Abstract

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 suficient 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.

Keywords

affinely flat manifold, (almost)-crystallographic, (almost)-Bieberbach group, (almost)-torsion-free, (virtually) polycyclic group, nilpotent Lie group, discrete cocompact subgroups, lattice, Malcev completion, cohomology of groups, complex (Kähler) structure, group action, group representation, flat Riemannian manifold, (infra)-nilmanifold

Mathematical Subject Classification

Primary: 22E40

Secondary: 32Q15, 14R20

Authors