Stallings' celebrated Fibration Theorem states that if a closed irreducible $3$-manifold $M$ admits a short exact sequence
\begin{equation}
1 \to N \to \pi\_1(M) \to \mathbb{Z} \to 1,
\end{equation}
where $N$ is finitely generated, then $M$ fibers over $S^1$.


My question is that whether it is possible to give the exact form of $M$ if we already know that $N$ is a surface group. More precisely, assume there is a short exact sequence
\begin{equation}
1 \to \pi\_1(S) \to \pi\_1(M) \to \mathbb{Z} \to 1,
\end{equation}
where $S$ is a closed surface. Let $t \in \pi\_1(M)$ be a preimage of $1 \in \mathbb{Z}$. Conjugation with $t$ induces an automorphism $\varphi$ of $\pi\_1(S)$ whose projection $[\varphi] \in Out(\pi\_1(S))$ in the outer automorphism group does not depend on the choice of $t$. By the Dehn-Nielsen-Baer Theorem the extended mapping class group $MCG^{\pm}(S)$ is isomorphic to $Out(\pi\_1(S))$. Let $f:S \to S$ be a homeomorphism so that $[f] \in MCG^{\pm}(S)$ corresponds to $[\varphi] \in Out(\pi\_1(S))$ under the Dehn-Nielsen-Baer isomorphism. My question is whether $M$ is diffeomorphic to the mapping torus $T\_f$ of $f:S \to S$. If this is true, does it just follow from Stallings' proof? Or, in case it is known but requires more work, is there a good reference for it?


More concretely, I am interested in the following situation (which might simplify things). If $M$ is a closed hyperbolic $3$-manifold so that $\pi\_1(M)$ is (abstractly) isomorphic to the fundamental group $\pi\_1(T\_f)$ of a mapping torus, then $M$ is diffeomorphic to $T\_f$. This is, of course, an easy consequence of the Geometrization Conjecture. However, I would like to know whether there is a proof of this fact that does *not* depend on the solution of the Geometrization Conjecture *nor* on Thurston's mapping torus theorem.



Stallings' fibration theorem - Explicit description
Your question ("is $M$ homeomorphic to $T\_f$?") is answered in the affirmative by Theorem 2 of Stallings' paper \*On fibering certain 3-manifolds". You will also need his Theorem 1. Here are the statements (slightly simplified).


**Theorem 1**: Suppose that $M$ is a compact connected three-manifold. Suppose that $\Gamma$ is a finitely generated normal subgroup of $\pi\_1(M)$ whose quotient group is $\mathbb{Z}$. Then there is a surface $F$ properly embedded in $M$ so that $\Gamma = \pi\_1(F)$.


**Theorem 2**: With hypotheses as in Theorem 1. Suppose that $M$ is irreducible. Suppose that $\Gamma$ is not $\mathbb{Z}/2\mathbb{Z}$. Then $M$ is a surface bundle over the circle, with $F$ isotopic to a fibre.


That is, your hypothesis on the short exact sequence (plus Theorem 1) gives the surface $F$. Your hypothesis that the manifold $M$ is hyperbolic then gives the additional hypotheses of Theorem 2.


Note that Stallings does not cite Waldhausen. I suppose that this is because his situation is a very very simple case of a Haken hierarchy. Once you have $F$ in your hands (and all the group theory hypotheses), it is "easy" to show that $M - F$ is homeomorphic to a product.


