
Journal of Lie Theory 17 (2007), No. 2, 379397 Copyright Heldermann Verlag 2007 Centralizers of Lie Algebras Associated to Descending Central Series of Certain PolyFree Groups Daniel C. Cohen Dept. of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. cohen@math.lsu.edu Frederick R. Cohen Dept. of Mathematics, University of Rochester, Rochester, NY 14225, U.S.A. cohf@math.rochester.edu Stratos Prassidis Dept. of Mathematics, Canisius College, Buffalo, NY 14208, U.S.A. prasside@canisius.edu Polyfree groups are constructed as iterated semidirect products of free groups. The class of polyfree groups includes the classical pure braid groups, fundamental groups of fibertype hyperplane arrangements, and certain subgroups of the automorphism groups of free groups. The purpose of this article is to compute centralizers of certain natural Lie subalgebras of the Lie algebra obtained from the descending central series of polyfree groups Γ including some of the geometrically interesting classes of groups mentioned above. The main results here extend the result in F. R. Cohen and S. Prassidis ["On injective homomorphisms for pure braid groups and associated Lie algebras", J. Algebra 298 (2006) 363370] for such groups. These results imply that a homomorphism f from Γ to G is faithful, essentially, if it is faithful when restricted to the level of Lie algebras obtained from the descending central series for the product of F_{T} and Z, where F_{T} is the "top" free group in the semidirect products of free groups and Z is the center of Γ. The arguments use a mixture of homological, and Lie algebraic methods applied to certain choices of extensions. The limitations of these methods are illustrated using the "poison groups" of E. Formanek and C. Procesi ["The automorphism group of a free group is not linear", J. Algebra 149 (1992) 494499] polyfree groups whose Lie algebras do not have certain properties considered here. Keywords: polyfree group, descending central series, Lie algebra centralizer, McCool group, orbit configuration space, fibertype arrangement. MSC: 20E22, 20F14; 20F28, 20F36, 20F40, 32S22, 55R80 [ Fulltextpdf (240 KB)] for subscribers only. 