Homological stability for Hurwitz spaces and the Cohen-Lenstra   conjecture over function fields, II
  We prove a version of the Cohen--Lenstra conjecture over function fields (completing the results of our prior paper). This is deduced from two more general theorems, one topological, one arithmetic: We compute the direct limit of homology, over puncture-stabilization, of spaces of maps from a punctured manifold to a fixed target; and we compute the Galois action on the set of stable components of Hurwitz schemes. 
Jordan S. Ellenberg, Akshay Venkatesh, and Craig Westerland
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