Fibering flat manifolds of diagonal type and their fundamental groups

An $n$-dimensional closed flat manifold is said to be of diagonal type if the standard representation of its holonomy group $G$ is diagonal. An $n$-dimensional Bieberbach group of diagonal type is the fundamental group of such a manifold. We introduce the diagonal Vasquez invariant of $G$ as the least integer $n_d(G)$ such that every flat manifold of diagonal type with holonomy $G$ fibers over a flat manifold of dimension at most $n_d(G)$ with flat torus fibers. Using a combinatorial description of Bieberbach groups of diagonal type, we give both upper and lower bounds for this invariant. We show that the lower bounds are exact when $G$ has low rank. We apply this to analyse diffuseness properties of Bieberbach groups of diagonal type. This leads to a complete classification of Bieberbach groups of diagonal type with Klein four-group holonomy and to an application to Kaplansky's Unit Conjecture.