Quasi-symmetries between metric spaces and rough quasi-isometries between their infinite hyperbolic cones

Abstract

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 $\omega$ in the Gromov boundary of its infinite hyperbolic cone such that $Z$ can be seen as the Gromov boundary relative to $\omega$ of its infinite hyperbolic cone. Third, we prove that for a visual Gromov hyperbolic metric space $X$ and a Gromov boundary point $\omega$, $X$ is roughly similar to the infinite hyperbolic cone of its Gromov boundary relative to $\omega$. 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.