Discrete Subgroups of the Euclidean and Hyperbolic Planes

Home , Frieze group, Point group, Point reflection, Wallpaper group

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

Susannah C. Coates

Dr. Valentin Deaconu/Thesis Advisor

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 symmetry 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). The discrete subgroups of Isom(E) are discussed, as are the finite subgroups, and the cyclic and dihedral groups. Chapter 3 characterizes the frieze groups, gives a presentation and the group structure, and discusses the symmetries of each group. Chapter 4 characterizes the Euclidean wallpaper groups, with discussion of the crystallographic restriction, which determines that a wallpaper group may contain a rotation of order 1, 2, 3, 4 or 6, and no other; and of lattices determined by the translation subgroup of each wallpaper group. This chapter ends 3 by presenting each wallpaper group as the extension of a point group by Z2, and then recharacterizes some of the wallpaper groups as triangle groups. Chapter 5 introduces the hyperbolic plane and considers two models, the Poincarédisk, and the Poincaré upper half-plane. Chapter 6 classifies the isometries of the hyperbolic plane, building them from compositions of reflections, and finding an isometry that does not exist in

Isom(E), the parallel displacement.

Chapter 7 investigates the structure of the group of isometries of the hyperbolic plane,

Isom(H). In the Poincar´eupper half-plane model, elements in Isom(H) may be represented by linear fractional transformations, which in turn may be treated as 2 × 2 matrices. Beginning with GL (2, R), the general linear group of 2 × 2 matrices having entries in R and nonzero determinant, we trace through the subgroups to PSL (2, R), the projective special linear group of 2 × 2 real matrices having determinant 1, which describes all direct isometries in the Poincar´eupper half-plane model. Orientation reversing transformations are elements having determinant −1 in P GL (2, R), the projective general linear group of 2×2 real matrices. The trace of each matrix determines the isometry-type it represents, and translations in Isom(H) are not closed. This sets up an analysis of how the relative translation distance determines the isometry-type of composed translations, and of composed parallel displacements. The discrete and

ﬁnite subgroups of Isom(H) are characterized.

Chapter 8 describes the frieze groups in the hyperbolic plane. They are structurally identical to the frieze groups in the Euclidean plane. Chapter 9 addresses the impos- sibility of groups in Isom(H) which are structurally identical to the wallpaper groups in Isom(E), and then discusses some groups which do tile the hyperbolic plane, including triangle and regular polygonal groups. The paper ends by noting questions which have been raised and not yet addressed in this project. 4

Chapter 1

Isometries in E

1.1 Fundamental Deﬁnitions and Axioms

In this thesis, Euclid’s axioms are assumed, as well as Hilbert’s axioms of betweenness, congruence and continuity. For the convenience of the reader and to establish notation, we recall some of these axioms and deﬁnitions.

The undeﬁned terms are point, line, lie on, between and congruent. Given a point P that lies on line l, this can be said as, “P is incident with l,” or “l passes through P .” If P lies on both line l and line m, then “l and m intersect at P ,” or “l and m have P in common.” In Euclidean geometry, line is synonymous with “straight line,” and is considered to be the unique shortest path between any two points lying on it. In the Euclidean metric, this shortest path is the line segment connecting A and B.

Notationally, A ∗ B ∗ C reads “point B is between points A and C.”

Axiom 1.1 (Betweenness Axiom I). If A ∗ B ∗ C, then A,B, and C are three distinct 5

points all lying on the same line, and C ∗ B ∗ A ([4], p73).

Axiom 1.2 (Betweenness Axiom II). Given any two distinct points B and D, there ←→ exist points A, C and E lying on BD such that A ∗ B ∗ D, B ∗ C ∗ D, and B ∗ D ∗ E ([4], p74).

Axiom 1.3 (Betweenness Axiom III). If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two ([4], p74).

Axiom 1.4 (Euclid’s Postulate I). For every point P and for every point Q not equal to P there exists a unique line l that passes through P and Q ([4], p14).

Deﬁnition 1.5. Given two points A and B, the segment AB is the set whose mem- ←→ bers are A and B, and all points that lie on the line AB and are between A and B. The two given points A and B are called the endpoints of the segment AB ([4], p14).

Axiom 1.6 (Euclid’s Postulate II). For every segment AB and for every segment CD there exists a unique point E such that B is between A and E, and segment CD is congruent to the segment BE ([4], p15).

Deﬁnition 1.7. Given two points O and A, the set of all points P such that the segment OP is congruent to a segment OA is called a circle with O as center, and each of the segments OP is called a radius of the circle ([4], p15).

Axiom 1.8 (Euclid’s Postulate III). For every point O and every point A not equal to O there exists a circle with center O and radius OA ([4], p15).

−→ ←→ Deﬁnition 1.9. The ray AB is the following set of points lying on the line AB: ←→ those points that belong to the segment AB and all points C on AB such that B is 6

−→ between A and C. The ray AB is said to emanate from the vertex A and to be part ←→ of the line AB ([4], p16).

−→ −→ Deﬁnition 1.10. Rays AB and AC are said to be opposite if they are distinct, if ←→ they emanate from the same point A, and if they are part of the same line AB.

Deﬁnition 1.11. An angle with vertex A is a point A together with two distinct −→ −→ nonopposite rays AB and AC (called the sides of the angle) emanating from A ([4], p17).

−−→ Deﬁnition 1.12. If two angles ∠BAD and ∠CAD have a common side AD, and −→ −→ the other two sides AB and AC form opposite rays, then the angles are supplements of each other, or are supplementary angles ([4], p17).

Deﬁnition 1.13. An angle ∠BAD is a right angle if it has a supplementary angle to which it is congruent ([4], p17).

Axiom 1.14 (Euclid’s Postulate IV). All right angles are congruent to each other.

Deﬁnition 1.15. Two lines l and m are parallel if they do not intersect, i.e., if no point lies on both of them. We denote this by l k m ([4], p19).

Axiom 1.16 (The Euclidean Parallel Postulate). For every line l and for every point P that does not lie on l, there exists a unique line m through P that is parallel to l ([4], p19).

A weaker version of this parallel postulate is attributed to David Hilbert.

Axiom 1.17 (Hilbert’s Axiom of Parallelism). For every line l and every point P not lying on l there is at most one lie m through P such that m is parallel to l ([4], p102). 7

1.2 Classiﬁcation of Isometries of E

Deﬁnition 1.18. An automorphism of the Euclidean plane E is a bijective mapping f : E → E which preserves incidence, betweenness and congruence of angles ([4], p321).

The automorphisms of E, Aut(E), form a group under function composition, and elements of Aut(E) do not necessarily preserve distance. For instance, 4ABC and its image 4A0B0C0 under f must have equal angles, and as a result, the distances A0B0 and B0C0 must only maintain the same proportion to each other as AB and BC. Equality is not required.

Deﬁnition 1.19. The 4ABC and 4A0B0C0 are similar if and only if the lengths of their corresponding sides are proportional ([11], p4).

Since automorphisms preserve betweenness, then given f ∈ Aut (E) and points A, B, C with A ∗ B ∗ C, so A, B, C are collinear, then f (A) ∗ f (B) ∗ f (C), and so f (A) , f (B) , f (C) are also collinear.

