2022/11/28 by Huang, Manzi, Xu, Zhihao · 1 citation
#FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2211.15020
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.