On the Gromov-Hausdorff distance between the cloud of bounded metric spaces and a cloud with nontrivial stabilizer
B. A. Nesterov
TL;DR
The paper extends the Gromov-Hausdorff framework to clouds—equivalence classes of unbounded metric spaces at finite $d_{GH}$—and introduces the stabilizer and center concepts to study how these clouds transform under scaling. It proves that all clouds are proper classes and develops a center-image bound: the image of $\Delta_1$ under finite-distortion correspondences to a cloud with nontrivial stabilizer must lie near the center, leading to infinite cloud-to-cloud distances under certain conditions. It further shows that the ultrametric inequality holds in the $[\Delta_1]$ cloud but can fail in clouds containing $\mathbb{R}$, and establishes a dichotomy for distances between clouds that share stabilizers, with the concrete result that the cloud containing the real line lies at infinite distance from the cloud of bounded metric spaces. These results illuminate the geometric and set-theoretic structure of clouds and highlight how stabilizers govern inter-cloud distances.
Abstract
The paper studies the class of all metric spaces considered up to zero Gromov-Hausdorff distance between them. In this class, we examine clouds - classes of spaces situated at finite Gromov-Hausdorff distances from a reference space. We prove that all clouds are proper classes. The Gromov-Hausdorff distance is defined for clouds similarly with the case of that for metric spaces. A multiplicative group of transformations of clouds is defined which is called stabilizer. We show that under certain restrictions the distance between the cloud of bounded metric spaces and a cloud with a nontrivial stabilizer is finite. In particular, the distance between the cloud of bounded metric spaces and the cloud containing the real line is calculated.