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 moreover compute the dominant part of the stable homology after inverting finitely many primes. This has applications to Malle’s conjecture over function fields, the Cohen—Lenstra—Martinet heuristics over function fields, as well as to the Picard rank conjecture.
Dustin Clause (IHES)
