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Discrete Subgroups of the Euclidean and Hyperbolic Planes

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Discrete Subgroups of the Euclidean and Hyperbolic Planes University of Nevada, Reno Wallpapering the Hyperbolic Plane: Discrete Subgroups Of the Euclidean and Hyperbolic Planes A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics by Susannah C. Coates Dr. Valentin Deaconu/Thesis Advisor December 2014 c by Susannah C. Coates 2014 All Rights Reserved THE GRADUATE SCHOOL We recommend that the thesis prepared under our supervision by SUSANNAH C. COATES entitled Wallpapering the Hyperbolic Plane: Discrete Subgroups Of the Euclidean and Hyperbolic Planes be accepted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Valentin Deaconu, Ph.D., Advisor Swatee Naik, Ph.D., Committee Member Frederick Harris, Ph.D., Graduate School Representative David Zeh, Ph.D., Dean, Graduate School December 2014 i Abstract The question driving this thesis is, \What are the hyperbolic analogues of the Eu- clidean wallpaper groups?" We first investigate symmetry groups of the Euclidean plane, the frieze and wallpaper groups, understanding both in terms of generators and relations. Each wallpaper group has a presentation as an extension of a point group by Z2. Some also have presentations in terms of reflections in the sides of a triangle. Since triangle groups also exist in the hyperbolic plane, we discuss the Euclidean triangle groups. We then examine the isometries of the hyperbolic plane, finding that Isom(H) differs structurally from Isom(E). In particular, translations are not closed in the hyperbolic plane. Thus, though the frieze groups are structurally the same in the hyperbolic as in the Euclidean context, Isom(H) has no translation subgroup, and so the Euclidean wallpaper groups do not transfer. There do exist tilings of the hyperbolic plane, however. We close with a discussion of the Fuchsian groups, and some hyperbolic triangle groups. The original results of this thesis analyze the composition of certain translations and of certain parallel displacements in Isom(H). ii Acknowledgements First and foremost, I wish to thank my advisor, Dr. Valentin Deaconu who has tire- lessly guided and encouraged this research over the past year and a half. My thanks also to my committee members Dr. Swatee Naik, for her rigorous and considered critiques, and her support and counsel outside of the thesis itself, and Dr. Frederick Harris for his immediate attention, detailed comments, and cheerful and freely given guidance. Acknowledgement is due to the following individuals, in no particular order, both inside and outside of academia, who have aided and sustained me throughout this project: Philip Raath, Mary-Margaret Coates, Sarah Tegeler, Dr. Anna Panorska, Jordan Blocher, Xander Henderson et al., Dr. Slaven Jabuka, Dr. Lou Talman, Dr. Chris Herald, Mike and Jayde Gilmore, Joe and Ashley Thibadeau, Banafsheh Rekabdar, Anthony LaFleur and Annalee Gomm. This list of names is of course incomplete. iii Contents Introduction 1 1 Isometries in E 4 1.1 Fundamental Definitions and Axioms . .4 1.2 Classification of Isometries of E .....................7 2 Group Structure of Isom(E) 12 2.1 Describing Isometries Using Complex Numbers . 13 2.2 Isometries As An Extension of O (2) by R2 .............. 15 2.3 Discrete Subgroups of Isom(E)...................... 20 2.4 Finite Subgroups of Isom(E)....................... 22 3 Frieze Groups 25 4 Wallpaper Groups 32 4.1 The Crystallographic Restriction . 34 4.2 Lattices of R2 ............................... 38 4.3 Wallpaper Group Presentations . 41 4.4 Triangle Groups . 55 5 Isometries in H 64 iv 5.1 The Hyperbolic Plane . 65 5.2 Models of the Hyperbolic Plane . 66 6 Classification of Isometries in H 71 7 Group Structure of Isom(H) 76 7.1 Linear Fractional Transformations and Fixed Points . 83 7.2 Matrices and Trace . 84 7.3 Composing Translations and Parallel Displacements . 90 7.4 Discrete Subgroups of Isom(H)..................... 95 7.5 Finite Subgroups of Isom(H)....................... 96 8 Frieze Groups in H 100 9 Analogues of Wallpaper Groups in H 108 9.1 Dirichlet Region . 109 9.2 Triangle Groups . 113 9.3 Conclusions . 116 9.4 Questions and Future Work . 117 Appendix: Semidirect Product of Groups 118 v List of Figures 1.1 Illustration of Proposition 1.33 . 11 3.1 Partial orbit of O under the translation subgroup h(I; v)i ....... 26 3.2 Hop . 27 3.3 Jump . 27 3.4 Sidle . 28 3.5 Step . 29 3.6 Spinhop . 29 3.7 Spinjump . 30 3.8 Spinsidle . 31 4.1 Fundamental region and translation lattice of a wallpaper group . 33 4.2 Wallpaper group W1 ........................... 43 1 4.3 Wallpaper group W1 ........................... 43 1 4.4 Example of wallpaper group W1 , including fundamental region Image credit: [8], Plate 11, Egyption No 8, Image 11 . 44 2 4.5 Wallpaper group W1 ........................... 44 3 4.6 Wallpaper group W1 ........................... 