We define an infinite sequence of new invariants, $delta_n$, of agroup G that measure the size of the successive quotients of the derivedseries 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 relatedto the topology of the 3-manifold. They give lower bounds for theThurston norm which provide better estimates than the bound establishedby McMullen using the Alexander norm. We also show that the $delta_n$ giveobstructions to a 3-manifold fibering over $S^1$ and to a 3-manifold beingSeifert fibered. Moreover, we show that the $delta_n$ give computablealgebraic obstructions to a 4-manifold of the form $X x S^1$ admitting asymplectic structure even when the obstructions given by theSeiberg-Witten invariants fail. There are also applications to theminimal ropelength of knots and links in $S^3$. In addition, wediscuss the applications to the cut number of a 3-manifold (this is alsoknown as the corank of the fundamental group of the 3-manifold)