Quasiconformal Maps between Bowditch Boundaries of Relatively Hyperbolic Groups

TL;DR

The paper generalizes Paulin's boundary rigidity to relatively hyperbolic groups by introducing shadow-preserving $\eta$-quasiconformal maps on Bowditch boundaries and proving a two-way correspondence with coarse cusp-preserving quasi-isometries. It defines ring structures via shadows of horoballs, analyzes horoball shadows, and shows that a $C$-coarsely cusp-preserving $(\lambda,K)$-quasi-isometry induces a boundary $\eta$-quasiconformal map with explicit distortion $\eta(t)=e^{A\ln t + B}$, while conversely a boundary $\eta$-quasiconformal map that preserves parabolic endpoints yields a quasi-isometry between cusped spaces and, under a shadow-preservation condition, a coarse cusp-preserving quasi-isometry between the groups. The results connect quasiconformal, relative quasi-Möbius, and shadow-respecting quasisymmetric notions, establishing boundary rigidity principles for relatively hyperbolic groups. Overall, the work advances the quasi-isometric classification of relatively hyperbolic groups via their Bowditch boundaries and horoball geometry, providing a unified framework for boundary-to-group rigidity in this setting.

Abstract

Classifying groups up to quasi-isometry is a fundamental 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. F. Paulin proved that two hyperbolic groups are quasi-isometric if and only if their Gromov boundaries are quasiconformally equivalent. In this article, we extend Paulin's result to relatively hyperbolic groups and their Bowditch boundaries. A notion of quasiconformal map preserving the shadows of horoballs relative to a point at the Bowditch boundary is defined and we have shown that every coarsely cusp-preserving quasi-isometry between two relatively hyperbolic groups induces a shadow-preserving quasiconformal map between their Bowditch boundaries. Conversely, we have shown that if the Bowditch boundaries of two relatively hyperbolic groups are quasiconformally equivalent and the quasiconformal map coarsely preserves the shadows of horoballs relative to each boundary point, then the quasiconformal map induces a coarsely cusp-preserving quasi-isometry between those groups.

Figures (9)

• Figure 2.1:
• Figure 3.1: Shadow of a ball
• Figure 3.2:
• Figure 3.3: Ring
• Figure 3.4:

Theorems & Definitions (58)

• Theorem 1
• Theorem 2
• Corollary 1.1
• Definition 2.1
• Definition 2.2: Hyperbolic Metric Spaces
• Definition 2.3: Quasi-isometries and Quasigeodesics
• Proposition 2.4: Stability of Quasigeodesics, BH99
• Theorem 2.5: Quasi-isometries Preserve Hyperbolicity
• Definition 2.6: Hyperbolic Groups, Gro87BH99
• Proposition 2.7: BH99*Chapter III.$\Gamma$, Proposition 2.2