces in the ambient stratum. We evaluate various Siegel-Veech constants associated to the geometry of configurations of periodic cylinders and their area, and study extremal properties of such configurations in a fixed stratum and in all strata of a fixed genus.
Teichmuller curves play an important role in the study of dynamics in polygonal billiards. In this article, we provide a criterion similar to the original Mollers criterion, to detect whether a complex curve, embedded in the moduli space of Riemann s
urfaces and endowed with a line subbundle of the Hodge bundle, is a Teichmuller curve, and give a dynamical proof of this criterion.
