Coarse embeddings of quotients by finite group actions
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913491195002880 |
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| author | Weighill, Thomas |
| author_facet | Weighill, Thomas |
| contents | We prove that for a metric space $X$ and a finite group $G$ acting on $X$ by isometries, if $X$ coarsely embeds into a Hilbert space, then so does the quotient $X/G$. A crucial step towards our main result is to show that for any integer $k > 0$ the space of unordered $k$-tuples of points in Hilbert space, with the $1$-Wasserstein distance, itself coarsely embeds into Hilbert space. Our proof relies on establishing bounds on the sliced Wasserstein distance between empirical measures in $\mathbb{R}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_09369 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Coarse embeddings of quotients by finite group actions Weighill, Thomas Metric Geometry 51F30 We prove that for a metric space $X$ and a finite group $G$ acting on $X$ by isometries, if $X$ coarsely embeds into a Hilbert space, then so does the quotient $X/G$. A crucial step towards our main result is to show that for any integer $k > 0$ the space of unordered $k$-tuples of points in Hilbert space, with the $1$-Wasserstein distance, itself coarsely embeds into Hilbert space. Our proof relies on establishing bounds on the sliced Wasserstein distance between empirical measures in $\mathbb{R}^n$. |
| title | Coarse embeddings of quotients by finite group actions |
| topic | Metric Geometry 51F30 |
| url | https://arxiv.org/abs/2310.09369 |