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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2211.15020 |
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| _version_ | 1866911829111865344 |
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| author | Huang, Manzi Xu, Zhihao |
| author_facet | Huang, Manzi Xu, Zhihao |
| contents | In this paper, we first prove that any power quasi-symmetry of two metric spaces induces a rough quasi-isometry between their infinite hyperbolic cones. Second, we prove that for a complete metric space $Z$, there exists a point $ω$ in the Gromov boundary of its infinite hyperbolic cone such that $Z$ can be seen as the Gromov boundary relative to $ω$ of its infinite hyperbolic cone. Third, we prove that for a visual Gromov hyperbolic metric space $X$ and a Gromov boundary point $ω$, $X$ is roughly similar to the infinite hyperbolic cone of its Gromov boundary relative to $ω$. These are the generalizations of Theorem 7.4, Theorem 8.1 and Theorem 8.2 in [3] since the underlying spaces are not assumed to be bounded and the hyperbolic cones are infinite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_15020 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quasi-symmetries between metric spaces and rough quasi-isometries between their infinite hyperbolic cones Huang, Manzi Xu, Zhihao Metric Geometry In this paper, we first prove that any power quasi-symmetry of two metric spaces induces a rough quasi-isometry between their infinite hyperbolic cones. Second, we prove that for a complete metric space $Z$, there exists a point $ω$ in the Gromov boundary of its infinite hyperbolic cone such that $Z$ can be seen as the Gromov boundary relative to $ω$ of its infinite hyperbolic cone. Third, we prove that for a visual Gromov hyperbolic metric space $X$ and a Gromov boundary point $ω$, $X$ is roughly similar to the infinite hyperbolic cone of its Gromov boundary relative to $ω$. These are the generalizations of Theorem 7.4, Theorem 8.1 and Theorem 8.2 in [3] since the underlying spaces are not assumed to be bounded and the hyperbolic cones are infinite. |
| title | Quasi-symmetries between metric spaces and rough quasi-isometries between their infinite hyperbolic cones |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2211.15020 |