Extending and improving conical bicombings
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866910492183756800 |
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| author | Basso, Giuliano |
| author_facet | Basso, Giuliano |
| contents | We study metric spaces that admit a conical bicombing and thus obey a weak form of non-positive curvature. Prime examples of such spaces are injective metric spaces. In this article we give a complete characterization of complete metric spaces admitting a conical bicombing by showing that every such space is isometric to a closed $σ$-convex subset of some injective metric space. In addition, we show that every proper metric space that admits a conical bicombing also admits a consistent bicombing that satisfies certain convexity conditions. This can be seen as a strong indication that a question from Descombes and Lang about improving conical bicombings might have a positive answer. As an application, we prove that any group acting geometrically on a proper metric space with a conical bicombing admits a $\mathcal{Z}$-structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2005_13941 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Extending and improving conical bicombings Basso, Giuliano Metric Geometry 53C23 (Primary) 20F65, 51F99 (Secondary) We study metric spaces that admit a conical bicombing and thus obey a weak form of non-positive curvature. Prime examples of such spaces are injective metric spaces. In this article we give a complete characterization of complete metric spaces admitting a conical bicombing by showing that every such space is isometric to a closed $σ$-convex subset of some injective metric space. In addition, we show that every proper metric space that admits a conical bicombing also admits a consistent bicombing that satisfies certain convexity conditions. This can be seen as a strong indication that a question from Descombes and Lang about improving conical bicombings might have a positive answer. As an application, we prove that any group acting geometrically on a proper metric space with a conical bicombing admits a $\mathcal{Z}$-structure. |
| title | Extending and improving conical bicombings |
| topic | Metric Geometry 53C23 (Primary) 20F65, 51F99 (Secondary) |
| url | https://arxiv.org/abs/2005.13941 |