Counting Components of Hurwitz Spaces

For a finite group $G$, we describe the asymptotic growth of 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, as the number of branch points grows to infinity. More precisely, we compute both the exponent and (in many cases) the coefficient of the leading monomial in the count of components containing covers whose monodromy group is a given subgroup of $G$. By the work of Ellenberg, Tran, Venkatesh and Westerland, this asymptotic behavior is related to the distribution of field extensions of~$\mathbb{F}_q(T)$ with Galois group $G$.