Stability of lower central series of compact 3-manifold groups

The length of a group G is the least ordinal alpha such that G(alpha) = G(alpha+1) where G(alpha) is the alpha th term of the transfinite lower central series. We begin by establishing connections between lower central series length and the Parafree Conjecture, four-dimensional topological surgery, and link concordance. We prove that the length of all surface groups and most Fuchsian groups is at most omega. We show that the length of the group a Seifert fibration over a base of non-positive even Euler characteristic is at most omega. Our major result is the existence of closed hyperbolic 3-manifolds with length at least 2 omega. We observe that any closed orientable 3-manifold group has the same lower central series quotients as a hyperbolic one. (C) 1997 Elsevier Science Ltd. All rights reserved.