Fibering flat manifolds of diagonal type and their fundamental groups

Abstract

An [Formula: see text]-dimensional closed flat manifold is said to be of diagonal type if the standard representation of its holonomy group [Formula: see text] is diagonal. An [Formula: see text]-dimensional Bieberbach group of diagonal type is the fundamental group of such a manifold. We introduce the diagonal Vasquez invariant of [Formula: see text] as the least integer [Formula: see text] such that every flat manifold of diagonal type with holonomy [Formula: see text] fibers over a flat manifold of dimension at most [Formula: see text] 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 [Formula: see text] has low rank. We apply this to analyze 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.