Maps between Boundaries of Relatively Hyperbolic Groups

Abstract: F. Paulin proved that if the Gromov boundaries of two hyperbolic groups are quasi-M\"obius equivalence, then those two hyperbolic groups are quasi-isometric to each other. This article aims to extend Paulin's results to relatively hyperbolic groups by introducing the notion of `relative quasi-M\"obius maps' between Bowditch boundaries of relatively hyperbolic groups. A coarsely cusp-preserving quasi-isometry between two relatively hyperbolic groups induces a homeomorphism between their Bowditch boundaries. We will show that the induced homeomorphism is relative quasi-M\"{o}bius and linearly distorts the exit points of bi-infinite geodesics to combinatorial horoballs. Conversely, we will show that if a homeomorphism between Bowditch boundaries of two relatively hyperbolic groups, preserving parabolic endpoints, is either relative quasi-M\"{o}bius or distorts the exit points of bi-infinite geodesics to combinatorial horoballs linearly, then that homeomorphism induces a coarsely cusp-preserving quasi-isometry between the relatively hyperbolic groups.