Quotient Groups of the Fundamental Groups of Certain Strata of the Moduli Space of Quadratic Differentials

In [6], Kontsevich and Zorich conjecture that the fundamental groups of strata of the moduli space of abelian differentials are commensurable with various mapping class groups. In this paper we consider a similar question for strata of quadratic differentials, and in particular we construct a quotient group of the fundamental group for a certain family of strata. We do so by mapping a stratum into a larger configuration space of points on surfaces, and showing that the image of the fundamental group of the stratum under this map is in the kernel of a version of the Abel-Jacobi map. We then construct a set of generators for the kernel of the Abel-Jacobi map, and show that in some cases the image of the fundamental group of the stratum in the fundamental group of the configuration space is equal to this kernel.