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Abstract

In this article we study the relation between flat solvmanifolds and \(G_2\)-geometry. First, we give a classification of 7-dimensional flat splittable solvmanifolds using the classification of finite subgroups of \(\mathsf{GL}(n,\mathbb {Z})\) for \(n=5\) and \(n=6\). Then, we look for closed, coclosed and divergence-free \(G_2\)-structures compatible with the flat metric on them. In particular, we provide explicit examples of compact flat manifolds with a torsion-free \(G_2\)-structure whose finite holonomy is cyclic and contained in \(G_2\), and examples of compact flat manifolds admitting a divergence-free \(G_2\)-structure.

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Notes

  1. Throughout this article, a \(n\times n\) matrix A will be said to be similar (or conjugated) to B if there exists \(P\in \mathsf{GL}(n,\mathbb {R})\) such that \(P^{-1}AP=B\) and integrally similar if \(P\in \mathsf{GL}(n,\mathbb {Z})\).

  2. A spin manifold is an oriented Riemannian manifold with a spin structure.

  3. Throughout the article we will denote the block diagonal matrix \(\begin{pmatrix}A&{}\quad 0\\ 0&{}\quad B\end{pmatrix}\) by \(A\oplus B\).

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Acknowledgements

Special thanks go to Jorge Lauret. This article originated from a suggestion of his. Also, I want to thank Agustín Garrone, Andrés Moreno and Henrique Sá Earp for very fruitful conversations. Thanks to my advisor Adrián Andrada for his careful reading of the article. Finally, I’m very grateful to the IMEC at UNICAMP for the warm hospitality during my visit.

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Correspondence to Alejandro Tolcachier.

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Communicated by Vicente Cortés.

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Tolcachier, A. \(G_2\)-structures on flat solvmanifolds. Abh. Math. Semin. Univ. Hambg. 92, 179–207 (2022). https://doi.org/10.1007/s12188-022-00261-7

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