Hurwitz spaces and the asymptotic Picard rank conjecture

I will explain work joint with Aaron Landesman where we prove that for a finite group G and conjugacy invariant subset c, Hurwitz spaces parameterizing connected G-covers of the complement of a configuration of points on a disk with monodromy in c satisfy homological stability. We use this to prove that the Hurwitz stack parameterizing simply branched degree d covers of P^1 with n branch points has trivial rational Picard group when n is much larger than d.