Hyperbolic dimension of metric spaces

Abstract:

A new quasi-isometry invariant of metric spaces, called the hyperbolic dimension ($\operatorname {hypdim}$) is introduced; this is a version of Gromov’s asymptotic dimension ($\operatorname {asdim}$). The inequality $\operatorname {hypdim}\le \operatorname {asdim}$ is always fulfilled; however, unlike the asymptotic dimension, $\operatorname {hypdim}\mathbb {R}^n=0$ for every Euclidean space $\mathbb {R}^n$ (while $\operatorname {asdim}\mathbb {R}^n=n$). This invariant possesses the usual properties of dimension such as monotonicity and product theorems. The main result says that the hyperbolic dimension of any Gromov hyperbolic space $X$ (under mild restrictions) is at least the topological dimension of the boundary at infinity plus 1, $\operatorname {hypdim} X\ge \dim \partial _{\infty }X+1$. As an application, it is shown that there is no quasi-isometric embedding of the real hyperbolic space $\operatorname {H}^n$ into the metric product of $n-1$ metric trees stabilized by any Euclidean factor, $T_1\times \dots \times T_{n-1}\times \mathbb {R}^m$, $m\ge 0$.