$P{D}_3$-groups and HNN extensions

Jonathan A. Hillman

Abstract

We show that if a $P{D}_3$-group $G$ splits as an HNN extension $A{\ast }_C\phi$ where $C$ is a $P{D}_2$-group then the Poincaré dual in $H^1(G;\mathbb{Z})=\mathrm{Hom}(G,\mathbb{Z})$ 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 $P{D}_3$-groups which split over $P{D}_2$-groups.

Keywords: fundamental class, HNN extension, $P{D}_3$-group, surface group.

AMS Subject Classification: Primary 57N13.