GEOMETRY CRYSTAL GROUPS (2ND ED)

by Andrzej Szczepanski

Overview

It is eleven years since the First Edition of Geometry of Crystallographic Groups appeared. This Second Edition expands on the first, providing details of a new result of automorphism of crystallographic groups, and on Hantzsche-Wendt groups/manifolds.

Crystalographic groups are groups which act via isometries on some n-dimensional Euclidean space, so-named because in three dimensions they occur as the symmetry groups of a crystal. There are short introductions to the theme before every chapter, and a list of conjectures and open projects at the end of the book.

Geometry of Crystallographic Groups is suitable as a textbook for students, containing basic theory of crystallographic groups. It is also suitable for researchers in the field, discussing in its second half more advanced and recent topics.

Contents:

• Definitions
• Bieberbach Theorems
• Classification Methods
• Flat Manifolds with b₁ = 0
• Outer Automorphism Groups
• Spin Structures and Dirac Operator
• Flat Manifolds with Complex Structures
• Crystallographic Groups as Isometries of ℍⁿ
• Hantzsche-Wendt Groups
• Combinatorial Hantzsche-Wendt Groups
• Open Problems

Readership: Researchers in geometry and topology, algebra and theory students, Institutes of Crystallography, University Chemistry departments.

Review of the First Edition:'This very precise and well written text is an extended version of the notes of the lectures given by the author at Gdańsk University for graduate students.' - The European Mathematical Society

Key Features:

• This is a mathematical book, but crystallography is also a popular topic within Chemistry and Physics. It is therefore a useful book for students of all three of these
• This book builds on the work of L S Charlap, Bieberbach Groups and Flat Manifolds, with many fresh and important insights and results
• New materials from the last two decades are detailed clearly

Product Details

ISBN-13:	9789811286612
Publisher:	WSPC
Publication date:	07/30/2024
Series:	ALGEBRA AND DISCRETE MATHEMATICS , #5
Sold by:	Barnes & Noble
Format:	eBook
Pages:	272
File size:	21 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface v

1 Definitions 1

1.1 Exercises 8

2 Bieberbach Theorems 11

2.1 The first Bieberbach Theorem 11

2.2 Proof of the second Bieberbach Theorem 16

2.2.1 Cohomology group language 16

2.3 Proof of the third Bieberbach Theorem 24

2.4 Exercises 25

3 Classification Methods 29

3.1 Three methods of classification 30

3.1.1 The methods of Calabi and Auslander-Vasquez 31

3.2 Classification in dimension two 39

3.3 Platycosms 42

3.4 Exercises 48

4 Flat Manifolds with b¹ = 0 51

4.1 Examples of (non)primitive groups 55

4.2 Minimal dimension 57

4.3 Exercises 62

5 Outer Automorphism Groups 63

5.1 Some representation theory and 9-diagrams 63

5.2 Infinity of outer automorphism group 70

5.3 R₁ - groups 77

5.4 Exercises 83

6 Spin Structures and Dirac Operator 85

6.1 Spin(n) group 85

6.2 Vector bundles 88

6.3 Spin structure 91

6.3.1 Case of cyclic holonomy 98

6.4 The Dirac operator 103

6.5 Exercises 109

7 Flat Manifolds with Complex Structures 111

7.1 Kähler flat manifolds in low dimensions 114

7.2 The Hodge diamond for Kähler flat manifolds 117

7.3 Exercises 123

8 Crystallographic Groups as Isometries of Hⁿ 125

8.1 Hyperbolic space Hⁿ 125

8.2 Exercises 130

9 Hantzsche-Wendt Groups 131

9.1 Definitions 131

9.2 Non-oriented GHW groups 135

9.3 Graph connecting GHW manifolds 142

9.4 Abelianization of HW group 144

9.5 Relation with Fibonacci groups 148

9.6 An invariant of GHW 152

9.7 Complex Hantzsche-Wendt manifolds 158

9.8 Exercises 160

10 Open Problems 163

10.1 The classification problems 163

10.2 The Anosov relation for flat manifolds 165

10.3 Generalized Hantzsche-Wendt flat manifolds 166

10.4 Flat manifolds and other geometries 167

10.5 The Auslander conjecture 169

Appendix A Alternative Proof of Bieberbach Theorem 171

Appendix B Burnside Transfer Theorem 175

Appendix C Example of a Flat Manifold without Symmetry 179

Bibliography 187

Index 193