Hatcher and Wahl conjectured that automorphism groups Aut(H*G*...*G) of free products of groups are homologically stable.  This conjecture was based on their proof of the same result for groups G and H arising as fundamental groups of 3-manifolds.  Their proof was highly geometric and so does not apply for general G and H.

 

After explaining what all of this means, I'll describe a proof for the general case which uses a moduli space of vertex labelled graphs and some functor homology to prove that stability results for carefully chosen G and H can be lifted to stability results for all G and H.  Hatcher and Wahl's work covers the chosen case and so the conjecture is proved.