Deﬁnition 1.20. A collineation is a function f that maps lines to lines ([4], p310).

The Euclidean plane can be represented as R2, equipped with the Euclidean metric where for any two points P (x1, y1), Q (x2, y2) ∈ E, the distance between them is q 2 2 deﬁned as d (P,Q) = (x2 − x1) + (y2 − y1) .

Deﬁnition 1.21. An isometry is a distance-preserving automorphism ([4], p321).

For each isometry f : E → E, d (f (P ) , f (Q)) = d (P,Q), ∀P,Q ∈ E. The practical upshot of this is that an isometry is, “a collineation that preserves betweenness, midpoints, segments, rays, triangles, angles, angle measure and perpendicularity ([11], 8

p28).” The preservation of distance results in the preservation of these other structural aspects as well.

Since every isometry is an automorphism, then as sets Isom(E) ⊂ Aut (E). Under composition of functions, Isom(E) is a subgroup of Aut(E). To denote the subgroup, we write Isom(E) ≤ Aut (E) ([4], p321).

Greenberg [4] establishes the Euclidean isometries, beginning with reﬂections, and constructing all other isometries by composing no more than three reﬂections. He

demonstrates that the only non-identity isometric transformations of E are the re- ﬂection, glide reﬂection, rotation and translation.

Definition 1.22. A reflection Rm in a line m is an isometry which fixes m and transforms each point P/∈ m as follows. Let M be the foot of the perpendicular from

0 0 0 P to m. Then Rm (P ) is the unique point P such that P ∗M ∗P and PM = P M([4], p111).

Deﬁnition 1.23. An involution is a transformation which is its own inverse.

0 0 Given that Rm (P ) = P , a second reﬂection across m sends P 7→ P , so RmRm = I, the identity transformation. Thus Rm is an involution. Given any line l ⊥ m, and

any point P ∈ l, then Rm (P ) ∈ l, so the line l is invariant under Rm. Note that if

P ∈ l is also an element of m, P is ﬁxed by Rm.

Deﬁnition 1.24. A translation is constructed by composing two reﬂections in distinct lines l and m, which have a common perpendicular t. That is, Tt = RmRl. Then the axis of translation is t, and the translation is said to be along t or to have taken place by a vector parallel to t ([4], p326).

The translation isometry has no fixed points. In the first reflection Rl, the line l 9

remains fixed, and all other points are reflected about l. In the second reflection Rm, points of m remain fixed, and all other points are reflected about m. Given that l

and m are distinct, then the ﬁxed point set Fix(Rm) of Rm is reﬂected by Rl, and

Fix(Rl) is reﬂected by Rm. So Fix(RmRl) = Fix (Rm) ∩ Fix (Rl) = ∅. Invariant sets

under Tt are those lines parallel to t.

Deﬁnition 1.25. If lines l and m intersect at a point P , then RmRl ﬁxes P , and

RmRl = SP,θ is a rotation about P by an angle θ. The point P is the center of rotation.

Deﬁnition 1.26. If l and m are not perpendicular, they form a pair of obtuse and

a pair of acute angles at their point of intersection. Then for SP,θ = RmRl, the angle of rotation θ = 2φ where φ is the magnitude of the acute angle.

The direction of the rotation depends on the order of reﬂection. If SP,θ = RmRl, then

RlRm = SP,−θ, a rotation in the opposite direction. So SP,θSP,−θ = RmRlRlRm =

−1 RmRm = I, whence SP,−θ = SP,θ.

π Deﬁnition 1.27. If l ⊥ m, then φ = , and given S = R R , then θ = π. This 2 P,θ m l particular rotation is called a half-turn about P .

0 0 0 Given any point Q 6= P , then SP,π (Q) = Q where Q ∗ P ∗ Q and QP = PQ .

Thus any line intersecting P is invariant under SP,π. Consider the composition of two half-turns about distinct centers, A and B.

Proposition 1.28. Given a point B on a line t, then T is a nontrivial translation along t if and only if there exists a unique point A 6= B such that T is the composition of the half-turns SA,π and SB,π ([4], Proposition 9.14).

The composition of two half-turns is a translation along the line containing the two 10

centers of rotation.

Proposition 1.29. In the Euclidean plane, the result of the composition of three

half-turns SC,πSB,πSA,π is again a half-turn, whether or not A, B, C are collinear ([4], Proposition 9.15).

Deﬁnition 1.30. Any isometry which is a composition of two reﬂections is an orientation preserving, even or direct isometry.

Translations and rotations are the even elements of Isom(E). The Angle Addition Theorem identiﬁes the isometry type of a composition of rotations.

Theorem 1.31 (Angle Addition Theorem). A rotation by θ1 followed by a rotation by θ2 is a rotation of θ1 + θ2 unless θ1 + θ2 = 2kπ for k ∈ Z, in which case the product is a translation. A translation followed by a nonidentity rotation by θ is a rotation by θ. A nonidentity rotation by θ followed by a translation is a rotation by θ. A translation followed by a translation is a translation ([11], Theorem 7.9).

Deﬁnition 1.32. A glide reﬂection D is the composition of a nonidentity translation

Tt having axis t, and a reﬂection Rt across t. Given translation Tt = RmRl where l ⊥ t and m ⊥ t, then glide reﬂection D = RtTt = RtRmRl ([4], p338).

Concerning the characteristics of the glide reﬂection, Greenberg gives the following proposition (Figure reference, [4], Figure 9.18).

Proposition 1.33. (See Figure 1.1.) Let l ⊥ t at A, m ⊥ t at B, Tt = RmRl,

D = RtTt. Then

(a) RtTt = D = TtRt

(b) SA,πRm = D = RlSB,π 11

(d) D has no ﬁxed points

(e) The only line invariant under D is t

(f) Conversely, given a line l and a point B, let the line t be the perpendicular to l and contain B. Then

(f.1) if B does not lie on l, RlSB,π is a glide reﬂection having axis t

(f.2) if B lies on l, RlSB,π = Rt

P Rl (P ) Tt (P ) = RmRl (P )

t A B

SA,π (P ) D (P ) = SA,πRm (P ) = RlSB,π (P ) l m

Figure 1.1: Illustration of Proposition 1.33

Since the identity transformation may be considered a “composition of zero reflections,” and a reflection itself considered a “composition of one reflection,” then any

isometry in Isom(E) may be constructed from a composition of no more than three reﬂections ([4], p326).

Deﬁnition 1.34. Any isometry which is a composition of one or three reﬂections is an orientation reversing, odd or opposite isometry.

Reﬂections and glide reﬂections are the odd elements of Isom(E). 12

Chapter 2

Group Structure of Isom(E)

Consider isometries in terms of coordinates in the Cartesian model of the Euclidean plane E. A translation “moves each point a ﬁxed distance in a ﬁxed direction ([4], p344).” Represent the distance and direction of a translation T by a vector emanating from the origin. Then the translation T (P ) of a point P (x, y), is

T (P ) = (x, y) + (a, b) = (x + a, y + b) where (a, b) are the coordinates of the endpoint of the translation vector ha, bi. Note that for a nonidentity translation, no point is mapped to itself, so Fix{T } = ∅.

