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 ▪. Such groups are torsion free and even satisfy Poincare' duality. We study their commensurability classes and establish criteria which enable us to rule out certain group extensions from membership of ▪. In consequence, we are able to show that ▪ 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.
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