A Bieberbach theorem for crystallographic group extensions

Abstract

In this paper we prove that for each dimension $n$ there are only finitely many isomorphism classes of pairs of groups $(\Gamma,\mathrm{N})$ such that $\Gamma$ is an $n$-dimensional crystallographic group and $\mathrm{N}$ is a normal subgroup of $\Gamma$ such that $\Gamma/\mathrm{N}$ is a crystallographic group.