Flat Pseudo-Riemannian Homogeneous Spaces with Non-Abelian Holonomy Group

TL;DR

This paper constructs examples of non-compact, non-complete flat pseudo-Riemannian manifolds with non-abelian holonomy groups, showing such structures exist starting from dimension 8, unlike the compact case.

Contribution

It provides the first known non-compact, non-complete homogeneous flat pseudo-Riemannian manifolds with non-abelian holonomy, and establishes criteria for proper affine group actions.

Findings

Non-compact examples with non-abelian holonomy exist in dimension ≥8.

Complete example of dimension 14 with non-abelian holonomy.

Compact homogeneous flat pseudo-Riemannian manifolds have abelian holonomy.

Abstract

We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear holonomy is non-abelian, starting in dimensions $\geq 8$, which is the lowest possible dimension. We also construct a complete flat pseudo-Riemannian homogeneous manifold of dimension 14 with non-abelian linear holonomy. Furthermore, we derive a criterion for the properness of the action of an affine transformation group with transitive centralizer.