Fundamental groups of moduli spaces of quadratic differentials

Kontsevich and Zorich have conjectured that the fundamental groups of strata of quadratic differentials with zeroes of fixed orders are commensurable to mapping class groups. In this thesis we will consider that conjecture. To do so, we begin by working with strata of quadratic differentials over Teichmuller space and considering such a stratum as a subspace of some larger configuration space of points over surfaces in Teichmuller space. We show that the image of fundamental groups of strata in the fundamental group of the larger configuration space is in the kernel of the Abel-Jacobi map. We then construct a set of generators for the kernel of the Abel-Jacobi map and show that if sufficiently many of the zeroes are of equal weight then the kernel of the Abel-Jacobi map is equal to the image of the fundamental group of the stratum in the fundamental group of the configuration space. Finally we relate the fundamental group of a stratum of quadratic differentials over Teichmuller space to the fundamental group of the same stratum over moduli space by considering the connected components of both of them.