\(PD_3\)-groups and HNN extensions

Jonathan A. Hillman

Abstract

We show that if a \(PD_3\)-group \(G\) splits as an HNN extension \(A*_C\varphi\) where \(C\) is a \(PD_2\)-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 \(PD_3\)-groups which split over \(PD_2\)-groups.

Keywords: fundamental class, HNN extension, \(PD_3\)-group, surface group.

AMS Subject Classification: Primary 57N13.