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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.15020 |
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Table of 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.