A fibering theorem for 3-manifolds (Q6566727)

The paper intends to generalize results of \textit{M. Moon} [Topology Appl. 149, No. 1--3, 17--32 (2005; Zbl 1068.57003)] on the fibering of certain compact \(3\)-manifolds over the circle, and also to generalize a theorem of \textit{H. B. Griffiths} [Acta Math. 110, 1--17 (1963; Zbl 0119.18902)] on the fibering of certain \(2\)-manifolds over the circle. The author mentions in the introduction that his arguments run in parallel to Moon's. The main theorem stated in the paper reads as follows.\N\NLet \(M\) be a compact \(3\)-manifold with empty or toroidal boundary. If \(G=\pi_1(M)\) contains a finitely generated subgroup \(U\) of infinite index in \(G\) which contains a nontrivial subnormal subgroup \(N\) of \(G\), then: (a) \(M\) is irreducible, (b) if further:\N\begin{itemize}\N\item[1.] \(N\) has a subnormal series of length \(n\) in which \(n-1\) terms are assumed to be finitely generated,\N\item[2.] \(N\) intersects nontrivially the fundamental groups of the splitting tori of some decomposition \({\mathcal D}\) of \(M\) into geometric pieces, and\N\item[3.] the intersections of \(N\) with the fundamental groups of the geometric pieces are not isomorphic to \({\mathbb Z}\),\N\end{itemize}\Nthen, \(M\) 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.\N\NIn the meantime an erratum to this article has been posted at \url{https://gcc.episciences.org/page/errata} announcing a rather significant gap in [Moon, loc. cit.]. While the statement of Part (a) of the main theorem remains valid, Part (b) is significantly impacted by what according to the author appears to be an oversight in Theorem 2.10 in [loc. cit.] and therefore Part (b) must be stated contingent on the successful proof of a theorem analogous to Theorem 2.9 in [loc. cit.] for compact manifolds with boundary. (Reviewers remark: Theorem 2.10 in [loc. cit.] was stated for compact manifolds, but this did not mean to include compact manifolds with boundary. In fact, the manifolds considered in [loc. cit.] were obtained by gluing closed manifolds along tori.) Also the proof of Part (a) of the main theorem requires a correction which is adressed in Section 4 of the erratum.