Holonomy Groups of Complete Flat Pseudo-Riemannian Homogeneous Spaces

TL;DR

This paper investigates the structure and bounds of holonomy groups in complete flat pseudo-Riemannian homogeneous spaces, establishing minimal dimension bounds for non-abelian cases and characterizing fundamental groups in low dimensions.

Contribution

It provides a sharp lower bound of dimension 14 for non-abelian linear holonomy and develops a structure theory for fundamental groups in dimensions less than 7.

Findings

Complete flat pseudo-Riemannian homogeneous manifolds with non-abelian holonomy have dimension at least 14.

A structure theory for fundamental groups in dimensions less than 7 is established.

Any finitely generated torsion-free 2-step nilpotent group can be realized as a fundamental group with abelian holonomy.

Abstract

We show that a complete flat pseudo-Riemannian homogeneous manifold with non-abelian linear holonomy is of dimension at least 14. Due to an example constructed in a previous article by Oliver Baues and the author, this is a sharp bound. Also, we give a structure theory for the fundamental groups of complete flat pseudo-Riemannian manifolds in dimensions less than 7. 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.