On groups with $S^2$ Bowditch boundary

TL;DR

This paper characterizes when a relatively hyperbolic pair has a Bowditch boundary homeomorphic to a 2-sphere, linking it to 3-dimensional Poincaré duality pairs through boundary analysis.

Contribution

It establishes a precise equivalence between Bowditch boundary being a 2-sphere and the group being a 3-dimensional Poincaré duality pair, using boundary relationship studies.

Findings

Bowditch boundary is a 2-sphere iff the pair is a 3D Poincaré duality pair

Relationship between Bowditch and Dahmani boundaries analyzed

Provides a topological characterization of certain relatively hyperbolic groups

Abstract

We prove that a relatively hyperbolic pair $(G,P)$ has Bowditch boundary a 2-sphere if and only if it is a 3-dimensional Poincare duality pair. We prove this by studying the relationship between the Bowditch and Dahmani boundaries of relatively hyperbolic groups.