Hyperbolic dimension of metric spaces

TL;DR

The paper introduces the hyperbolic dimension, a new quasi-isometry invariant for metric spaces, which differs from asymptotic dimension by assigning zero to Euclidean spaces and relates to boundary dimensions in hyperbolic spaces.

Contribution

It defines the hyperbolic dimension, explores its properties, and establishes its relation to boundary dimensions in hyperbolic spaces, providing new insights into metric space invariants.

Findings

Hyperbolic dimension is at most the asymptotic dimension.

Hyperbolic dimension of Euclidean space R^n is zero.

Hyperbolic dimension of Gromov hyperbolic space exceeds boundary topological dimension by one.

Abstract

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.