Holonomy groups of complete flat pseudo-Riemannian homogeneous spaces

Abstract

We show that a complete flat pseudo-Riemannian homogeneous manifold with non-abelian linear holonomy is of dimension ≥14. Due to an example constructed in a previous article (Baues and Globke, 2012 [2]), this is a sharp bound. Also, we give a structure theory for the fundamental groups of complete flat pseudo-Riemannian manifolds in dimensions ≤6. Finally, we observe that every finitely generated torsion-free 2-step nilpotent group can be realized as the fundamental group of a complete flat pseudo-Riemannian manifold with abelian linear holonomy.