On Space Groups and Dirichlet{Voronoi Stereohedra Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) am Fachbereich Mathematik und Informatik der Freien Universit¨atBerlin von Moritz W. Schmitt Berlin 2016 Betreuer: Prof. G¨unter M. Ziegler, Ph.D. Zweitgutachter: Prof. Dr. Achill Sch¨urmann Tag der Disputation: 14. Juli 2016 Preface This thesis is about space groups and their applications in discrete geometry. It is structured into three different parts. Each part corresponds to one of the three chapters, which we will describe in the following. In Chapter 1 an introduction to n-dimensional space groups is given. While there are many other texts on space groups, most of them cover only dimension n ≤ 3, focus very much on crystallographic aspects, or are simply not up- to-date. We therefore tried to remedy the situation by collecting the most important results from this area, especially those that are related to tilings and the algebraic structure of space groups. We made a great effort to include the most important references on space groups. Our hope is that future researchers might find this survey a good starting point for their explorations. In the second chapter we present a detailed investigation of which Dirichlet{ Voronoi stereohedra the tetragonal, trigonal, hexagonal, and cubic groups can generate. A stereohedron is a convex n-dimensional polytope that tiles Rn by the action of some space group. If this polytope is a Dirichlet{Voronoi domain of some orbit of some space group, the stereohedron is called a Dirichlet{Voronoi stereohedron. Such stereohedra are examples of so-called convex monohedral tiles, whose possible shapes and combinatorial properties are poorly understood in general. In particular it is a long-standing open question whether the number of facets of a convex monohedral tile can be bounded by a function that only depends on the dimension n. With our investigations we want on the one hand to complement the work of Santos et al. [BS01; BS06; SS08; SS11] and on the other hand hope to give a realistic picture of what can be expected from 3- dimensional space groups. To carry out these computations we developed the extensive software suite plesiohedron, which is an important part of this thesis and can be found at github.com/moritzschmitt/plesiohedron. Finally, Chapter 3 is devoted to the number s(n) of isomorphism classes of space groups of Rn. It follows from a theorem by Bieberbach that there are only finitely many nonisomorphic space groups in each dimension. Schwarzenberger showed by counting the orthogonal space groups that 2 s(n) = 2Ω(n ) and Buser later proved O(n2) s(n) = 22 : We were able to show that O(n log n) s(n) = 22 : For this we develop a new bound on s(n), which we also use to bound the number of conjugacy classes of finite subgroups in GL(n; Z). This bound is then evaluated by applying the mass formula of Smith{Minkowski{Siegel. In Appendix A we concisely provide the necessary background on group cohomology for Chapter 1. This is not new but can also not easily be found in the overwhelming body of literature of homological algebra. Appendix B gives details on how exactly the computations for Chapter 2 were carried out. ii Acknowledgements First and foremost I would like to thank my two advisors G¨unter M. Ziegler and Ivan Izmestiev. G¨unter was always available for questions and helpful dis- cussion, and his great mathematical intuition gave me guidance on more than one occasion. I am also grateful for the funding I received from his grants. Ivan listened to my ideas and presentations for hours on end with much patience, and his pointed questions and insightful comments were immensely useful. Besides my two advisors there are also many other people I wish to thank for discussing questions related to this thesis: Mois Aroyo, Benjamin Assarf, Pavle Blagojevi´c,Nikolai Dolbilin, Alexander Engstr¨om,Tobias Finis, Moritz Firsching, Elke Koch, Benjamin Lorenz, Arnau Padrol, Julian Pfeifle, Francisco Santos, Raman Sanyal, Jan-Christoph Schlage-Puchta, Benjamin Schr¨oter,and Christian Stump. Achill Sch¨urmannagreed to be the external referee for this thesis, for which I am much obliged to him. For this thesis many computations were carried out on the allegro and ZE- DAT/HPC clusters of Freie Universit¨at. Both clusters are managed by very competent administrators who patiently answered my many emails and fulfilled all requests I had. Without them a large part of this thesis would not have been possible. For much needed moral and emotional support I am deeply grateful to my partner Melanie Win Myint. Her contributions to the success of this thesis are immeasurable. Our children Ella and Noah are always enjoyable distractions from less livelier things such as the preparation of this document. Finally, I would like to thank my parents Astrid Bismark-Schmitt and Andreas Schmitt for their constant support and the confidence they equipped me with. iii Contents Preface . ii 1 A Primer on Euclidean Space Groups 1 1.1 Introduction and Bieberbach's theorems . .4 1.2 Geometric properties . .9 1.3 Algebraic properties . 