Abstract :
The fundamental groups of closed 4-manifolds which fibre over a hyperbolic surface, with fibre also a hyperbolic surface, constitute a natural class of geometrical groups, herein denoted by image. Such groups are torsion free and even satisfy Poincareʹ duality. We study their commensurability classes and establish criteria which enable us to rule out certain group extensions from membership of image. In consequence, we are able to show that image is not closed under torsion-free extension by finite groups. At the geometrical level, this leads to the construction of certain closed 4-manifolds which whilst not themselves fibring in the desired manner, nevertheless, possess finite coverings which do.
