Calculating Gromov-Hausdorff distance by means of asymptotic dimension

TL;DR

The paper tackles the challenge of computing Gromov–Hausdorff distances between unbounded metric spaces by leveraging asymptotic dimension and stabilizers. It develops a main theorem that yields a lower bound $d_{GH}(A,X) \ge \frac{r}{2}$ under asdim constraints and a nontrivial stabilizer, and uses a contradiction argument via small-distortion correspondences. As an application, it shows $d_{GH}(\mathbb{R}^2, \mathbb{Z}^2) = d_H(\mathbb{R}^2, \mathbb{Z}^2)$ by constructing explicit $\varepsilon$-nets and employing a chess-coloring argument to bound distortions, along with a concrete subset $A$ achieving the same distance. The results provide a concrete, geometry-driven route to exact GH distances in certain unbounded settings and suggest a framework for analyzing similar pairs via nets and disjoint coverings.

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: $d_{GH}(\mathbb{R}^2, \mathbb{Z}^2) = d_H(\mathbb{R}^2, \mathbb{Z}^2)$.

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Theorems & Definitions (17)

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