Fibering flat manifolds of diagonal type and their fundamental groups

Petrosyan, Nansen and Chung, Ho Yiu (2020) Fibering flat manifolds of diagonal type and their fundamental groups. International Journal of Algebra and Computation.

Record type: Article

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 nd(G) such that every flat manifold of diagonal type with holonomy G fibers over a flat manifold of dimension at most nd(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. For example, we prove that a Bieberbach group of diagonal type with holonomy group isomorphic to the Klein four-group is non-diffuse if and only if it is a direct product of its center with a semi-direct product of a free abelian subgroup by Promislow group ΔP.

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