# Holonomy groups of compact flat solvmanifolds

## TL;DR

This paper investigates the holonomy groups of flat solvmanifolds, proving they are abelian, can realize any finite abelian group, and classifying possible groups in low dimensions.

## Contribution

It provides an elementary proof that holonomy groups are abelian and constructs examples for all finite abelian groups as holonomy groups.

## Key findings

- Holonomy groups of flat solvmanifolds are abelian.
- Any finite abelian group can be realized as a holonomy group.
- Classification of possible holonomy groups in dimensions 3 to 6.

## Abstract

This article is concerned with the study of the holonomy group of flat solvmanifolds. It is known that the holonomy group of a flat solvmanifold is abelian; we give an elementary proof of this fact and moreover we prove that any finite abelian group is the holonomy group of a flat solvmanifold. Furthermore, we show that the minimal dimension of a flat solvmanifold with holonomy group $\mathbb{Z}_n$ coincides with the minimal dimension of a compact flat manifold with holonomy group $\mathbb{Z}_n$. Finally, we give the possible holonomy groups of flat solvmanifolds in dimensions 3, 4, 5 and 6; exhibiting in the latter case a general construction to show examples of non cyclic holonomy groups.