On non-vanishing of the cohomology of \(\mathrm{Aut}(F_n)\) and \(\mathrm{Out}(F_n)\) in top dimensions

Few results are know about the \(L^2\)-Betti numbers of \(\mathrm{Aut}(F_n)\) and \(\mathrm{Out}(F_n)\), the groups of automorphisms (resp. outer automorphisms) of the free group \(F_n\). 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 \(L^2\)-Betti numbers of \(\mathrm{Aut}(F_n)\) and  \(\mathrm{Out}(F_n)\) 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.