To compose translations, the coordinates (a0, b0) of the endpoint of a second translation vector ha0, b0i are added to the once-translated point:

T 0 (x + a, y + b) = (x + a, y + b) + (a0, b0) = (x + a + a0, y + b + b0) .

So the composed translation T 0T corresponds to translation by the vector ha, bi + 13

ha0, b0i = ha + a0, b + b0i = ha0 + a, b0 + bi. Thus translations in E form an abelian group isomorphic to the group (R2, +) of real-valued vectors under vector addition ([4], p345). Depending on the context of the discussion, translations will be denoted either by T as a transformation, or v as a vector.

The other even isometry, rotation, fixes a point; reflection fixes a line; and glide reflection maps the axis of reflection to itself, leaving the line invariant (but no point fixed) while reflecting the rest of the plane. Description of these isometries requires something beyond vectors. Two ways of doing this are to use the complex numbers, or to use the group of real-valued 2 × 2 orthogonal matrices.

2.1 Describing Isometries Using Complex Num-

bers

In the complex plane, a translation vector is represented by the complex number

zt = at + bti corresponding to the endpoint of the vector emanating from the origin.

Then for the complex point z = a + bi, T (z) = z + zt = a + bi + at + bti = (a + at) +

(b + bt) i = (at + a) + (bt + b) i, and the group of translations of the complex plane is isomorphic to the additive abelian group (C, +) of complex numbers.

Recall that rotations SP,θ about center P by angle θ are the other type of orientation preserving isometry. By the following argument, we may reduce to considering rotations about the origin O. The ﬁxed point set of a nonidentity rotation S about P is Fix(S) = {P }. Let T be the translation mapping P 7→ O, and S0 a rotation of angle θ about O. Then the conjugation T −1S0T of S0 is equivalent to the rotation S about P ([4], p346). Since P and θ are arbitrary, we may consider a rotation of any angle 14

θ about any center P 6= O as a conjugated rotation about O.

Consider the point z = a + bi in the complex plane. A half-turn about the origin θ takes z 7→ −z. If the two axes of reﬂection l and m are not perpendicular, and 2 is the angle measure of the acute angle formed by l and m, with 0 < θ < π, then

the rotation S = RmRl may be written in terms of complex numbers in one of two ways, depending on the slope of l. If l has positive slope, then S (z) = eiθz. If l has negative slope, then S (z) = e−iθz ([4], p347). Stated more generally, S (z) = eiθz, −π < θ ≤ π. Given two rotations about O, S (z) = eiθz and S0 = eiϕz, their composition S0S = eiϕ eiθz = eiϕeiθ z = ei(ϕ+θ)z = ei(θ+ϕ)z. Then, “the group of rotations about a ﬁxed point is isomorphic to the one-parameter multiplicative group S1 of complex numbers eiθ of absolute value 1 ([4], p347).” Note that rotations about diﬀerent centers do not in general commute, and such a composition may be either a rotation or a translation.

Reﬂections and glide reﬂections are the orientation reversing transformations. They

iθ can be achieved with the mapping z 7→ e z + z0, where z is the complex conjugate of z, and the mapping z 7→ z is a reﬂection in the x axis ([4], p348). For a reﬂection,

z0 = 0. For a glide reﬂection, z0 6= 0.

To see how two arbitrary isometries behave when composed, let Φ, Φ0 ∈ Isom (E)

iθ 0 iθ0 0 with Φ (z) = e z + z0, and Φ (z) = e z + z0. Then

0 0 iθ Φ Φ(z) = Φ e z + z0

iθ0 iθ 0 = e e z + z0 + z0

iθ0 iθ iθ0 0 = e e z + e z0 + z0.

Thus the translation element of Φ is rotated by the rotation element of Φ0, so Φ and 15

Φ0 do not commute.

To elucidate the group structure, let us return to the real numbers and use matrices to describe rotations and reﬂections.

2.2 Isometries As An Extension of O (2) by R2

Translation of a point P = (x, y) by T = (a, b) ∈ R2 can be represented by vector addition, as before: T (P ) = (x + a, y + b).

←→ The line OP deﬁned by the point P and the origin O makes an angle φ with the positive x-axis. Consider the rotation SO,θ (P ) of P by θ about O. Deﬁne SO,θ (P ) = P 0 = (x0, y0). Then OP and OP 0 lie on the circle with center O and radius r = OP = OP 0; and so P = (x, y) = (r cos φ, r sin φ) and P 0 = (r cos (φ + θ) , r sin (φ + θ)). Recall the trigonometric identities cos (α + β) = (cos α) (cos β) − (sin α) (sin β), and sin (α + β) = (sin α) (cos β) + (cos α) (sin β). Then

x0 = r cos (φ + θ)

= r (cos φ) (cos θ) − r (sin φ) (sin θ)

= x cos θ − y sin θ and

y0 = r sin (φ + θ)

= r (sin φ) (cos θ) + r (cos φ) (sin θ)

= x sin θ + y cos θ. 16

    x x0 0     Writing the points P and P as the column vectors   and   respectively, the y y0 mapping P 7→ P 0 can be expressed in terms of matrix multiplication:

      cos θ − sin θ x x0           =   . sin θ cos θ y y0

  cos θ − sin θ   So the rotation SO,θ is expressed as the matrix A =  , giving SO,θ (P ) = sin θ cos θ AP = P 0.   1 0 0   A reﬂection Rx in the x-axis maps P = (x, y) 7→ P = (x, −y) using B =   0 −1 by       1 0 x x           =   . 0 −1 y −y

The composition SO,θRx of a rotation and reﬂection is given by

      cos θ − sin θ 1 0 cos θ sin θ       AB =     =   . sin θ cos θ 0 −1 sin θ − cos θ

This reverses the orientation of the object.

Deﬁnition 2.1. An orthogonal matrix is a matrix A such that AT A = I.

It follows that det(A) = ±1. The above deﬁned A, B and AB are orthogonal matrices, and so the set deﬁned by A and AB,

        cos θ − sin θ cos θ sin θ    ,   0 ≤ θ ≤ 2π      sin θ cos θ sin θ − cos θ  17

becomes a group of orthogonal matrices under matrix multiplication.

Deﬁnition 2.2. The orthogonal group (O (2, R) , ·) is the group of all 2×2 orthogonal matrices having real entries, with the group operation of matrix multiplication.

In the rest of this thesis, the group (O (2, R) , ·) is written O (2).          cos θ − sin θ cos θ sin θ  Clearly    ,   0 ≤ θ ≤ 2π , · ⊂ O (2).        sin θ cos θ sin θ − cos θ 

To prove containment the other way, we show that any A ∈ O (2) may be written in     a b a c   T   the above form. Let A =   ,A =   ∈ O (2). Since c d b d

      a c a b 1 0 T       A A =     =   , b d c d 0 1

then a2 + c2 = 1, and there exists an angle θ such that a = cos θ and c = sin θ; b2 + d2 = 1, and there exists and angle φ such that b = cos φ and d = sin φ; and ab+cd = 0, which gives that (cos θ) (cos φ)+(sin θ) (sin φ) = cos (θ − φ) = 0. So then

π π θ − φ = ± 2 , giving that φ = θ ± 2 . Thus,

    cos θ cos φ cos θ − sin θ     A =   =   , sin θ sin φ sin θ cos θ

since cos (φ) = cos θ ± π = − sin θ, and sin (φ) = sin θ ± π = cos θ. So there is 2    2      cos θ − sin θ cos θ sin θ  equality of groups,    ,   0 ≤ θ ≤ 2π , · = O (2)        sin θ cos θ sin θ − cos θ  ([12], pp250-253). 18

  a b   Let   = A ∈ O (2), and consider its action on the origin O = (0, 0): c d

      a b 0 0           =   . c d 0 0

Then any A preserves the origin, and so O (2) consists of transformations that have at least one ﬁxed point: rotations (det(A) = 1) and reﬂections (det(A) = −1).

