Journal of Lie Theory 17 (2007), No. 2, 379--397
Copyright Heldermann Verlag 2007
Centralizers of Lie Algebras Associated to Descending Central Series of Certain Poly-Free Groups
Daniel C. Cohen
Dept. of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.
Frederick R. Cohen
Dept. of Mathematics, University of Rochester, Rochester, NY 14225, U.S.A.
Dept. of Mathematics, Canisius College, Buffalo, NY 14208, U.S.A.
Poly-free groups are constructed as iterated semidirect products of free groups. The class of poly-free groups includes the classical pure braid groups, fundamental groups of fiber-type 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 poly-free 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) 363--370] 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 FT and Z, where FT 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) 494--499] poly-free groups whose Lie algebras do not have certain properties considered here.
Keywords: poly-free group, descending central series, Lie algebra centralizer, McCool group, orbit configuration space, fiber-type arrangement.
MSC: 20E22, 20F14; 20F28, 20F36, 20F40, 32S22, 55R80
[ Fulltext-pdf (240 KB)] for subscribers only.