16 2 Dirichlet{Voronoi Stereohedra 23 2.1 Setting the stage . 23 2.2 Face vectors of DV-stereohedra . 25 2.2.1 Triclinic groups . 26 2.2.2 Tetragonal groups . 27 2.2.3 Trigonal groups . 81 2.2.4 Hexagonal groups . 102 2.2.5 Cubic groups . 119 2.3 Discussion of the results . 151 3 On the Number of Space Groups 157 3.1 Known results . 157 3.2 A new upper bound . 158 3.3 On the number of Z-classes . 162 Appendices 164 A Cohomology of Groups . 165 B Background on Computations . 168 Bibliography 174 Index of Space Groups 182 Index 186 Deutsche Zusammenfassung 189 Erkl¨arung 190 Chapter 1 A Primer on Euclidean Space Groups This chapter serves a double purpose. First and foremost it shall serve as a concise introduction to Euclidean space groups. This beautiful part of group theory deserves to be known more widely, and when we started working on it, we quickly found ourselves in agreement with Yale who wrote \(. .) we struggled to correlate the various notations, incomplete proofs, and partial presentations found in other introductions (. .)" [Yal68, p. 119] Yale somewhat remedies the situation for dimensions n ≤ 3 but does not discuss space groups in higher dimensions. However, higher dimensional space groups are of importance, also for this thesis, and that is why we treat the general case. The other purpose { and this sets our introduction apart from Yale's even in dimension three { is to present a survey of the most important results from the area. There are many other introductions to space groups. Let us list those known to us together with short comments on what they cover. We refrain from listing the many books on crystallography, even though almost all of them also have space groups as their topic. But on the one hand we found none that covers groups in dimensions higher than three, and often they are not as rigorous as needed by mathematicians. On the other hand they naturally concentrate on topics that are of interest to crystallographers and solid-state scientists but maybe less so to mathematicians. Marcel Berger. Geometry I. Universitext. Springer, 2009: This is the treatment closest in spirit to ours. Space groups are understood as a mean to generate monohedral tilings. Unfortunately, only the planar case is treated and the classification is done only for space groups consisting of positive isometries. Harold Brown et al. Crystallographic groups of four-dimensional space. Wiley, 1978: On the first few pages a readable but incomplete in- troduction to space group theory is given. The bulk of the book are tables that 1 present the classification of four-dimensional space groups as a tedious list that nowadays would not appear in print. Johann Jakob Burckhardt. Die Bewegungsgruppen der Kristal- lographie. 2nd ed. Birkh¨auser,1966: Burckhardt gives a very concrete introduction to space groups with strong emphasis on dimensions n = 2; 3. He does not discuss any advanced theorems such as the ones by Bieberbach. John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss. The symmetries of things. A K Peters, 2008: In this very interesting book an introduction to the powerful orbifold signature is given. This signature is used to classify all three-dimensional space groups. Higher-dimensional space groups are not mentioned in the book. Harold S. M. Coxeter. Introduction to geometry. Wiley, 1989: This classical work on geometry only provides a superficial introduction to pla- nar space groups in its Chapter 4. Nonetheless it is a nice starting point for further explorations. Harold S. M. Coxeter and William O. J. Moser. Generators and relations for discrete groups. 4th ed. Springer, 1980: Before all 17 two-dimensional space groups are given abstractly as finite presentations in Chap. 4, the authors quickly discuss finite symmetry and rotation groups. It is nicely explained how they arrive at each presentation by using Cayley graphs. Leonard S. Charlap. Bieberbach groups and flat manifolds. Spring- er, 1986: A torsion-free space group is called a Bieberbach group. The orbit space of a Bieberbach group Γ is a manifold M n = Rn=Γ with funda- mental group Γ and Riemannian structure inherited from Rn. It is a so-called flat manifold, i.e., a manifold with sectional curvature zero. Moreover, every compact flat manifold can be obtained as an orbit space of a Bieberbach group. Charlap's book starts to discuss Bieberbach's theorems in detail and then pro- ceeds by discussing the algebraic and geometric properties of Bieberbach groups.