Hyperbolic dimension of metric spaces
Hyperbolic dimension of metric spaces
复制标题
度量空间的双曲维数
DOI：
10.1090/s1061-0022-07-00986-7
复制
发表时间：
2004
影响因子：
0.8
通讯作者：
V. Schroeder
中科院分区：
文献类型：
--
作者：
S. Buyalo;V. Schroeder
我们引入了度量空间的一个新的拟等距不变量，称为双曲维，它是Gromov的渐近维asdim的一个版本。双曲维至多是渐近维，但与渐近维不同的是，任意欧氏空间R^n的双曲维为零(当R^n=n时)。这种不变量具有单调性和乘积定理等常见的量纲性质。我们的主要结果是：任何Gromov双曲空间X(有温和的限制)的双曲维度至少是无穷远处边界的拓扑维+1。作为应用，我们得到了实双曲空间H^n不存在拟等距嵌入到由任何欧氏因子稳定的度量树的(n-1)重度量积中。
We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however, unlike the asymptotic dimension, the hyperbolic dimension of any Euclidean space R^n is zero (while asdim R^n=n.) This invariant possesses usual properties of dimension like monotonicity and product theorems. Our main result says that the hyperbolic dimension of any Gromov hyperbolic space X (with mild restrictions) is at least the topological dimension of the boundary at infinity plus 1. As an application we obtain that there is no quasi-isometric embedding of the real hyperbolic space H^n into the (n-1)-fold metric product of metric trees stabilized by any Euclidean factor.