Amenable isometry groups of Hadamard spaces
Amenable isometry groups of Hadamard spaces
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哈达玛空间的顺应等距群
DOI：
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
W. Ballmann
中科院分区：
文献类型：
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作者：
S. Adams;W. Ballmann
非正弯曲空间理论中不断出现的主题之一是几何与拓扑之间的关系。由于非正曲率完备空间的通用覆盖空间是可收缩的，因此这相当于该空间（或其通用覆盖空间）的几何形状与其基本群的代数结构之间的关系。一个范例是 Avez [Av] 的众所周知的结果，即非正截面曲率的紧致黎曼流形 M 的基本群当且仅当 M 不平坦时才具有指数增长。格罗莫夫观察到，阿韦兹的证明产生了一个更强的陈述：当且仅当 M 是平坦的时，M 的基本群才成立，参见 [G1, p. 11]。 93]。 Zimmer 将其推广到 M 是完整且有限体积 [Zi] 的情况。 Anderson [An] 以及 Burger 和 Schroeder [BS] 获得了进一步的概括。本文的目的是在较弱的假设下讨论这些结果，即基础空间是非正曲率的 Alexandrov 空间。为了研究基本群与非正曲率空间几何之间的关系，可以方便地研究基本群在通用覆盖空间上的等距作用。在许多情况下，没有必要假设群作用以这种特定方式出现，并且可以研究非正曲率的完全单连通空间上的等距作用。我们下面的大多数论点都是这种情况。设X为Hadamard空间，即Alexandrov意义上的非正曲率的完全单连通测地线空间。 X 中的平面是
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