Gromov-Hausdorff distances between quotient metric spaces
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arXiv
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| Main Authors: | , , , , , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916445801152512 |
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| author | Adams, Henry Albornoz, Armando Bruda, Glenn Du, Jianda van Rensburg, Lodewyk Jansen Leon, Alejandro Narayanan, Saketh Panish, Connor Rugenstein, Chris Wall, Martin |
| author_facet | Adams, Henry Albornoz, Armando Bruda, Glenn Du, Jianda van Rensburg, Lodewyk Jansen Leon, Alejandro Narayanan, Saketh Panish, Connor Rugenstein, Chris Wall, Martin |
| contents | 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 $Y$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14938 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gromov-Hausdorff distances between quotient metric spaces Adams, Henry Albornoz, Armando Bruda, Glenn Du, Jianda van Rensburg, Lodewyk Jansen Leon, Alejandro Narayanan, Saketh Panish, Connor Rugenstein, Chris Wall, Martin Metric Geometry 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 $Y$. |
| title | Gromov-Hausdorff distances between quotient metric spaces |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2410.14938 |