Isometries, rigidity and universal covers

The goal of this paper is to describe all closed, aspherical Riemannian manifolds M whose universal covers M̃ have have a nontrivial amount of symmetry. By this we mean that Isom(M̃) is not discrete. By the well-known theorem of Myers-Steenrod [MS], this condition is equivalent to [Isom(M̃) : π1(M)] = ∞. Also note that if any cover of M has a nondiscrete isometry group, then so does its universal cover M̃ . 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 M̃ 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 §1.1 the statement of Theorem 1.2, we give in §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.