Counting Components of Hurwitz Spaces

For a finite group $G$, we obtain asymptotics for the number of connected components of Hurwitz spaces of marked $G$-covers (of both the affine and projective lines) whose monodromy classes are constrained in a certain way, when the number of branch points grows to infinity. More precisely, we compute both the degree and (in many cases) the coefficient of the leading monomial in the count of components of marked $G$-covers whose monodromy group is a given subgroup of $G$. By the work of Ellenberg, Tran, Venkatesh and Westerland, these asymptotics are related to the distribution of field extensions of $\mathbb{F}_q(T)$ with Galois group $G$.