Rigidity of nonpositively curved graphmanifolds

Abstract

There are strong relations between the metric structure of a complete Riemannian manifold M of nonpositive sectional curvature and the algebraic structure of its fundamental group. In particular, we have the following rigidity results for compact M: The fundamental group ~I(M) determines whether the universal covering space X of M is a nontrivial Riemannian product or a symmetric space of higher rank. The product case follows from the work of Lawson-Yau [LY] and GromollWolf [GW] combined with the results in [E 1]. The statement is also true, i fM is complete and has finite volume IS]. The case of symmetric spaces is stated in [BE]. It is based on the rigidity results of Mostow ['M], Gromov [BGS], and Eberlein [E2] and the investigation of higher rank manifolds by Ballmann, Brin, Burns, Eberlein, and Spatzier [BBE], ['BBS], [B]. This result also holds in the finite volume case. These examples, products and higher rank symmetric spaces have rank > 2 in the sense of ['BBE]. On the other hand, manifolds of higher rank are products or symmetric spaces [B]. Thus the above rigidity results reflect the rigidity properties of higher rank spaces. In this paper we study the rigidity of a special class of rank 1 manifolds, namely the 3-dimensional graphmanifolds of non-positive curvature. Our main result says, that the whole metric structure of nonpositively curved graphmanifolds is determined by the fundamental group (see Theorem 2 in Sect. 4 for a precise statement). We should say a few words about the relevance of nonpositively curved graphmanifolds. Waldhausen ['W] studied graphmanifolds from a topological viewpoint. Gromov [G] mentioned, that a special class of graphmanifolds allow complete Riemannian metrics of nonpositive curvature. In this way, he obtained the first examples of complete Riemannian manifolds with bounded nonpositive curvature a 2 < K < 0 and finite volume with infinite topological type.