Let \(G\supseteq G' \supseteq G''\) denote the first terms of the derived series of a finitely-generated group \(G\). The Alexander module of \(G\) is the commutator subgroup \(G'/G''\) of \(G/G''\), a module over the group ring of \(G/G'=H_1(G)\). Following an observation of W. Massey, the Hilbert series of this module relative to its filtration by powers of the augmentation ideal determines the ranks \(\theta_q(G)\), \(q \geq 0\), of the factors in the lower central series of \(G/G''\), called the Chen ranks of \(G\).
In this context the authors introduced a graded module \(W(V,K)\), the Koszul module, associated to a \(n\)-dimensional complex vector space \(V\) and a subspace \(K\) of \(\bigwedge^2(V)\), in earlier work on Green’s conjecture about syzygies of algebraic curves [M. Aprodu et al., Invent. Math. 218, No. 3, 657–720 (2019; Zbl 1430.14074)]. For the application to Chen ranks, one takes \(V=H_1(G,{\mathbb C})\) and \(K\) to be the image of the dual of the cup product map \(\bigwedge^2(V^\vee) \to H^2(G,{\mathbb C})\). Then \(\theta_q(G) \leq \dim W_q(V,K)\) with equality if and only if \(G\) is a 1-formal group.
Using this approach, in the present paper the authors prove precise vanishing results and upper bounds for \(\theta_q(G)\) under the assumption that \(\theta_q(G)=0\) for \(q\gg 1\), that is, the lower central series factors of \(G/G''\) are finite abelian groups for \(q\) sufficiently large. This is equivalent to nonexistence of nonzero decomposable elements in \(K^\vee \subseteq \bigwedge^2(V^\vee)\), connecting the Koszul module to the resonance variety of \(G\). This condition is shown to be equivalent to the vanishing of \(W_{n-3}(V,K)\), where \(n=\dim(V)\), resulting in upper bounds on \(\theta_q(G)\) for \(1\leq q \leq n-4\) in terms of the rank of \(H_1(G)\), which become equalities for 1-formal groups. These results are then applied to determine growth rates and bounds on nilpotence classes of Alexander modules in several families of groups, including fundamental groups of certain compact Kähler manifolds and Torelli groups of mapping class groups and free groups.