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

Abstract

Let script A sign be a complex hyperplane arrangement, with fundamental group G and holonomy Lie algebra script h sign. Suppose script h sign 3 is a free abelian group of minimum possible rank, given the values the Möbius function μ: ℒ2 → ℤ takes on the rank 2 flats of script A sign. 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 φr(G) = ∑X∈ℒ2 φr (Fμ(X)(X)), for r ≥ 2. We illustrate this new Lower Central Series formula with several families of examples. © Swiss Mathematical Society.