= Normal finitely presented subgroups of three-dimensional duality groups
Let $N$ be a normal <finitely presented group> inside a dimension-three <Poincare duality group> $G$, with $G/N$ containing an element of infinite order. Then $N$ is either a <free group> or a dimension-two <Poincare duality group>. The <Strebel infinite-index subgroup theorem> gives $\operatorname{cd}N\leq2$. If the dimension is two, take the preimage of an infinite cyclic subgroup of the quotient. The <cyclic-extension shift of group-ring cohomology> gives dimension three, forcing this preimage to have finite index by Strebel. Its duality then gives $H^*(N;\mathbb ZN)$ concentrated in degree two with top group $\mathbb Z$. The quotient is virtually cyclic, and surface automorphism realization gives a finite-index surface mapping-torus group.
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