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 $X \times S^1$ 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 $S^3$.