$G_2$-structures on flat solvmanifolds

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

m a t h s f G L (n, m a t h b b 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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