On torsion free crystallographic groups

Abstract

The torsion free crystallographic groups arise as fundamental groups of compact flat Riemannian manifolds. Let R be a crystallographic group with point group G and translation group T. In this paper we consider the Q G-module T⊗ z Q , for which we prove: If R is torsion free, then G does not act irreducibly on T⊗ z Q . A proof of this theorem for solvable groups G was first given by G. Cliff. The theorem proves a conjecture made by the second author. The proof of the theorem uses the classification of the finite simple groups.