Thus, similar to above with complex notation, each isometry of the Euclidean plane may be written as a rotation or reﬂection followed by a translation, TA (x) = T (Ax) =

Ax + v = (A, v)(x) where the T is translation by the vector v. So as a set Isom(E) = {(A, v) | A ∈ O(2), v ∈ R2}.

To ﬁnd the group operation for Isom(E), let (A, v) , (B, u) ∈ Isom (E), and x ∈ E. Then ([12], p252)

(A, v)(B, u)(x) = (A, v)(Bx + u)

= A (Bx + u) + v

= ABx + Au + v

= (AB, Au + v) x. (2.1)

For notational purposes, denote this operation ∗. Then the set Isom(E) is closed under ∗. The identity element (I, 0) uses the identity matrix and zero vector, and the inverse of any element (A, v) is (A, v)−1 = (A−1, −A−1v) ([12], p252). Since O (2) is noncommutative, then under ∗, the set Isom(E) becomes the noncommutative group (Isom (E) , ∗). From now on Isom(E) refers to this group. 19

Consider the inclusion of R2 into Isom(E) by v 7→ (I, v). Then for any (A, u) ∈ Isom (E),

(A, u)(I, v)(A, u)−1 = (A, u)(I, v) A−1, −A−1u

= (A, u) A−1, v − A−1u

= AA−1, u + A v − A−1u

= (I, u + Av − u)

= (I, Av) . (2.2)

A conjugated translation is again a translation, so (R2, +) / (Isom (E) , ∗) is a normal subgroup. Since the original translation vector may have been rotated or reﬂected, then O (2) acts on R2 by conjugation. For each A ∈ O (2), deﬁne an automorphism

2 2 −1 ϕA : R → R by ϕA ((I, v)) = (A, 0) (I, v)(A, 0) = (I, Av). Now O (2) may be

2 2 mapped into Aut(R ) by ϕ : O (2) → Aut (R ) where ϕ (A) = ϕA. Hence Isom(E) 2 ∼ 2 is the semidirect product of R by O (2), Isom(E) = R oϕ O (2) ([6], p165). For further discussion of the semidirect product of two groups, refer to the Appendix, page 118.

To understand the group structure in terms of exact sequences, consider the sequence

ι 2 2 R π 0 → R , + → (Isom (E) , ∗) → (O (2) , ·) → 1 of groups with the inclusion

2 ιR2 : R → Isom (E) by ιR2 (v) = (I, v) , 20 and the canonical epimorphism

π : Isom (E) → O (2) by π ((A, v)) = A.

Then im (ιR2 ) = ker (π), so we have a short exact sequence of group homomorphisms.

There also exists the inclusion ιO(2) : O (2) → Isom (E) by ιO(2) (A) = (A, 0). Then

π◦ιO(2) = idO(2), the identity map in (O (2) , ·) and so the sequence is split exact:

ι 2 π 2 R 0 → R , + → (Isom (E) , ∗) (O (2) , ·) → 1. ιO(2)

∼ 2 Then (O (2) , ·) = (Isom (E) , ∗) / (R , +), with the binary operation as in (9.1) where 2 ∼ 2 O (2) acts on R . Thus, Isom(E) = R o O (2) ([6], p163).

2.3 Discrete Subgroups of Isom(E)

Deﬁnition 2.3. A subgroup G ≤ Isom (E) is discrete if, for each arbitrary pair of points x, y ∈ E, there exist neighborhoods U of x and V of y such that the following set is ﬁnite: {g ∈ G|g (U) ∩ V 6= ∅} [13].

Topologically, the subgroup G ≤ Isom (E) is discrete if the induced topology on G is the discrete topology ([9], p26).

The discrete subgroups of Isom(E) having inﬁnite order contain h(I, v)i generated by a translation by a nonzero vector v, or h(I, u) , (I, v)i with u and v nonzero and linearly independent. 21

Deﬁnition 2.4. A point group is an isometry group of ﬁnite order, and therefore can contain no translations ([11], p211).

Given the group G ≤ Isom (E), the point group of G is

∼ G0 = π (G) = {A ∈ O (2) | (A, v) ∈ G} ⊂ O (2) ,

whose isometries ﬁx at least one point.

Let G be an arbitrary discrete subgroup of Isom(E). If G ∩ R2 = {(I, 0)}, then ∼ G = G0 = π (G) ⊂ O (2), and so G contains only rotations and reﬂections. Reﬂections 2π have order 2, and rotations of θ = have order n. Suppose there exists a rotation n

SP,θ ∈ G with θ not a rational fraction of π. Then G contains a rotation of inﬁnite order, and the orbit of a point Q under such a rotation is dense on the circle centered

at P and containing Q. Thus, hSP,θi is not discrete, and so G is not discrete, which contradicts the assumption that G is discrete. Therefore, for any rotation SP,θ ∈ G, θ is a rational fraction of π.

If G∩R2 6= {(I, 0)} then the translation subgroup of G either has one generator (I, v), or two generators (I, u) , (I, v), with u, v linearly independent. In both cases, G is inﬁnite. If the translation subgroup is cyclic, denote the orbit of a reference point P under the translation subgroup by h(I, v)i (P ).

Deﬁnition 2.5. (See Figure 4.1.) When the translation subgroup of a group G of isometries has two generators, the orbit of a reference point P under the translation subgroup is h(I, u) , (I, v)i (P ), called the translation lattice of G, and denoted by L ([11], p88).

The actions of the point group G0 must map x ∈ h(I, v)i (P ) back to h(I, v)i (P ), and 22

lattice points to lattice points. Though not noted at the time, Equation 2.2 (page 19) shows not only that a conjugated translation is again a translation, but that the

conjugated translation has been acted upon by the point group element A ∈ G0. Therefore, (I, Av)(P ) ∈ h(I, v)i (P ), if the translation subgroup of G is cyclic, or (I, Av)(P ) ∈ L, if the translation subgroup of G has two generators.

One way to classify these discrete isometry groups is by the point group G0 and the

intersection of the group G with R2.

2.4 Finite Subgroups of Isom(E)

Theorem 2.6. A ﬁnite group of isometries is either a cyclic group Cn or a dihedral

group Dn ([11], Theorem 8.8).

The cyclic and dihedral groups are subgroups of O (2), so necessarily have a ﬁxed

2π point. Each cyclic group Cn is generated by a rotation S of angle θ = n about a fixed point, for n ∈ Z, n > 0, and each dihedral group Dn is generated by a 2π rotation S of angle θ = n about a fixed point, along with a reflection R such that R−1SR = S−1.

