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International Mathematics Research Notices (2004) 2004:1057-1086, doi:10.1155/S1073792804132017
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Copyright © 2004 Hindawi Publishing Corporation. All rights reserved.

Chen Lie algebras

Stefan Papadima and Alexander I. Suciu

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 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 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 link complements in S3, and fundamental groups of complements of hyperplane arrangements in C{ell}. 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.


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