Fibering flat manifolds of diagonal type and their fundamental groups

TL;DR

This paper introduces the diagonal Vasquez invariant for flat manifolds of diagonal type, providing bounds, classifications, and applications to group properties and conjectures in geometric topology.

Contribution

It defines the diagonal Vasquez invariant, establishes bounds, and classifies Bieberbach groups of diagonal type with Klein four-group holonomy, linking to group properties and conjectures.

Findings

Exact lower bounds for low-rank holonomy groups.

Complete classification of groups with Klein four-group holonomy.

Application to Kaplansky's Unit Conjecture.

Abstract

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.