Aspects of positively Ricci curved spaces: New examples and the fundamental group

For a simply connected nilpotent Lie group $L$, we construct a complete metric with positive Ricci curvature on the product manifold $L\times\IR\sp{p}$, where $p$ is taken sufficiently large. The construction uses a warped product method and involves subtle choices of functions. We endow $L$ with a family of almost flat metrics, and the little "negativeness" of $L$ can be compensated by warping the euclidean $\IR\sp{p}$ factor. From the construction one also sees that the isometry group of the resulting manifold contains the original group $L$.

A basic consequence of this construction is that every finitely generated torsion-free discrete nilpotent group can be realized as the fundamental group of a complete manifold with positive Ricci curvature.

We also establish an a priori bound on the growth of the fundamental group for a class of compact near elliptic manifolds (in the sense of Gromov) whose volume is uniformly bounded from below.