度量空间的流与并
Flows and joins of metric spaces
摘要：
我们引入函子 *，它给每个度量空间X分配它的对称并 *X。作为一个集合，*X是连接X中的点的序偶对的区间的并集。在拓扑上，*X是X与自身的通常并的自然商。我们定义一个Isom（X）-不变度量d* on *X。对于任何双曲复形X，例如格罗莫夫双曲群的凯莱图，都可以精确地定义H^n和负曲流形的经典概念。在紧化X-bar= X并bdry X中定义了二重差分、交比和时函数.它们是连续的，Isom（X）不变的，并且满足尖锐的恒等式。我们刻画了Isom（X）中双曲等距g的平移长度.对任意双曲复形X，定义了X-棒的对称联 * X-棒及其上的（广义）度量d*.测地流空间F（X）是 *X-bar的一部分。(F(X)，d*）是单连通负曲流形上单位切丛（的全空间）的一个模拟.这个流空间被定义为任何双曲复形X，并且具有尖锐的性质。给出了两个度量空间的非对称并X*Y的构造。这些概念是规范的，即X中的函子，并且不涉及“准”语言。应用和关系的Borel猜想和其他人进行了讨论。
英文摘要：
We introduce the functor * which assigns to every metric space X its symmetric join *X. As a set, *X is a union of intervals connecting ordered pairs of points in X. Topologically, *X is a natural quotient of the usual join of X with itself. We define an Isom(X)-invariant metric d* on *X. Classical concepts known for H^n and negatively curved manifolds are defined in a precise way for any hyperbolic complex X, for example for a Cayley graph of a Gromov hyperbolic group. We define a double difference, a cross-ratio and horofunctions in the compactification X-bar= X union bdry X. They are continuous, Isom(X)-invariant, and satisfy sharp identities. We characterize the translation length of a hyperbolic isometry g in Isom(X). For any hyperbolic complex X, the symmetric join *X-bar of X-bar and the (generalized) metric d* on it are defined. The geodesic flow space F(X) arises as a part of *X-bar. (F(X),d*) is an analogue of (the total space of) the unit tangent bundle on a simply connected negatively curved manifold. This flow space is defined for any hyperbolic complex X and has sharp properties. We also give a construction of the asymmetric join X*Y of two metric spaces. These concepts are canonical, i.e. functorial in X, and involve no `quasi'-language. Applications and relation to the Borel conjecture and others are discussed.
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