$PD_3$-groups and HNN Extensions

Abstract

We show that if a $PD_3$-group $G$ splits as an HNN extension $A*_Cφ$ where $C$ is a $PD_3$-group then the Poincaré dual in $H^1(G;\mathbb{Z})=Hom(G,\mathbb{Z})$ of the homology class $[C]$ is the epimorphism $f:G\to\mathbb{Z}$ with kernel the normal closure of $A$. We also make several other observations about $PD_3$-groups which split over $PD_2$-groups.