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

Let A be a complex hyperplane arrangement, with fundamental group G and holonomy Lie algebra H . Suppose H_3 is a free abelian group of minimum possible rank, given the values the Möbius function μ L_2→ Z takes on the rank 2 flats of A . Then the associated graded Lie algebra of G decomposes (in degrees ≥ 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 φ_r(G)=∑ X∈ L_2 φ_r(Fμ(X)) , for r≥ 2 . We illustrate this new Lower Central Series formula with several families of examples.