CAT(0) spaces quasi-isometric to Euclidean spaces
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915889270489088 |
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| author | Cavallucci, Nicola Sambusetti, Andrea |
| author_facet | Cavallucci, Nicola Sambusetti, Andrea |
| contents | We show that if a proper, geodesically complete, CAT(0) homology manifold is quasi-isometric to the Euclidean space R^n then it is homeomorphic to R^n. On the other hand, we show that there exist proper, geodesically complete, CAT(0) spaces quasi-isometric to R^n, which are not homeomorphic to it. We prove that our example is sharp in a suitable sense. Finally, we provide an example of a sequence of proper, geodesically complete, CAT(0) spaces that are not homology manifolds and that converge in the Gromov-Hausdorff sense to a topological manifold: this shows that the set of topological manifolds is not open in the class of proper, geodesically complete, CAT(0) spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23768 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | CAT(0) spaces quasi-isometric to Euclidean spaces Cavallucci, Nicola Sambusetti, Andrea Metric Geometry Differential Geometry Geometric Topology We show that if a proper, geodesically complete, CAT(0) homology manifold is quasi-isometric to the Euclidean space R^n then it is homeomorphic to R^n. On the other hand, we show that there exist proper, geodesically complete, CAT(0) spaces quasi-isometric to R^n, which are not homeomorphic to it. We prove that our example is sharp in a suitable sense. Finally, we provide an example of a sequence of proper, geodesically complete, CAT(0) spaces that are not homology manifolds and that converge in the Gromov-Hausdorff sense to a topological manifold: this shows that the set of topological manifolds is not open in the class of proper, geodesically complete, CAT(0) spaces. |
| title | CAT(0) spaces quasi-isometric to Euclidean spaces |
| topic | Metric Geometry Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2603.23768 |