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. Frédéric 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.