2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/174028We 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.16 pages. In the new version, the property on the homeomorphism originally used to characterize quasi-isometry between the hyperbolic spaces is proved to be equivalent to being PQ-symmetric. Therefore, is no longer named as a new property and several changes are made following from thisGeometric Topologytext