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

TL;DR

This paper investigates the homological properties of the Johnson filtration in the automorphism group of a free group, revealing finiteness results and structural insights into associated invariants.

Contribution

It demonstrates the finiteness of the first Betti number for a key subgroup and explores the structure of the Alexander invariant and resonance varieties.

Findings

First Betti number of the second Johnson subgroup is finite

Alexander invariant is a non-trivial module over Laurent polynomial ring

First resonance variety of outer Torelli group is trivial

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.