Proof of Theorem 2.6. Let G be a ﬁnite subgroup of Isom(E). It will be shown that G can contain no nonidentity translations or glide reﬂections, and that it must be

one of Cn or Dn, a cyclic or dihedral group.

Suppose there exists the translation T ∈ G where T = RlRm. Then lkm. Deﬁne

2 the line k as the reﬂection of m across l, and we may write T = RkRl. Then T =

2 (RkRl)(RlRm) = RkRm. Deﬁne j as the reﬂection of l across k. Then RjRl = T , 23

3 and T = (RjRl)(RlRm) = RjRm with jkm. By induction on this algorithm, for

n −n any n ∈ N, T = RjRm 6= I. A similar argument shows that T 6= I. Suppose the group containing T were ﬁnite and T m = T n with m 6= n. Then T n−m = I for n 6= 0 which contradicts the above conclusion. Therefore, a ﬁnite subgroup G ≤ Isom(E) can contain no translations ([4], p366).

Let l, m ⊥ t, so lkm, and suppose there exists a glide reﬂection D ∈ G where D =

RtRlRm. Deﬁne k as the reﬂection of m across l, then kkm and D = RkRlRt. Then

2 n D = (RkRlRt)(RtRlRm) = RkRm, a translation. If n ∈ N is even, then D is a translation, and we reduce to the previous case. If n ∈ N is odd, then Dn is a glide reflection, therefore an odd isometry, and so not the identity. Thus, no finite subgroup of G ≤ Isom(E) may contain a glide reflection.

Thus, G may contain only rotations SP,θ and reﬂections Rl. Both I and rotations are even, and the product of even isometries and the inverse of an even isometry are even. Therefore, let us begin with the group containing only even isometries, I and rotations.

Let G contain only rotations. Then SP,θ ∈ G with 0 ≤ θ ≤ 2π. Suppose SQ,φ ∈ G ←→ such that P 6= Q, and θ, φ 6= 2kπ for k ∈ Z. Let t = PQ. Then there exist lines p through P and q through Q such that SP,θ = RpRt and SQ,φ = RtRq. G also contains the composition

−1 −1 SP,θSQ,φSP,θSQ,φ = RtRpRqRtRpRtRtRq

= RtRpRqRtRpRq

2 = (RtRpRq)

Since p, q and t do not all intersect at the same point, then RtRpRq is a glide reﬂection 24

([4], p340), the square of which is a translation. But G contains no translations, so P = Q. Thus, all nonidentity rotations in G have center P , and Fix(G) = {P }.

Since G is ﬁnite and contains only rotations, then it is discrete. Since it is discrete,

then there exists SP,ψ ∈ G where ψ is the minimal rotation. Let SP,φ ∈ G. If there

exists k ∈ Z such that 0 < φ − kψ < ψ, then this contradicts the minimality of k ψ. Therefore, φ = kψ, and SP,φ = (SP,ψ) for all SP,φ ∈ G, for some k ∈ Z. Thus

G = Cn, the cyclic group having n elements.

Suppose the ﬁnite discrete group G contains reﬂections in lines l and m with l 6= m.

If lkm, then RlRm is a translation, so it must be that l and m intersect at some point

P , and the rotation RlRm fixes P . Suppose G contains a reflection in another line k 6= l 6= m. If kkl then RkRl is a translation, and similarly for if kkm. If k is skew to l and m, then k intersects each of them. Suppose k intersects l at some point Q, then the rotation RkRl leaves Q fixed. By the previous argument, the rotations of G have a common fixed point, so P = Q, and all lines of reflection intersect at a common point, which is also the center of rotation.

Let G be a finite discrete group which contains at least one reflection. The identity transformation I ∈ G, and products and inverses of even isometries are even, so the even isometries of G form a subgroup. By the previous argument, this is Cn, and the center of rotation P is a fixed point of G. Let G contain a reflection Rl,

2 n then l contains P . The products SP,θRl, (SP,θ) Rl, ..., (SP,θ) Rl form n distinct odd isometries in G. These odd isometries are reflections; otherwise, they would be glide reflections, which contradicts the finiteness of G. Thus in addition to n rotations, G contains n reflections, in which case G = Dn, the dihedral group having 2n elements ([11], p66, [4], pp367-8). 25

Chapter 3

Frieze Groups

Consider a frieze on a building, the band of sculpted or painted decoration, often near the top of a wall. Note that it propagates horizontally around the building, but not vertically towards the ground or the roof. If the pattern is a strictly repeating pattern, then it can repeat in only ﬁnitely many ways. The symmetries of this frieze are those of one of the mathematical frieze groups.

Deﬁnition 3.1. A frieze group is a discrete group of isometries which leaves a given ∼ line t invariant, and whose translations form an inﬁnite cyclic group hT i = Z, generated by a minimal translation T by a vector v ∈ R2 ([11], p78).

In Isom(E), the set {(I, nv) | n ∈ Z} ⊂ G of translations in G is the inﬁnite cyclic group h(I, v)i. Given a point P , consider the orbit h(I, v)i (P ) of P under h(I, v)i. These points are structural reference points for the other isometries of the group.

Since a vector can be rotated by any angle, we may reduce to considering the case where v is collinear with the x-axis, oriented in the positive x direction. Likewise, since a point may be translated to any other point, we may reduce to considering the 26

case where P = O, the origin. Thus, h(I, v)i (O) may be identiﬁed with the subgroup h(I, v)i ≤ G. The orbit of this translation subgroup is as in Figure 3.1:

(I, −v)(O) (I, v)(O) (I, 3v)(O) O (I, 2v)(O)

Figure 3.1: Partial orbit of O under the translation subgroup h(I, v)i

Elements of the point group G0 = π (G) ⊂ O (2) must leave the axis of translation invariant, and map h(I, v)i (O) to itself. With the x-axis the axis of translation, the elements of O (2) which do this are the   1 0   • I =  : Identity transformation 0 1

  1 0   • Rx =  : Reﬂection or glide reﬂection across the x-axis 0 −1

  −1 0   • Ry =  : Reﬂection across the y-axis 0 1

  −1 0   • Sπ =  : Half-turn (rotation by π about the origin) 0 −1

These transformations along with v then generate the seven frieze groups. For each frieze group G,

ι 2 R π 0 → (Z, +) → (G, ∗) → (G0, ·) → 1

is a short exact sequence, and G is an extension of its point group G0 by its translation

subgroup Z. If G0 and Z commute as subgroups of G, then G is a direct product. If

they do not, then G0 acts on Z and G is a semidirect product. 27

Note that Rx,Ry,Sπ are all involutions, and that any two generate the third, so if two are in a group, then all three are. Conway ([2], p68) has given descriptive names to the frieze groups, according to what one would have to do in order to create the patterns with footprints. Described in terms of generators and relations, they are as follows:

∼ • G = Hop: h(I, v)i = Z

The point group G0 = {I} is trivial, so Hop contains only translations, translating an object by integer multiples of the minimal vector v, Figure 3.2.

