Homological stability for Hurwitz spaces and applications

We show the homology of the Hurwitz space associated to an arbitrary finite rack stabilizes integrally in a suitable sense. We also compute the dominant part of its stable homology after inverting finitely many primes. This proves a conjecture of Ellenberg–Venkatesh–Westerland and improves upon our previous results for non-splitting racks. We obtain applications to Malle’s conjecture, the Picard rank conjecture, and the Cohen–Lenstra–Martinet heuristics.