1-Uryson width and covers
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915448556093440 |
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| author | Alpert, Hannah Banerjee, Arka Papasoglu, Panos |
| author_facet | Alpert, Hannah Banerjee, Arka Papasoglu, Panos |
| contents | We investigate the following question: Do there exist Riemannian polyhedra $X$ such that the 1-Uryson width of their universal covers $\mathrm{UW}_1(\widetilde{X})$ is bounded but $\mathrm{UW}_1(X)$ is arbitrarily large? We rule out two specific cases: when $π_1(X)$ is virtually cyclic and when $X$ is a Riemannian surface. More specifically, we show that if $X$ is a compact polyhedron with a virtually cyclic fundamental group, then its 1-Uryson width is bounded by the 1-Uryson width of its universal cover $\widetilde{X}$. Precisely: $$\mathrm{UW}_1(X) \leq 6 \cdot \mathrm{UW}_1(\widetilde{X}).$$ We show that if $X$ is a Riemannian surface with boundary then $$\mathrm{UW}_1(X) \leq \mathrm{UW}_1(\widetilde{X}).$$ Furthermore, we show that if there exist spaces $X$ for which $\mathrm{UW}_1(\widetilde{X})$ is bounded while $\mathrm{UW}_1(X)$ is arbitrarily large, then such examples must already appear in low dimensions. In particular, such $X$ can be found among Riemannian $2$-complexes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_21126 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | 1-Uryson width and covers Alpert, Hannah Banerjee, Arka Papasoglu, Panos Metric Geometry Differential Geometry 53C23, 53C20 We investigate the following question: Do there exist Riemannian polyhedra $X$ such that the 1-Uryson width of their universal covers $\mathrm{UW}_1(\widetilde{X})$ is bounded but $\mathrm{UW}_1(X)$ is arbitrarily large? We rule out two specific cases: when $π_1(X)$ is virtually cyclic and when $X$ is a Riemannian surface. More specifically, we show that if $X$ is a compact polyhedron with a virtually cyclic fundamental group, then its 1-Uryson width is bounded by the 1-Uryson width of its universal cover $\widetilde{X}$. Precisely: $$\mathrm{UW}_1(X) \leq 6 \cdot \mathrm{UW}_1(\widetilde{X}).$$ We show that if $X$ is a Riemannian surface with boundary then $$\mathrm{UW}_1(X) \leq \mathrm{UW}_1(\widetilde{X}).$$ Furthermore, we show that if there exist spaces $X$ for which $\mathrm{UW}_1(\widetilde{X})$ is bounded while $\mathrm{UW}_1(X)$ is arbitrarily large, then such examples must already appear in low dimensions. In particular, such $X$ can be found among Riemannian $2$-complexes. |
| title | 1-Uryson width and covers |
| topic | Metric Geometry Differential Geometry 53C23, 53C20 |
| url | https://arxiv.org/abs/2505.21126 |