Stallings's Fibring Theorem and ${\mathrm{P}\mathrm{D}}^3$-pairs

We give a relatively self-contained proof that if a group $G$ fibres algebraically and is part of a ${\mathrm{P}\mathrm{D}}^3$-pair, then $G$ is the fundamental group of a fibred compact aspherical 3-manifold. This yields a homological proof of a classical theorem of Stallings: if $G={\pi }_1(M^3)$ is the fundamental group of a compact irreducible 3-manifold $M^3$ and $\phi \text{\hspace{0.17em}⁣}:G\to \mathbb{Z}$ is a surjective homomorphism with finitely generated kernel, then $\phi$ is induced by a topological fibration of $M^3$ over the circle.