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)