Gromov-Hausdorff distances between quotient metric spaces

TL;DR

This paper investigates how the Gromov-Hausdorff distance between metric spaces behaves under quotient operations by group actions, revealing that unlike Hausdorff distance, it can vary arbitrarily relative to the original spaces.

Contribution

The study demonstrates that the Gromov-Hausdorff distance between quotient spaces can be arbitrarily scaled compared to the original spaces, highlighting fundamental differences from the Hausdorff distance.

Findings

Hausdorff distance is preserved under quotients for G-invariant subsets.

Gromov-Hausdorff distance can be arbitrarily scaled between quotient and original spaces.

The ratio of Gromov-Hausdorff distances between quotients and original spaces is unbounded.

Abstract

The Hausdorff distance measures how far apart two sets are in a common metric space. By contrast, the Gromov-Hausdorff distance provides a notion of distance between two abstract metric spaces. How do these distances behave for quotients of spaces under group actions? Suppose a group $G$ acts by isometries on two metric spaces $X$ and $Y$. In this article, we study how the Hausdorff and Gromov-Hausdorff distances between $X$ and $Y$ and their quotient spaces $X/G$ and $Y/G$ are related. For the Hausdorff distance, we show that if $X$ and $Y$ are $G$-invariant subsets of a common metric space, then we have $d_{\mathrm{H}}(X,Y)=d_{\mathrm{H}}(X/G,Y/G)$. However, the Gromov-Hausdorff distance does not preserve this relationship: we show how to make the ratio $\frac{d_{\mathrm{GH}}(X/G,Y/G)}{d_{\mathrm{GH}}(X,Y)}$ both arbitrarily large and arbitrarily small, even if $X$ is an arbitrarily dense $G$-invariant subset of $Y$.