An extension of a theorem of Griffiths to 3-manifold groups.

Abstract

The aim of this paper is to show that a closed Haken 3-manifold is virtually a closed surface bundle over $S\sp1$ under certain conditions. First we prove that a compact 3-manifold M with geometric structure is a surface bundle over $S\sp1$ if the fundamental group G contains a finitely generated subgroup U of infinite index in G and U contains a nontrivial normal subgroup N of G which is not isomorphic to $\doubz$. Next we prove that a closed Haken 3-manifold M is virtually a surface bundle over $S\sp1$ if the fundamental group of M satisfies the above condition and the group N in the condition intersects each splitting torus group nontrivially.