Hyperbolic dimension of metric spaces

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, $T_1\times\dots\times T_{n-1}\times\mathbb{R}^m$, $m\ge 0$.