Topology of open nonpositively curved manifolds

Abstract

Abstract This is a survey of topological properties of open, complete nonpositively curved manifolds which may have infinite volume. Topics include topology of ends, restrictions on the fundamental group, as well as a review of known examples. Among many monographs and surveys on aspects of nonpositive curvature, none deals with topology of open complete nonpositively curved manifolds, and this paper aims to fill the void. Most of the material discussed here is not widely-known. A number of questions is posed, ranging from naive to hopelessly difficult. Proofs are supplied when there is no explicit reference, and as always, the author is solely responsible for mistakes. This survey has a narrow focus and does not discuss topological properties of • hyperbolic 3-manifolds [161, 112, 51], • open negatively pinched manifolds [19, 26], • non-Riemannian nonpositively curved manifolds [61], • compact nonpositively curved manifolds [78, 124], • higher rank locally symmetric spaces [125, 73, 28, 87] and their compactifications [35], which are covered in the above references. We choose to work in the Riemannian setting which leads to some simplifications, even though many results hold in a far greater generality, and the references usually point to the strongest results available. Special attention is given to rank one manifolds, such as manifolds of negative curvature. Nonpositively curved manifolds are aspherical so we focus on groups of finite cohomological dimension (i.e. fundamental groups of aspherical manifolds), or better yet, groups of type F (i.e. the fundamental groups of compact aspherical manifolds with boundary). Conventions: unless stated otherwise, manifolds are smooth, metrics are Riemannian, and sectional curvature is denoted by K . Acknowledgments: The author is grateful for NSF support (DMS- 1105045). Thanks are due to Jim Davis, Denis Osin, Yunhui Wu, and the referee for correcting misstatements in the earlier version. Flavors of negative curvature A Hadamard manifold is a connected simply-connected complete manifold of K ≤ 0. By the Cartan-Hadamard theorem, any Hadamard manifold is diffeomorphic to a Euclidean space. Thus any complete manifold of K ≤ 0 is the quotient of a Hadamard manifold by a discrete torsion-free isometry group (torsion-freeness can be seen geometrically: any finite isometry group of a Hadamard manifold fixes the circumcenter of its orbit, or topologically: a nontrivial finite group has infinite cohomological dimension so it cannot act freely on a contractible manifold).