For any group \(G\), one has a graded Lie algebra \(\text{gr}\,G=\oplus\Gamma_kG/\Gamma_{k+1} G\), where \(\Gamma_\bullet G\) is the lower central series. When \(X\) is a connected CW-complex (with finite 2-skeleton), Chen introduced a Lie algebra \({\mathcal H}(X;\mathbb Q)=L(H_1(X;\mathbb Q))/{\mathcal R}\), where \({\mathcal R}\) is generated by the dual of the cup product. The Lie algebra \({\mathcal H}(X;\mathbb Q)\) corresponds to the case \(X=K(G,1) \) if \(G\) is a finitely presented group.
In the paper under review, the authors generalize previous results of Chen and prove:
\(\text{gr}(G/G^{(i)})\otimes \mathbb Q\cong {\mathcal H}(G;\mathbb Q)/{\mathcal H}^{(i)}(G;\mathbb Q)\), if \(i\geq 2\) and \(G^{(i)}\) denotes the derived series of \(G\). A version over the integer exists also if \(G/G^{(1)}\) and \({\mathcal H}(G)/{\mathcal H}(G)^{(i)}\) are torsion free for \(i\geq 1\). Applications to linking numbers and arrangements of hyperplanes are given.