Isometry groups of simply connected manifolds of nonpositive curvature

Abstract

Let H denote a complete simply connected Riemannian manifold of nonpositive sectional curvature, and let I(H) denote the group of isometries of H. In this paper we consider density properties of subgroups D~_I(H) that satisfy the duality condition (defined below), These density properties also yield characterizations of Riemannian symmetric spaces of noncompact type and results about lattices in H that strengthen several of the results of [ 11 ] and [ 15]. If H is a symmetric space of noncompact type and if D is a subgroup of Io(H), then the duality condition for D is implied by the Selberg property (S) for D [20, pp. 4-6] or [10]. A partial converse is obtained in [10]. It is an interesting question whether the two conditions are equivalent in this context. Our density results are very similar to those of [5]. In Proposition 4.2 we obtain a differential geometric version of the Borel density theorem (cf. Corollary 4.2 of [5]): Let H admit no Euclidean de Rham factor, and let G~_I(H) be a subgroup whose normalizer D in I(H) satisfies the duality condition. Then either (1) G is discrete or (2) there exist manifolds Hi , / /2 such that (a) H is isometric to the Riemannian product HlXH2, (b) H1 is a symmetric space of noncompact type, (c) ((~)0=Io(Hl) and (d) there exists a discrete subgroup B~_I(Hz), whose normalizer in 1(//2) satisfies the duality condition, such that Io(HO• is a subgroup of t) of finite index in 0 . Using the result just quoted or the main theorem of section 3 we then obtain the following decomposition of a manifold H whose isometry group I(H) satisfies the duality condition (Proposition 4.1): Let I(H) satisfy the duality condition. Then there exist manifolds H0, Ht and H2, two of which may have dimension zero, such that (1) H is isometric to