Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

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We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This has the following arithmetic consequence: let l>2 be prime and A a finite abelian l-group. Then there exists Q = Q(A) such that, for q greater than Q and not congruent to 1 modulo l, a positive fraction of quadratic extensions of Fq(t) have the l-part of their class group isomorphic to A.