Chen Lie algebras

Abstract

The Chen groups of a finitely-presented group G are the lower central series quotients of its maximal metabelian quotient, G/G''. The direct sum of the Chen groups is a graded Lie algebra, with bracket induced by the group commutator. If G is the fundamental group of a formal space, we give an analog of a basic result of D. Sullivan, by showing that the rational Chen Lie algebra of G is isomorphic to the rational holonomy Lie algebra of G modulo the second derived subalgebra. Following an idea of W.S. Massey, we point out a connection between the Alexander invariant of a group G defined by commutator-relators, and its integral holonomy Lie algebra. As an application, we determine the Chen Lie algebras of several classes of geometrically defined groups, including surface-like groups, fundamental groups of certain classical link complements, and fundamental groups of complements of complex hyperplane arrangements. For link groups, we sharpen Massey and Traldi's solution of the Murasugi conjecture. For arrangement groups, we prove that the rational Chen Lie algebra is combinatorially determined.

23 pages

References in corpus (3)

• Fundamental groups of line arrangements: Enumerative aspects
• Lower central series and free resolutions of hyperplane arrangements
• When does the associated graded Lie algebra of an arrangement group decompose?

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