On fundamental groups of manifolds of nonnegative curvature

Abstract

We will characterize the fundamental groups of compact manifolds of (almost) nonnegative Ricci curvature and also the fundamental groups of manifolds that admit bounded curvature collapses to manifolds of nonnegative sectional curvature. Actually it turns out that the known necessary conditions on these groups are sufficient as well. Furthermore, we reduce the Milnor problem—are the fundamental groups of open manifolds of nonnegative Ricci curvature finitely generated?—to manifolds with abelian fundamental groups. Moreover, we prove for each positive integer n that there are only finitely many non-cyclic, finite, simple groups acting effectively on some complete n -manifold of nonnegative Ricci curvature. Finally, sharping a result of Cheeger and Gromoll [6], we show for a compact Riemannian manifold (M,g 0) of nonnegative Ricci curvature that there is a continuous family of metrics (g λ),λ∈[0,1] such that the universal covering spaces of (M,g λ) are mutually isometric and (M,g 1) is finitely covered by a Riemannian product N×T d , where T d is a torus and N is simply connected.