Hurwitz spaces and braid group representations

Abstract

We give a new construction of a Hurwitz space, which is a moduli space of all branched covers of the Riemann sphere having a given combinatorial description. The action of the fundamental group of the Hurwitz space on the homology of the branched cover gives rise to a linear represenation of a finite index subgroup of the spherical braid group, or of a finite extension of such a subgroup. We construct examples of each of these two cases. Using a result of Fried, we use these representations to extract information about the dimension of the image of the Hurwitz space in the genus g moduli space. © 2004 Rocky Mountain Mathematics Consortium.