Szczepański A. Geometry of Crystallographic Groups

World Scientific Publishing Co. Pte. Ltd., 2012. – 178 p. – ISBN: 9814412252.
Crystallographic groups are groups which act in a nice way and via isometries on some n-dimensional Euclidean space. They got their name, because in three dimensions they occur as the symmetry groups of a crystal (which we imagine to extend to infinity in all directions). The book is divided into two parts. In the first part, the basic theory of crystallographic groups is developed from the very beginning, while in the second part, more advanced and more recent topics are discussed. So the first part of the book should be usable as a textbook, while the second part is more interesting to researchers in the field. There are short introductions to the theme before every chapter. At the end of a book is a list of conjectures and open problems. Moreover there are three appendices. The last one gives an example of the torsion free crystallographic group with a trivial center and a trivial outer automorphism group. This volume omits topics about generalization of crystallographic groups to nilpotent or solvable world and classical crystallography. We want to emphasize that most theorems and facts presented in the second part are from the last two decades. This is after the book of L Charlap "Bieberbach groups and flat manifolds" was published.Definitions.
Bieberbach Theorems.
The first Bieberbach Theorem.
Proof of the second Bieberbach Theorem.
Proof of the third Bieberbach Theorem.
Classification Methods.
Three methods of classification.
Classification in dimension two.
Platycosms.
Flat Manifolds with 6_a = 0.
Examples of (non)primitive groups.
Minimal dimension.
Outer Automorphism Groups.
Some representation theory and 9-diagrams.
Infinity of outer automorphism group.
IZi - groups.
Spin Structures and Dirac Operator.
Spin(n) group.
Vector bundles.
Spin structure.
The Dirac operator.
Flat Manifolds with Complex Structures.
Kahler flat manifolds in low dimensions.
The Hodge diamond for Kahler flat manifolds.
Crystallographic Groups as Isometries of Hⁿ.
Hyperbolic space Hⁿ.
Hantzsche-Wendt Groups.
Definitions.
Non-oriented GHW groups.
Graph connecting GHW manifolds.
Abelianization of HW group.
Relation with Fibonacci groups.
An invariant of GHW.
Complex Hantzsche-Wendt manifolds.
Open Problems.
The classification problems.
The Anosov relation for flat manifolds.
Generalized Hantzsche-Wendt flat manifolds.
Flat manifolds and other geometries.
The Auslander conjecture.Alternative Proof of Bieberbach Theorem.
Burnside Transfer Theorem.
Example of a Flat Manifold without Symmetry.