Asymptotic dimension for covers with controlled growth
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866912371444809728 |
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| author | Hume, David Mackay, John M. Tessera, Romain |
| author_facet | Hume, David Mackay, John M. Tessera, Romain |
| contents | We prove various obstructions to the existence of regular maps (or coarse embeddings) between commonly studied spaces. For instance, there is no regular map (or coarse embedding) $\mathbb H^n\to\mathbb H^{n-1}\times Y$ for $n\geq 3$, or $(T_3)^n \to (T_3)^{n-1}\times Y$ whenever $Y$ is a bounded degree graph with subexponential growth, where $T_3$ is the $3$-regular tree.
We also resolve a question of Benjamini-Schramm-Timár, proving that there is no regular map $\mathbb H^2 \to T_3 \times Y$ whenever $Y$ is a bounded degree graph with at most polynomial growth, and no quasi-isometric embedding whenever $Y$ has subexponential growth.
Finally, we show that there is no regular map $F^n\to \mathbb Z\wr F^{n-1}$ where $F$ is the free group on two generators.
To prove these results, we introduce and study generalizations of asymptotic dimension which allow unbounded covers with controlled growth. For bounded degree graphs, these invariants are monotone with respect to regular maps (hence coarse embeddings). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_01969 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotic dimension for covers with controlled growth Hume, David Mackay, John M. Tessera, Romain Metric Geometry 51F30, 20F69 We prove various obstructions to the existence of regular maps (or coarse embeddings) between commonly studied spaces. For instance, there is no regular map (or coarse embedding) $\mathbb H^n\to\mathbb H^{n-1}\times Y$ for $n\geq 3$, or $(T_3)^n \to (T_3)^{n-1}\times Y$ whenever $Y$ is a bounded degree graph with subexponential growth, where $T_3$ is the $3$-regular tree. We also resolve a question of Benjamini-Schramm-Timár, proving that there is no regular map $\mathbb H^2 \to T_3 \times Y$ whenever $Y$ is a bounded degree graph with at most polynomial growth, and no quasi-isometric embedding whenever $Y$ has subexponential growth. Finally, we show that there is no regular map $F^n\to \mathbb Z\wr F^{n-1}$ where $F$ is the free group on two generators. To prove these results, we introduce and study generalizations of asymptotic dimension which allow unbounded covers with controlled growth. For bounded degree graphs, these invariants are monotone with respect to regular maps (hence coarse embeddings). |
| title | Asymptotic dimension for covers with controlled growth |
| topic | Metric Geometry 51F30, 20F69 |
| url | https://arxiv.org/abs/2303.01969 |