Quasiconformal Maps between Bowditch Boundaries of Relatively Hyperbolic Groups

Abstract

Classifying finitely generated groups up to quasi-isometry is a central problem in geometric group theory. In the context of hyperbolic and relatively hyperbolic groups, one of the key invariants in this classification is the boundary at infinity. Fr\'{e}d\'{e}ric Paulin proved that two hyperbolic groups are quasi-isometric if and only if their Gromov boundaries are quasiconformally equivalent. In this article, we extend this correspondence to relatively hyperbolic groups via their Bowditch boundaries. We introduce a notion of quasiconformal maps on Bowditch boundaries that coarsely preserve shadows of horoballs relative to boundary points. We prove that any coarsely cusp-preserving quasi-isometry between relatively hyperbolic groups induces such a quasiconformal boundary map. Conversely, we prove that every quasiconformal homeomorphism of Bowditch boundaries that coarsely preserves shadows of horoballs arises from a coarsely cusp-preserving quasi-isometry between the groups.