Torsion-free crystallographic groups with indecomposable holonomy group

Abstract

Let K be a principal ideal domain, G a finite group, and M a KG-module which as K-module is free of finite rank, and on which $G$ acts faithfully. A generalized crystallographic group (introduced by the authors in volume 5 of Journal of Group Theory) is a group $\frak C$ which has a normal subgroup isomorphic to M with quotient G, such that conjugation in $\frak C$ gives the same action of G on M that we started with. (When $K=\Bbb Z$, these are just the classical crystallographic groups.) The K-free rank of M is said to be the dimension of $\frak C$, the holonomy group of $\frak C$ is G, and $\frak C$ is called indecomposable if M is an indecomposable KG-module. Let K be either $\Bbb Z$, or its localization $\Bbb Z_{(p)}$ at the prime p, or the ring $\Bbb Z_p$ of p-adic integers, and consider indecomposable torsionfree generalized crystallographic groups whose holonomy group is noncyclic of order p^2. In Theorem 2, we prove that (for any given p) the dimensions of these groups are not bounded. For $K=\Bbb Z$, we show in Theorem 3 that there are infinitely many non-isomorphic indecomposable torsionfree crystallographic groups with holonomy group the alternating group of degree 4. In Theorem 1, we look at a cyclic G whose order |G| satisfies the following condition: for all prime divisors p of |G|, p^2 also divides G, and for at least one p, even p^3 does. We prove that then every product of |G| with a positive integer coprime to it occurs as the dimension of some indecomposable torsionfree crystallographic group with holonomy group G.