Figure 3.2: Hop

2 • Jump: h(I, v) , (Rx, 0) | (Rx, 0) = (I, 0) , (Rx, 0) (I, v)(Rx, 0) = (I, v)i

∼ = Z × Z2

∼ The point group G0 = hRxi = Z2. Jump translates an object by integer multiples of v, and reﬂects the object across the x-axis, which is the axis of translation, Figure 3.3. These two actions commute, so Jump is abelian, and

m every element has the form ((Rx) , nv) for m, n ∈ Z. Jump is a direct product. The x-axis is its line of symmetry, and it has no point of symmetry ([11], p78).

Figure 3.3: Jump

2 −1 −1 • Sidle: h(I, v) , (Ry, 0) | (Ry, 0) = (I, 0) , (Ry, 0) (I, v)(Ry, 0) = (I, v) i 28

∼ = Z o Z2

∼ The point group G0 = hRyi = Z2. Sidle translates an object by integer multiplies of v, and reﬂects the object across the y-axis, Figure 3.4. Since

−1 (Ry, 0) (I, v) = (I, v) (Ry, 0) = (I, −v)(Ry, 0), then each element has the form

m ((Ry) , nv) ([11], p80). The reﬂection and the translation do not commute. Sidle is a semidirect product. Sidle has no point of symmetry, but does have a line of symmetry orthogonal to the x-axis.

Figure 3.4: Sidle

v v 2 • Step: h Rx, 2 | Rx, 2 = (I, v)i

∼ = Z

∼ The point group G0 = hRxi = Z2 6= {I}, but G0 does not act on h(I, v)i by conjugation. In Step, the object is translated by integer multiples of v on one side of the x-axis, and reﬂected across the x-axis and also translated by

v integer multiples of v, but offset by a distance of 2 , Figure 3.5. The reflection 2 in the x-axis is from a glide reflection where (Rx, a) is a translation along the x-axis which preserves h(I, v)i (O). To see how a relates to the minimal

2 translation v, note that (Rx, a) = (I, 2a) where 2a = nv with n ∈ Z. If n is n even, then ∈ , so a is an integer multiple of v. Thus (I, a) ∈ h(I, v)i (O), 2 Z −1 and (I, a) (Rx, a) = (Rx, 0), which contradicts that any reﬂection in G is a

glide reﬂection. Therefore, n is odd, n = 2k + 1 for some k ∈ Z. So (Rx, a) = nv R , = R , kv + v . Composing with (I, kv)−1, we get the glide reﬂection x 2 x 2 v v Rx, 2 as a group element. Since Rx, 2 generates (I, v), we need state only 29

v Rx, 2 as a generator. Step has no line nor point of symmetry, and the x-axis is preserved by the glide reﬂection ([11], p80).

Figure 3.5: Step

2 −1 −1 • Spinhop: h(I, v) , (Sπ, 0) | (Sπ, 0) = (I, 0) , (Sπ, 0) (I, v)(Sπ, 0) = (I, v) i

∼ = Z o Z2

∼ The point group is G0 = hSπi = C2. A half-turn is both the only rotation, and the only even isometry other than translation which preserves the x-axis. Spinhop translates the object by integer multiples of v, and rotates the object by π, Figure 3.6. Choose a center of rotation P to be an image of O under h(I, v)i. Then S preserves h(I, v)i (O), and (I, v)(S , 0) = S v , 0 , which also P,π P,π P + 2 ,π nv preserves h(I, v)i (O). So centers of rotation are the half-vector points . The 2 half-turn and translation elements do not commute. Spinhop is a semidirect product. It has a point of symmetry, but no line of symmetry ([11], p79).

Figure 3.6: Spinhop

2 2 • Spinjump:h(I, v) , (Rx, 0) , (Ry, 0) | (Rx, 0) = (Ry, 0) = (I, 0) ,

−1 −1 (Ry, 0) (I, v)(Ry, 0) = (I, v) , (Rx, 0) (I, v)(Rx, 0) = (I, v)i

∼ = (Z o Z2) × Z2 30

∼ The point group is G0 = hRx,Ryi = D2. In addition to translating, Spin- jump reﬂects the object across both the x- and y- axes, and rotates it by a half- nv turn with centers of rotation at half-vector points, , since (R , 0) (R , 0) = 2 x y

(Ry, 0) (Rx, 0) = (Sπ, 0), Figure 3.7. The translation (I, v) and the reﬂection

(Ry, 0) in the y-axis do not commute, giving the semidirect product Z o Z2.

However (Rx, 0) commutes with both (I, v) and (Ry, 0), and with their compo-

sition, giving the direct product (Z o Z2)×Z2. Spinjump has lines of symmetry in the x-axis and in lines orthogonal to th x-axis at half-vector points, and points of symmetry at half-vector points ([11], p80).

Figure 3.7: Spinjump

v 2 v 2 • Spinsidle: h Rx, 2 , (Sπ, 0) | (Sπ, 0) = (I, 0) = Rx, 2 (Sπ, 0) , v v −1 (Sπ, 0) Rx, 2 (Sπ, 0) = Rx, 2 i

∼ = Z o Z2

∼ The point group is G0 = hRx,Sπi = D2. The x-axis reflection in Spinsidle comes from a glide reflection, which generates the translation (I, v). Centers nv of half-turns are at half-vector points , and the composition R , v (S , 0) 2 x 2 π gives a reflection in a line perpendicular to the x-axis at a quarter-vector point v 3v nv + , or at a three-quarter-vector point nv + , depending on whether the 4 4 center of rotation is a vector point or a half-vector point, Figure 3.8. Half-turns and glide reflections do no commute. Spinsidle is a semidirect product. It has a point of symmetry, and a line of symmetry orthogonal to the x-axis ([11], p81). 31

Figure 3.8: Spinsidle 32

Chapter 4

Wallpaper Groups

2 ∼ 2 Consider now the case where G ∩ R = Z , then G is a wallpaper group. And before considering the formal deﬁnition, consider wallpaper. In particular, consider wallpaper bearing a repeating pattern which propagates in two nonopposite directions, covering the entire wall. The pattern can repeat and propagate in ﬁnitely many ways, and the symmetries of the wallpaper design are those of one of the mathematical wallpaper groups. The wallpaper groups are also called the two dimensional crystallographic groups, as these types of repeating symmetries in two and three dimensions have applications in crystallography.

Deﬁnition 4.1. A wallpaper group is a group of isometries whose translations are

exactly those in hT1,T2i where T1 and T2 are translations by linearly independent vectors ([11], p88).

Instead of generating a pattern repeating itself in one dimension along a line, the action of the group propagates the pattern in two dimensions, across the Euclidean plane. 33

Deﬁnition 4.2. (See Figure 4.1.) A closed region F ⊂ E, being the closure of a ◦ nonempty open set F ⊂ E, is a fundamental region for a group G of isometries if

S 1. g∈G g (F ) = E

◦ ◦ 2. F ∩ g(F ) = ∅ for all g ∈ G with g 6= I.

◦ The set ∂F = F −F is the boundary of F . Under the action of G, F tiles or tessellates the plane ([9], p49).

