  We define an infinite sequence of new invariants, delta_n, of a group G that
measure the size of the successive quotients of the derived series of G. In the
case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold
invariants. These invariants are closely related to the topology of the
3-manifold. They give lower bounds for the Thurston norm which provide better
estimates than the bound established by McMullen using the Alexander norm. We
also show that the delta_n give obstructions to a 3-manifold fibering over S^1
and to a 3-manifold being Seifert fibered. Moreover, we show that the delta_n
give computable algebraic obstructions to a 4-manifold of the form X x S^1
admitting a symplectic structure even when the obstructions given by the
Seiberg-Witten invariants fail. There are also applications to the minimal
ropelength and genera of knots and links in S^3.
