Rigidity and finiteness under Ricci curvature and volume controls

Abstract.

Consider the class of closed Riemannian manifolds M of dimension $\dim(M) \geqq 3$, Ricci curvature $\textrm{Ric}(M) \geqq -(n - 1)$, diameter diam(M) < D and almost maximal volume. We show that the isomorphism types of fundamental groups characterize the diffeomorphism types of manifolds in such a class. In particular, it can be viewed as a generalization of the well-known Mostow‚s rigidity theorem and a finiteness theorem.