When does the associated graded Lie algebra of an arrangement group decompose?

Abstract

Let $\mathcal{A}$ be a complex hyperplane arrangement, with fundamental group $G$ and holonomy Lie algebra $\mathfrak{H}$. Suppose ${\mathfrak{H}}_3$ is a free abelian group of minimum possible rank, given the values the Möbius function $\mu :{\mathcal{L}}_2\to \mathbb{Z}$ takes on the rank $2$ flats of $\mathcal{A}$. Then the associated graded Lie algebra of $G$ decomposes (in degrees $\ge 2$) as a direct product of free Lie algebras. In particular, the ranks of the lower central series quotients of the group are given by ${\varphi }_r(G)={\sum }_{X\in {\mathcal{L}}_2}{\varphi }_r(F_{\mu (X)})$, for $r\ge 2$. We illustrate this new Lower Central Series formula with several families of examples.