Vol. 219, No. 2, 2005

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On a special class of fibrations and Kähler rigidity

Nickolas J. Michelacakis

Vol. 219 (2005), No. 2, 311–322

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.

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 2000
Primary: 22E40
Secondary: 32Q15, 14R20
Received: 25 June 2003
Revised: 5 June 2004
Published: 1 April 2005
Nickolas J. Michelacakis
Mathematics and Statistics Department
University of Cyprus
P.O. Box 20537
Nicosia 1678