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The goal of this paper is to describe all closed, aspherical Riemannian manifolds M whose universal covers M have a nontrivial amount of symmetry. By this we mean that Isom(M ) is not discrete. By the well-known theorem of Myers-Steenrod [MS], this condition is equivalent to [Isom(M ) : π1 (M )] = ∞. Also note that if any cover of M has a nondiscrete isometry group, then so does its universal cover M . Our description of such M is given in Theorem 1.2 below. The proof of this theorem uses methods from Lie theory, harmonic maps,. | Annals of Mathematics Isometries rigidity and universal covers By Benson Farb and Shmuel Weinberger Annals of Mathematics 168 2008 915 940 Isometries rigidity and universal covers By Benson Farb and Shmuel Weinberger 1. Introduction The goal of this paper is to describe all closed aspherical Riemannian manifolds M whose universal covers M have a nontrivial amount of symmetry. By this we mean that Isom M is not discrete. By the well-known theorem of Myers-Steenrod MS this condition is equivalent to Isom f 1 M 1. Also note that if any cover of M has a nondiscrete isometry group then so does its universal cover M. Our description of such M is given in Theorem 1.2 below. The proof of this theorem uses methods from Lie theory harmonic maps large-scale geometry and the homological theory of transformation groups. The condition that M have nondiscrete isometry group appears in a wide variety of problems in geometry. Since Theorem 1.2 provides a taxonomy of such M it can be used to reduce many general problems to verifications of specific examples. Actually it is not always Theorem 1.2 which is applied directly but the main subresults from its proof. After explaining in Section 1.1 the statement of Theorem 1.2 we give in Section 1.2 a number of such applications. These range from new characterizations of locally symmetric manifolds to the classification of contractible manifolds covering both compact and finite volume manifolds to a new proof of the Nadel-Frankel Theorem in complex geometry. 1.1. Statement of the general theorem. The basic examples of closed aspherical Riemannian manifolds whose universal covers have nondiscrete isometry groups are the locally homogeneous Riemannian manifolds M i.e. those M whose universal cover admits a transitive Lie group action whose isotropy subgroups are maximal compact. Of course one might also take a product of such a manifold with an arbitrary manifold. To find nonhomogeneous examples which are not products one can perform the .
