Quasi-isometries between visual hyperbolic spaces
  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. 
\'Alvaro Mart\'inez-P\'erez