  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 G including
some of the geometrically interesting classes of groups mentioned above. The
main results here extend results of Cohen and Prassidis for such groups. These
results imply that a homomorphism out of 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 F x Z, where F is the "top" free
group in the semidirect products of free groups and Z is the center of G. 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 Formanek and Procesi, poly-free groups whose Lie
algebras do not have certain properties considered here.
