Geometry of Crystallographic Groups

The following sections are included:

• Preface

• Contents

In this introductory chapter we define a language of the isometry group of the Euclidean space. We also introduce an elementary world of covering spaces, group actions and discrete subgroups…

The central results on crystallographic groups were proved by Bieberbach in the years 1910-1912. The proof of the first Bieberbach theorem is the most difficult part of the chapter. In addition we prove a theorem of Zassenhaus from 1948 which treats crystallographic groups purely as abstract groups. Here we may apply the theory of group extensions using the language of finite group cohomology. Together with the Jordan - Zassenhaus theorem this is enough to prove the second Bieberbach theorem. Finally we prove the third Bieberbach theorem on the rigidity of crystallographic groups.

Since for each dimension there is only a finite number of isomorphism classes of crystallographic groups, we start this part with a review of the classification methods. We have to say that relations between the group extension and the second cohomology group will be very useful. We specially describe Auslander-Vasquez method, which seems to be completely forgotten. We finish the chapter with an elementary classification in dimensions 2 and 3. In dimension 3 we present a complete list of ten Bieberbach groups. First, we sketch an elementary, geometric algorithm which gives all ten groups. Next, we describe, step by step each of them as a subgroup of A(3)…

In this chapter we concentrate on the second step of the Calabi method of classification. We shall give a necessary and sufficient condition for a finite group to be a holonomy group of a Bieberbach group with finite abelianization (primitive groups). Here we shall use the Burnside transfer Theorem (Theorem B.2), which is formulated and proved in Appendix B. At the end we give a list of primitive groups…

This chapter, as the previous one, is related to the Calabi method of classification (see Section 3.2 and Proposition 3.1). This time we are interested in its third part. To classify the short exact sequences of groups

This chapter is mainly concerned with the existence of spin structures on a flat manifold. We want to convince the reader that such structure completely depends on some condition on the fundamental group of a manifold (see Corollary before Proposition 6.3). We have learned it from an article by F. Pfaffle, [96]. We start with an elementary definition of Spin(n) group and then introduce some basic information about vector bundles. It is necessary for a complete explanation of the above condition. Next, by using only algebraic techniques, we prove some classification results about the existence or non-existence of spin structures on flat manifolds. The last, most technical subsection is about a spectrum of the Dirac operator on flat manifolds. Here the approach essentially requires some knowledge of differential geometry which can be found, for example, in [39].

In this part we are interested in the classification of six dimensional flat manifolds, admitting the complex structure, which in this case is equivalent to the Kähler structure. In the beginning we give necessary and sufficient conditions on the existence of a complex structure on the flat manifold. We show that they depend on properties of a holonomy representation. At the end of this chapter we shall describe some homology properties of the flat Kähler manifolds…

In this chapter we consider a phenomenon of relations between hyperbolic and Euclidean geometries. It is well known that the boundary of hyperbolic space ℍⁿ is equal to ℝ^n-1. Using this observation, we show that any Bieberbach group of dimension n-1 is also a discrete subgroup of the group Isom(ℍⁿ). Most facts of this chapter are based on [74] and [75].

Only one 3-dimensional Bieberbach group Γ₉ has a non-cyclic holonomy group ℤ₂ × ℤ₂. Γ₉ is a finite presented group and has a cyclic presentation, see [63]. In particular it belongs to the family of Fibonacci groups, see [64]. We emphasize that Γ₉ is a member of another infinite class of groups, which we shall consider in this part. In fact, in the theory of torsion free crystallographic groups we define an infinite class of generalized Hantzsche-Wendt groups. Roughly speaking, a Bieberbach group of dimension n is the generalized Hanztsche-Wendt if and only if it has a holonomy group (ℤ₂)^n-1. Since this 3-dimensional group Γ₉ was first dicovered by German mathematicians W. Hantzsche and H. Wendt, these groups are called Hantzsche - Wendt groups. In the following Chapter we present a few interesting and exceptional properties of the above groups.

This part is related to the paper [118]. Let Γ ⊂ E(n) be a crystallographic group which defines the exact sequence

The following sections are included:

• Appendix A Alternative Proof of Bieberbach Theorem

• Appendix B Burnside Transfer Theorem

• Appendix C Example of a Flat Manifold without Symmetry

• Bibliography

• Index