Lipschitz precompactness for closed negatively curved manifolds

Abstract

We prove that, given a integer n ≥ 3 n\ge 3 and a group π \pi , the class of closed Riemannian n n -manifolds of uniformly bounded negative sectional curvatures and with fundamental groups isomorphic to π \pi is precompact in the Lipschitz topology. In particular, the class breaks into finitely many diffeomorphism types.