Résumé : Bestvina-Mess showed that the duality properties of a group $G$ are encoded in any boundary that gives a Z-compactification of $G$. For example, a hyperbolic group with Gromov boundary an $n$-sphere is a PD$(n+1)$ group. For relatively hyperbolic pairs $(G,P)$, the natural boundary - the Bowditch boundary - does not give a Z-compactification of G. Nevertheless we show that if the Bowditch boundary of $(G,P)$ is a 2-sphere, then $(G,P)$ is a PD(3) pair.
This is joint work with Genevieve Walsh.