A Bieberbach theorem for crystallographic group extensions

TL;DR

This paper proves that in each dimension, only finitely many isomorphism classes of pairs of groups exist where a crystallographic group contains a normal subgroup with a crystallographic quotient, advancing understanding of group extensions.

Contribution

It establishes a finiteness result for pairs of crystallographic groups and their normal subgroups, extending the classification of group extensions in geometric group theory.

Findings

Finiteness of isomorphism classes in each dimension

Classification of group extensions in crystallography

Advancement in understanding crystallographic group structures

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.