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

Abstract

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: Ł_2\to \Z takes on the rank 2 flats of \A. Then the associated graded Lie algebra of G decomposes (in degrees 2 and higher) 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)=\sum_{X\in Ł_2} ϕ_r(F_{μ(X)}), for r\ge 2. We illustrate this new Lower Central Series formula with several families of examples.

14 pages, accepted for publication by Commentarii Mathematici Helvetici

References in corpus (3)

• Fundamental groups of line arrangements: Enumerative aspects
• Lower central series and free resolutions of hyperplane arrangements
• Algebraic invariants for Bestvina-Brady groups

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• Homology, lower central series, and hyperplane arrangements
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• Holonomy Lie algebras and the LCS formula for subarrangements of A_n
• The holonomy Lie algebra of a matroid
• The homotopy Lie algebra of a complex hyperplane arrangement is not necessarily finitely presented
• A survey of combinatorial aspects in the topology of complex hyperplane arrangements
• Conjugation-free groups, lower central series and line arrangements
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