A fibering theorem for 3-manifolds

Abstract

We generalize a result of Moon on the fibering of certain 3-manifolds over the circle. Our main theorem is the following: Let $M$ be a closed 3-manifold. Suppose that $G=\pi_1(M)$ contains a finitely generated group $U$ of infinite index in $G$ which contains a non-trivial subnormal subgroup $N\neq \mathbb{Z}$ of $G$, and suppose that $N$ has a composition series of length $n$ in which at least $n-1$ terms are finitely generated. Suppose that $N$ intersects nontrivially the fundamental groups of the splitting tori given by the Geometrization Theorem and that the intersections of $N$ with the fundamental groups of the geometric pieces are non-trivial and not isomorphic to $\mathbb{Z}$. Then, $M$ has a finite cover which is a bundle over $\mathbb{S}$ with fiber a compact surface $F$ such that $\pi_1(F)$ and $U$ are commensurable.