Title: 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 $\mu\colon \mathcal{L}\_2\to \mathbb{Z}$ 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 $\phi\_r(G)=\sum \_{X\in \mathcal{L}2} \phi\_r(F{\mu(X)})$, for $r\ge 2$. We illustrate this new Lower Central Series formula with several families of examples.