A generalization of a theorem of Griffiths to 3-manifolds (Q1779227)

A result of Griffiths says that if the fundamental group of a surface contains a finitely generated subgroup of infinite index, containing on its turn a normal non-trivial subgroup, then the surface is either the torus or the Klein bottle, in other words the surface virtually fibres over \({\mathbf S}^1\). Sufficient conditions to ensure that a \(3\)-manifold virtually fibres over the circle, in terms of properties of the fundamental group, were studied by Stallings, and by Hempel and Jaco. Stallings showed that a compact \({\mathbb P}^2\)-irreducible \(3\)-manifold is a surface bundle over \({\mathbf S}^1\) if its fundamental group is an extension of a surface group by \({\mathbb Z}\) (the converse being obvious), while Hempel and Jaco proved that a compact \({\mathbb P}^2\)-irreducible \(3\)-manifold is virtually a surface bundle with fibre \(F\) if its fundamental group is an extension of a finitely generated non-cyclic group \(U\) by an infinite group, moreover \(\pi_1(F)\) and \(U\) are commensurable. These results suggest that it should be possible to generalise Griffiths result in dimension \(3\), and in this paper it is shown that this is indeed the case if the manifolds are required to be either geometric or to contain an incompressible torus (with some extra properties) which cuts the manifold into (one or two) Seifert fibred or hyperbolic pieces. More precisely, the author proves the following: Let \(M\) be a \(3\)-manifold and let \(G\) be its fundamental subgroup. Let \(U\) be a finitely generated subgroup of \(G\) of infinite index containing a non-trivial normal subgroup. Assume, moreover, that \(N\) is not \({\mathbb Z}\). Then \(M\) is a virtual fibre bundle with fibre \(F\), such that \(\pi_1(F)\) and \(U\) are commensurable, if \(M\) is either geometric or if \(M\) splits along an incompressible torus \(T\) into one or two manifolds which are hyperbolic or Seifert fibred, i.e. \(M=X_1\cup_TX_2\) or \(M=X_1\cup_T\), so that \(N\cap\pi_1(T)\) is non trivial and \(N\cap\pi_1(X_1)\) is not \({\mathbb Z}\). The proofs are based on a variety of different techniques and results ranging from group theory and combinatorics to low-dimensional topology and hyperbolic geometry.