2 ∼ 2 2 Since G ∩ R = Z , the translation subgroup is generated by ha, bi where a, b ∈ R are nonparallel vectors ([11], p88). Recall that the translation lattice of a wallpaper group G, as determined by a point P , is the set of all images of P under the trans-

lations in G, Figure 4.1. The lattice L = {ma + nb | m, n ∈ Z} maps out the vertices of a 2-dimensional grid, where a, b ∈ R2, a is minimal, and b is the minimal vector skew to a. The grid may be square, rectangular or rhombic.

l Translation lattice points Fundamental b region a

Figure 4.1: Fundamental region and translation lattice of a wallpaper group

As before, elements of the point group G0 = π (G) ⊂ O (2) preserve L. For each wallpaper group W = G,

ι 2 2 Z π 0 → Z , + → (G, ∗) → (G0, ·) → 1 34

∼ 2 is a short exact sequence which splits, giving that G0 = G/Z ; and similarly to the frieze groups, each wallpaper group W is structurally an extension of its point group

2 G0 by Z .

4.1 The Crystallographic Restriction

2 ∼ 2 Only rotations by specific angles preserve the lattice when G ∩ R = Z . The crystallographic restriction specifies these angles, and has been proven by many methods. One geometric, and one algebraic proof are presented here, after requisite definitions and supporting theorems.

Theorem 4.3 (Crystallographic Restriction). If G is a wallpaper group, and the

2π rotation SP,θ ∈ G with 0 ≤ θ ≤ 2π, then θ = n where n = 1, 2, 3, 4 or 6.

Deﬁnition 4.4. A point P is an n-center of a group G of isometries if the rotations of G having center P form a ﬁnite cyclic group Cn of order n > 1 ([11], p91).

Theorem 4.5. For a given n, if a point P is an n-center for a group G of isometries, and G contains an isometry that takes P to Q, then Q is an n-center for G. If l is a line of symmetry for a figure, and the symmetry group for the figure contains an isometry that takes l to m, then m is a line of symmetry for the figure ([11], p91).

Theorem 4.6. If S 2π and S 2π are in a wallpaper group W , with P 6= Q and P, n Q, n n > 1, then 2PQ is greater than or equal to the length of the shortest nonidentity translation in W ([11], p91).

In short, these two theorems give the result that centers of rotation map to centers of rotation having the same order, and that ([11], p91) 35

1. No two n-centers (same n) can be “too close”

2. A 2-center and a 4-center cannot be too close

3. A 2-center and a 6-center cannot be too close

4. A 3-center and a 6-center cannot be too close

Geometric Proof of Theorem 4.3. Let A and B be distinct n-centers, with AB as

small as possible. Deﬁne C = S 2π (A). Then C is an n-center, and AB = BC. B, n

Deﬁne D = S 2π (B). Then D is an n-center, and BC = CD. If A = D, then n = 6. C, n

D = A

n = 6 2π 6 C B

If A 6= D, then AD ≥ AB = BC = CD, so n < 6. Suppose n = 5. Then AD < AB, which contradicts the minimality of AB, so n 6= 5.

D A

n = 5 2π 5 C B

If n = 4, then AD = AB.

D A

n = 4 2π 4 C B 36

If n = 3 or n = 2, then AD > AB.

D A

n = 3 2π 3 C B

Thus, a wallpaper group W contains nonidentity rotations only of order n = 2, 3, 4, 6 ([11], p92).

The following algebraic proof of the crystallographic restriction relies on the fact that

for any A ∈ G0, the trace tr(A) of A must be an integer. To establish this fact, we consider a change of basis.

Relative to the standard basis {e1, e2} where e1 = (1, 0) and e2 = (0, 1), a vector

v may be expressed as v = xe1 + ye2 for x, y ∈ R. Then v has coordinates (x, y) with respect to the standard basis. Consider a, b ∈ L where a is the lattice element of smallest magnitude, and b is the lattice element of smallest magnitude skew to a. Since a and b are linearly independent, {a, b} is a basis of L. Let v ∈ L, then in terms of this new basis, v = ma + nb for m, n ∈ Z, so v has coordinates (m, n),   m   and as a column vector, v =  . Bases of lattices are not unique, so let (c, d) be n another basis of L. Then in particular, a and b may be written in terms of c and d, say a = x1c + y1d, b = x2c + y2d, and vice versa. 37

Beginning with v written with respect to {a, b},

v = ma + nb

= m (x1c + y1d) + n (x2c + y2d)

= (x1m + x2n) c + (y1m + y2n) d.

  p 0   Then our vector v with respect to the basis (c, d) is v =   where p = x1m + x2n q and q = y1m + y2n.   x1 x2   0 The matrix P =   is the matrix of transformation from v to v : y1 y2

      p x x m    1 1     =     q y1 y2 n

There exists a similar matrix of transformation Q from v0 to v. Then v = Q (P v) =

0 −1 Q (v ) = v, so Q = P , and so P is invertible. Since m, n, p, q ∈ Z, then x1, x2, y1, y2 ∈ Z, and so det(P ) ∈ Z. In particular, det(P ) = ±1 since det(PP −1) = det (I) = 1.

Deﬁnition 4.7. An invertible matrix P is called unimodular if both P and P −1 have integer entries. Equivalently, P is unimodular if it has integer entries and det(P ) = ±1 ([12], p258).

Theorem 4.8. Given a lattice L, the matrix of transformation of coordinates from one basis of L to another basis of L is a unimodular 2×2 matrix ([12],Theorem 6.3.1).

Theorem 4.9. Let G0 be the point group of a [wallpaper] group G with translation 38

lattice L. Let A ∈ G0, and let (b1, b2) be a basis of L. The mapping x 7→ Ax with

0 0 0 respect to the standard basis (e1, e2) transforms into x 7→ A x with respect to the

0 −1 basis (b1, b2), where A = P AP , and P is the matrix of transformation of coordinates

0 from (e1, e2) to (b1, b2). Moreover, the matrix A is unimodular. Consequently, with respect to the basis (b1, b2), G0 is a subgroup of [the unimodular group] ([12], Theorem 6.3.2).

Algebraic Proof of Theorem 4.3. Let W be a wallpaper group, and G0 its point group. Then   cos θ − sin θ   A =   ∈ G0 sin θ cos θ

for some 0 ≤ θ ≤ 2π. The trace of a matrix is invariant under change of basis, and tr(AB) = tr (BA), so tr(A)0 = tr (P AP −1) = tr (AP P −1) = tr (A). Since

0 A0 has integer entries, then tr(A) ∈ Z ⇒ tr (A) ∈ Z. For the above matrix A, 2π 2π 2π 2π 2π tr(A) = 2 cos θ ∈ {0, ±1, ±2}, so θ ∈ 1 , 2 , 3 , 4 , 6 . Thus G0 contains rotations of orders 1, 2, 3, 4 and 6 ([12], Ch 6.4).

4.2 Lattices of R2

Recall that for any wallpaper group W , elements of the point group must preserve the lattice, and that for a basis {a, b} of L, L = {ma + nb | m, n ∈ Z}. Only certain point groups are compatible with a given lattice type.

