Fibered orbifolds and crystallographic groups

TL;DR

This paper establishes a correspondence between normal subgroups of crystallographic groups and fibered orbifold structures on flat orbifolds, analyzing conditions for splitting of exact sequences and classifying fibrations in low dimensions.

Contribution

It characterizes fibered orbifold structures via normal subgroups of crystallographic groups and explores conditions for the splitting of associated exact sequences.

Findings

E^n/G is a fiber bundle over a torus with geodesic fibers.

Splitting of the exact sequence relates to the existence of an affine section.

Complete classification of fibrations for 2- and 3-dimensional crystallographic groups.

Abstract

In this paper, we prove that a normal subgroup N of an n-dimensional crystallographic group G determines a geometric fibered orbifold structure on the flat orbifold E^n/G, and conversely every geometric fibered orbifold structure on E^n/G is determined by a normal subgroup N of G, which is maximal in its commensurability class of normal subgroups of G. In particular, we prove that E^n/G is a fiber bundle, with totally geodesic fibers, over a b-dimensional torus, where b is the first Betti number of G. Let N be a normal subgroup of G which is maximal in its commensurability class. We study the relationship between the exact sequence 1 -> N -> G -> G/N -> 1 splitting and the corresponding fibration projection having an affine section. If N is torsion-free, we prove that the exact sequence splits if and only if the fibration projection has an affine section. If the generic fiber F = Span(N)/N has an ordinary point that is fixed by every isometry of F, we prove that the exact sequence always splits. Finally, we describe all the geometric fibrations of the orbit spaces of all 2- and 3-dimensional crystallographic groups building on the work of Conway and Thurston.