Quasi-isometries between visual hyperbolic spaces

TL;DR

This paper establishes a connection between boundary homeomorphisms and quasi-isometries in hyperbolic spaces, showing that PQ-symmetric homeomorphisms extend to quasi-isometries and characterize boundary quasi-isometries.

Contribution

It proves that PQ-symmetric homeomorphisms extend to quasi-isometries between hyperbolic approximations and characterizes when two visual Gromov hyperbolic spaces are quasi-isometric.

Findings

PQ-symmetric homeomorphisms extend to quasi-isometries

Boundary PQ-symmetry characterizes quasi-isometry of hyperbolic spaces

Provides a criterion for quasi-isometry based on boundary maps

Abstract

We prove that a PQ-symmetric homeomorphism between two complete metric spaces can be extended to a quasi-isometry between their hyperbolic approximations. This result is used to prove that two visual Gromov hyperbolic spaces are quasi-isometric if and only if there is a PQ-symmetric homeomorphism between their boundaries.