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

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

2010-11-24

arxiv.org/abs/1011.5292v1

Mathematics

Group Theory

32 pages

Scientific paper

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.

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