Polynomial stability of the homology of Hurwitz spaces

For a finite group G and a conjugation-invariant subset \(Q\subseteq G\) $Q\subseteq G$ , we consider the Hurwitz space \(\textrm{Hur}_n(Q)\) ${\text{Hur}}_n(Q)$ parametrising branched covers of the plane with n branch points, monodromies in G and local monodromies in Q. For \(i\ge 0\) $i\ge 0$ we prove that \(\bigoplus _n H_i(\textrm{Hur}_n(Q))\) ${⨁}_n{H}_i({\text{Hur}}_n(Q))$ is a finitely generated module over the ring \(\bigoplus _n H_0(\textrm{Hur}_n(Q))\) ${⨁}_n{H}_0({\text{Hur}}_n(Q))$ . As a consequence, we obtain polynomial stability of homology of Hurwitz spaces: taking homology coefficients in a field, the dimension of \(H_i(\textrm{Hur}_n(Q))\) $H_i({\text{Hur}}_n(Q))$ agrees for n large enough with a quasi-polynomial in n, whose degree is easily bounded in terms of G and Q. Under suitable hypotheses on G and Q, we prove classical homological stability for certain sequences of components of Hurwitz spaces. Our results generalise previous work of Ellenberg–Venkatesh–Westerland, and rely on techniques introduced by them and by Hatcher–Wahl.