Homological finiteness in the Johnson filtration of the automorphism group of a free group

Abstract

We examine the Johnson filtration of the (outer) automorphism group of a finitely generated group. In the case of a free group, we find a surprising result: the first Betti number of the second subgroup in the Johnson filtration is finite. Moreover, the corresponding Alexander invariant is a non-trivial module over the Laurent polynomial ring. In the process, we show that the first resonance variety of the outer Torelli group of a free group is trivial. We also establish a general relationship between the Alexander invariant and its infinitesimal counterpart.