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Isometry groups of simply connected manifolds of nonpositive curvature

1980; Duke University Press; Volume: 24; Issue: 1 Linguagem: Inglês

10.1215/ijm/1256047798

1945-6581

Su‐Shing Chen, Patrick Eberlein,

Geometric and Algebraic Topology

IntroductionLet H be a complete simply connected Riemannian manifold of nonpositive sectional curvature, and let D be a subgroup of I(H), the group of isometries of H. Let H(c)denote the set ofpoints at infinity for H(Section 1).In this paper we consider subgroups D that satisfy the duality condition and investigate the effects on and relationships between the algebraic structure of D, the structure of the orbits of D in H() and the geometry of H or HID if the latter is a smooth manifold.The idea of aflat point in H(c) (Section 3) plays an important part in this investigation.In the context of homogeneous or symmetric spaces it is interesting to ask if the duality condition on isometry groups and the idea of fiat points at infinity can be related to other properties of such spaces that have been studied.Heintze [18] has shown that if H is a symmetric space and if D Io(H)satisfies the Selberg property (S), then D satisfies the duality condition.It is unknown under what conditions the converse is true.The description of fiat points at infinity is trivial if H is symmetric (Section 3) but has not been considered if H is homogeneous but not symmetric.One may hope that the methods of Azencott-Wilson [2], [3] can provide such a description.A subgroup D _ I(H) satisfies the duality condition if for every geodesic of n there exists a sequence {qb,} _ D such that for any point p of H, b,(p) converges to (oo) and b-l(p) converges to y(-oo)(see Section 1 for definitions).If M HID is a smooth manifold, then D satisfies the duality condition if and only if every vector in SM, the unit tangent bundle of M, is nonwandering relative to the geodesic flow.In particular D satisfies the duality condition if HID is a smooth manifold that is either compact or has finite volume.The duality condition may appear at first glance to be a fairly mild restric- tion, but actually it is quite a strong one.For example, if H is a homogeneous space, then the full isometry group I(H) satisfies the duality condition if and only if H is the Riemannian product of a Euclidean space or line H1 and a