45 4.7 Wallpaper group W2 ........................... 46 1 4.8 Wallpaper group W2 ........................... 47 vi 2 4.9 Wallpaper group W2 ........................... 47 3 4.10 Wallpaper group W2 ........................... 48 4 4.11 Wallpaper group W2 ........................... 49 4.12 Wallpaper group W3 ........................... 50 1 4.13 Wallpaper group W3 ........................... 50 2 4.14 Wallpaper group W3 ........................... 51 4.15 Wallpaper group W4 ........................... 52 1 4.16 Wallpaper group W4 ........................... 52 2 4.17 Wallpaper group W4 ........................... 53 4.18 Wallpaper group W6 ........................... 54 1 4.19 Wallpaper group W6 ........................... 55 1 4.20 Example of wallpaper group W6 , including fundamental region Image credit: [8], Plate 30, Byzantine No 3, Image 19 . 55 4.21 Example of a (p; q; r) triangle . 57 4.22 Partial image of a fundamental triangle under the action of a full (p; q; r) triangle group . 57 4.23 Partial image of a fundamental quadrilateral under the action of an ordinary (p; q; r) triangle group . 58 4.24 Partial image of F under the action of the full (3; 3; 3) triangle group 59 4.25 Partial image of F under the action of the full (2; 4; 4) triangle group 61 4.26 Partial image of F under the action of the full (2; 3; 6) triangle group 62 5.1 Orthogonal circles . 67 5.2 Poincar´edisc model: divergently and asymptotically parallel lines . 67 5.3 Isomorphism between the Poincar´edisc and upper half plane models . 69 5.4 Distance from z to w equals distance from z1 to w1 .......... 70 6.1 Inverse P 0 of P in the circle γ ...................... 72 vii 6.2 Reflection of z across the geodesic l is the inverse z0 of z in l ..... 73 6.3 Partial orbits of z and w under translation with respect to t ..... 74 6.4 Partial orbit of i under parallel displacement about ideal point Σ = 2 74 7.1 d (z; D (z)) varies depending on the imaginary part of z ........ 88 8.1 Hop . 102 8.2 Jump . 103 8.3 Sidle . 104 8.4 Step . 105 8.5 Spinhop . 105 8.6 Spinjump . 106 8.7 Spinsidle . 107 9.1 Intersection of half-planes to determine a Dirichlet region . 111 9.2 Tesselation of the upper half-plane by the Modular group . 112 9.3 Tesselation of the upper half-plane by P GL (2; Z)........... 113 9.4 Triangles T1 and T2 generate the same group. 115 1 Introduction Euclid set down the Elements of Geometry more than 2000 years ago, beginning with the five axioms, and building up the structure of Euclidean geometry from there. Other axioms have been added over the millenia, and Euclidean geometry refined and better understood. The wallpaper groups, also called the 2-dimensional crystallographic groups, are sym- metry groups of the Euclidean plane wherein a convex polygonal fundamental region is translated in two non-parallel vectors, and possibly rotated or reflected as well, such that it tiles the plane. There are precisely 17 ways to do this, 17 wallpaper groups. The fifth axiom of Euclidean geometry is the Euclidean Parallel Postulate, that given a line, and a point not on that line, there exists exactly one line through the point that never intersects the given line. The Euclidean Parallel Postulate bedevilled mathe- maticians from Euclid's time on, seeming as if it should be provable, and therefore not a necessary assumption. In fact, the parallel postulate is independent of the other four axioms, and logically consistent geometries emerge when it is tinkered with. These are the non-Euclidean geometries. Hyperbolic geometry emerges with the assumption that given a line and a point not 2 on that line, there exist infinitely many lines through the point that never intersect the given line. So how does this one apparently simple change affect the tilings of the plane? What are the hyperbolic analogues of the Euclidean wallapaper groups? A review of current literature showed that while investigation into isometry groups of the hyperbolic plane is plentiful, no recently published papers addressed the question in the manner in which we wished to approach it. Thus, we turned to the fundamentals of group theory and the isometry groups of the Euclidean and hyperbolic planes in order to investigate. Chapter 1 lays out the fundamental definitions and axioms that underlie the rest of the discussion, and discusses the isometries of the Euclidean plane and how they are built from compositions of reflections. Chapter 2 investigates the structure of the group of isometries of the Euclidean plane, Isom(E). The Euclidean isometries may be represented in terms of complex numbers. They may also be described by vectors in the additive group (R2; +) and elements of the group of 2 × 2 orthogonal matrices, (O (2) ; ·). Since ι 2 π 2 R 0 ! R ; + ! (Isom (E) ; ∗) (O (2) ; ·) ! 1 ιO(2) is a short exact sequence of group homomorphisms which is split exact, then ∼ 2 Isom(E) = R o O (2).
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