Let be a complex hyperplane arrangement, with fundamental group and holonomy Lie algebra . Suppose is a free abelian group of minimum possible rank, given the values the Möbius function takes on the rank flats of . Then the associated graded Lie algebra of decomposes (in degrees ) as a direct product of free Lie algebras. In particular, the ranks of the lower central series quotients of the group are given by , for . We illustrate this new Lower Central Series formula with several families of examples.
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Alexander I. Suciu, Stefan Papadima, When does the associated graded Lie algebra of an arrangement group decompose?. Comment. Math. Helv. 81 (2006), no. 4, pp. 859–875DOI 10.4171/CMH/77