# Groups with 2-sphere Bowditch boundary

with G. Walsh

### Groups Geom. Dyn. 14 (2020) 3, 791-811.

Bestvina-Mess showed that the duality properties of a group are encoded in any boundary that gives a Z-set compactification; in particular, a hyperbolic group with Gromov boundary S^n is a PD(n+1) group. For relatively hyperbolic pairs (G,P), the natural boundary – the Bowditch boundary – does not give a Z-set 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.