A Hurwitz space H_{G,n} is an algebraic variety parametrizing branched covers of the projective line with some fixed finite Galois group G. We will prove that, under some hypotheses on G, the rational i'th homology of the Hurwitz spaces stabilizes when the number of branch points is sufficiently large compared to i.

This purely topological theorem has some interesting number-theoretic consequences. It implies, for instance, a weak form of the Cohen-Lenstra conjectures over rational function fields, and some quantitative inverse Galois results over function fields. For instance, we show that the average size of the p-part of the class number of a hyperelliptic genus-g curve over F_q is bounded independently of g, when q is large enough relative to p; the key point here is q can be held fixed while g grows.

I will try to give a general overview of the dictionary between conjectures about topology of moduli spaces, on the one hand, and arithmetic distribution conjectures (Cohen-Lenstra, Bhargava, Malle, inverse Galois...) on the other.