BETTI NUMBER ESTIMATES FOR NILPOTENT GROUPS

We prove an extension of the following result of Lubotzky and Magid on the rational cohomology of a nilpotent group G: b₁ < ∞ and G ⊗ ℚ ≠ 0, ℚ, ℚ² then b₂ > b²₁. Here the b_i are the rational Betti numbers of G ⊗ ℚ denotes the Malcev completion of G. In the extension, the bound is improved when we know that all relations of G all have at least a certain commutator length. As an application of the refined inequality, we show that each closed oriented 3-manifold falls into exactly one of the following classes: it is a rational homology 3-sphere, or it is a rational homology S¹ × S², or it has the rational homology of one of the oriented circle bundles over the torus (which are indexed by an Euler number n ε ℤ, e.g. n = 0 corresponds to the 3-torus) or it is of general type by which we mean that the rational lower central series of the fundamental group does not stabilize. In particular, any 3-manifold group which allows a maximal torsion-free nilpotent quotient admits a rational homology isomorphism to a torsion-free nilpotent group.