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arXiv · 1907.02021

Holonomy groups of compact flat solvmanifolds

Alejandro Tolcachier
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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.

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Alejandro Tolcachier. 2019-07-03. Holonomy groups of compact flat solvmanifolds. https://arxiv.org/abs/1907.02021

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