Fibered Orbifolds and Crystallographic Groups

In this paper, we prove that a normal subgroup N of an n-dimensional crystallographic group Γ determines a geometric fibered orbifold structure on the flat orbifold En/Γ, and conversely every geometric fibered orbifold structure on En/Γ is determined by a normal subgroup N of Γ, which is maximal in its commensurability class of normal subgroups of Γ. In particular, we prove that En/Γ is a fiber bundle, with totally geodesic fibers, over a β1-dimensional torus, where β1 is the first Betti number of Γ. Let N be a normal subgroup of Γ which is maximal in its commensurability class. We study the relationship between the exact sequence 1 → N → Γ → Γ/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 2and 3-dimensional crystallographic groups building on the work of Conway and Thurston.