Calculating Gromov-Hausdorff distance by means of asymptotic dimension

Abstract

In this paper, we apply the concept of asymptotic dimension to calculating Gromov-Hausdorff distances between some unbounded metric spaces. For example, we show that the Gromov--Hausdorff between ${\mathbb{R}}^2$ with the Euclidean metric and ${\mathbb{Z}}^2$ equals the Hausdorff distance between them: $d_{GH}({\mathbb{R}}^2,{\mathbb{Z}}^2)=d_H({\mathbb{R}}^2,{\mathbb{Z}}^2)$.

4 pages, 2 figures, working paper