Let 𝒜ℬn be the class of
torsion-free, discrete groups that contain a normal, at most n-step, nilpotent
subgroup of finite index. We give sufficient conditions for the fundamental group of a
fibration F → T → B, with base B an infra-nilmanifold, to belong to 𝒜ℬn. Manifolds
of this kind may, for example, appear as thin ends of nonpositively curved manifolds.
We prove that if, in addition, we require that T be Kähler, then T possesses a flat
Riemannian metric and the fundamental group π1(T) is necessarily a Bieberbach
group. Further, we prove that a torsion-free, virtually polycyclic group that can be
realised as the fundamental group of a compact, Kähler K(π,1)-manifold is
necessarily Bieberbach.
