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

TL;DR

This paper explores the deep connections between metric spaces and their hyperbolic cones, demonstrating how quasi-symmetries induce rough quasi-isometries and characterizing the boundary structures of hyperbolic spaces.

Contribution

It generalizes previous theorems to unbounded spaces and infinite hyperbolic cones, establishing new links between quasi-symmetries, hyperbolic cones, and Gromov boundaries.

Findings

Power quasi-symmetries induce rough quasi-isometries between hyperbolic cones

Existence of a boundary point linking a space to its hyperbolic cone

Rough similarity between hyperbolic spaces and their cones under certain conditions

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.