G2-structures on flat solvmanifolds

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 GL(n, 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.