Coloquio del IMUNAM - C.U.

Few results are know about the L2-Betti numbers of Aut(Fn) and Out(Fn), the groups of automorphisms (resp. outer automorphisms) of the free group Fn. Their virtual geometric dimension (smallest dimension of a K(G,1) for torsion-free finite index subgroups) are 2n−2, resp. 2n−3. I shall show that the top-dimensional L2-Betti numbers of Aut(Fn) and  Out(Fn) do not vanish.

By Lück approximation theorem, this implies that these groups admit finite index subgroups with non-vanishing top-dimensional  rational cohomology; in fact the usual Betti numbers for finite index subgroups grow linearly with the index.

I will review the basics of the theory and stay at an elementary level.