Amenable isometry groups of Hadamard spaces

Abstract

One of the ever recurring themes in the theory of nonpositively curved spaces is the relation between geometry and topology. Since the universal covering space of a complete space of nonpositive curvature is contractible, this amounts to relations between the geometry of the space (or its universal covering space) and the algebraic structure of its fundamental group. A paradigm is the well-known result of Avez [Av] that the fundamental group of a compact Riemannian manifold M of nonpositive sectional curvature has exponential growth if and only if M is not flat. Gromov observed that the proof of Avez yields a somewhat stronger statement: the fundamental group of M is amenable if and only if M is flat, see [G1, p. 93]. Zimmer generalized this to the case where M is complete and of finite volume [Zi]. Further generalizations were obtained by Anderson [An] and Burger and Schroeder [BS]. The objective of this paper is the discussion of these results under the weaker assumption that the underlying spaces are Alexandrov spaces of nonpositive curvature. For the investigation of relations between the fundamental group and the geometry of a space of nonpositive curvature, it is convenient to study the isometric action of the fundamental group on the universal covering space. In many situations, it is not necessary to assume that a group action arises in this particular way and one studies isometric actions on complete simply connected spaces of nonpositive curvature. This is the case in most of our arguments below. Let X be a Hadamard space, that is, a complete simply connected geodesic space of nonpositive curvature in the sense of Alexandrov. A flat in X is a