Let G be the point group, then G ∼= C or G ∼= D , so G contains a rotation A =  0  0 n 0 n 0 cos 2π − sin 2π n n ∼ ∼   where n ∈ {1, 2, 3, 4, 6}. If G0 = C1 or G0 = C2, then G0 contains  2π 2π  sin n cos n an identity rotation or a half-turn respectively, both of which are compatible with 39

any lattice, not just those stipulated under the crystallographic restriction. The point ∼ group G0 = Ci for i ∈ {1, 2} is associated with a general lattice which is generated

by two unrestricted linearly independent vectors. The point groups isomorphic to Cn

for n ∈ {3, 4, 6} and to Dn for n ∈ {2, 3, 4, 6} are compatible with a more restricted set of lattices.

∼ Let A ∈ G0 = Cn, and define P = (I, a)(O), the image of the origin under the minimal translation. Also define P 0 = AP = (A, 0) (I, a)(O) = O + Aa, the point 2π P rotated by about O. Define b = Aa. Since n > 2, then θ < π, so a and b are n linearly independent, and can be a basis of L. Note also that OP 0 = OP , so |b| = |a|. Consider the quadrilateral O,P,P 0 and Q = (I, b)( I, a)(O) = (I, a)(I, b)(O). Then

0 ∼ OPP Q is a rhombus having sides of equal length. If G0 = C3,C6, then b 6⊥ a, ∼ and the lattice is hexagonal. If G0 = C4, then b ⊥ a, and the lattice is square ([12], pp261-2).

All of the cyclic groups have been accounted for. The dihedral groups, containing reﬂections, contain other possibilities.

∼ Let G0 = Dn. Then G0 contains a matrix B such that det(B) = −1, a reﬂection Rl

across some line l, such that l contains the O. Let L = {ma + nb | m, n ∈ Z} where a is minimal in L and b is the minimal vector skew to a. It will be shown that the axis of reﬂection l is either parallel to a diagonal of a rhombus whose vertices are the image of O under the two basis elements a and b, and their sum or diﬀerence, in which case the lattice is hexagonal; or it is parallel to one of the basis elements, in which case the lattice is rectangular.

Let a be the minimal lattice element.

Case 1: a is neither parallel nor perpendicular to l. 40

Let b = Ba, the reﬂection of a across l. Then |b| = |a|, and by the minimality of a, |a + b| ≥ |a| and |a − b| ≥ |a|. Deﬁne P = (I, a)(O) ,Q = (I, b)(O) ,R = (I, a + b)(O). Then R lies on l, and a + b is one diagonal of the rhombus

OP QR. Since a is not perpendicular to l, then the rotation mapping a 7→ b cannot have order 4, so it must have order 3 or 6, so the lattice is hexagonal.

Case 2: a is parallel to l or a is perpendicular to l. ←→ Let a line m be perpendicular to a at O. Define P = (I, a)(O). Then OP is perpendicular to m. Let t be the perpendicular bisector of OP , and s = (B, 0) (t), the reflection of t across m. Let b be the minimal lattice element skew to a, and consider Q = (I, b)(O). If Q lies on the opposite side of t or s from m, then either |a + b| < |a| or |a − b| < |a|, so Q must lie on or between t and s. Without loss of generality, let Q lie between m and t or on m or t, and define Q0 = (B, 0) (Q). If a is parallel to the line of reflection l, then Q0 lies between t and m, and |Q + Q0| < |a|. If a is perpendicular to l, then Q0 lies between s and m, and |Q−Q0| < |a|. Thus Q must lie on t or on m. Suppose Q lies on t. Then (I, Bb − b)(O) = (I, a)(O) = P for a perpendicular to l. Define

R = (I, a − b)(O). Then l is parallel to a diagonal of the rhombus OQP R, and so by Case 1, the lattice is hexagonal. On the other hand, suppose Q lies on m. If a is perpendicular to l, then l is parallel to b, otherwise l is parallel to a. Either way, l is parallel to a basis element of L. Since a and b are perpendicular, and there is nothing to require that |b| = |a|, then the lattice is rectangular ([11], pp88-89). 41

4.3 Wallpaper Group Presentations

We now have our point groups, Cn and Dn for n ∈ {1, 2, 3, 4, 6}, and our lattices: general, rhombic, hexagonal, square and rectangular. They combine in various ways to produce the 17 wallpaper groups. As abstract groups, some of the wallpaper groups are isomorphic. However, when the groups are embedded in Isom(E), they are not the same. The isomorphism is not induced by an automorphism of Isom(E).

Translations, rotations and reﬂections are composed of one identity and one nonidentity element from the point group and translation subgroup of the wallpaper group,

(I, a) , (Sθ, 0) and (Rl, 0). For ease of notation, these isometries are denoted in the following presentations by their nonidentity component, e.g. a, Sθ and Rl. Composition of elements is understood as in (9.1). Multiplicative notation is used for all elements, keeping in mind that the group of translations is actually additive, so a−1 = −a. Since a glide reﬂection has nonidentity components from both the point group and

a translation subgroup, it continues to be denoted Rl, 2 . The identity transformation, which may be considered a translation, rotation or reﬂection, is denoted I.

The point group elements used in the following presentations are   1 0   • I =  : Identity transformation 0 1

        cos θ − sin θ 1 0 cos θ sin θ cos 2θ − sin 2θ         • Rl =       =  : sin θ cos θ 0 −1 − sin θ cos θ − sin 2θ − cos 2θ Reﬂection across a line l intersecting the origin and making an angle θ with the x-axis 42

  cos θ − sin θ   • Sθ =  : Rotation by angle θ about the origin sin θ cos θ

Clearly not all axes of reﬂection lie on the origin, and not all rotations are centered at the origin. This is addressed by conjugating an appropriate Rl or Sθ by appropriate translations.

∼ 2 ∼ ∼ We have L = Z and G0 = Cn or G0 = Dn for n ∈ {1, 2, 3, 4, 6}, depending on whether the point group contains reﬂections. The wallpaper groups are semidirect products

2 2 Z oϕ Cn or Z oϕ Dn of the lattice L, and the point group G0. The point group G0 acts on the lattice by automorphisms, described by the function ϕ : G0 → Aut (L), whose deﬁnition varies depending on the wallpaper group.

For further discussion of the presentations, see [7], p94; and for further discussion of fundamental regions and speciﬁc symmetries, see [11], pp93-107.

Wallpaper diagram key

Lattice point 3-center , ,

Axis of reﬂection 4-center ,

2-center 6-center

A polygon about a solid dot indicates an n-center which is also a lattice point.

∼ 1. W1: G0 = C1 = {I} Lattice: General.

∼ 2 Presentation: ha, b, | ab = bai = Z . The group W1 contains only translations by integer multiples of a and b, so a and b may be any two linearly independent vectors. The fundamental region is the smallest parallelogram determined by a and b, Figure 4.2. 43

b a

Figure 4.2: Wallpaper group W1

1 ∼ ∼ 2. W1 : G0 = D1 = {I,Rl} = Z2 Lattice: Rhombic. The axis of reﬂection l bisects the angle between a and b.

The reﬂection Rl maps lattice points to lattice points, so a and b must have equal magnitude, giving a rhombic lattice. There is no restriction on the angles of the rhombus. The symmetries and fundamental region of this group are shown in Figure 4.3, and an example wallpaper pattern in Figure 4.4.

2 −1 −1 ∼ 2 Presentation: ha, b, Rl| ab = ba, (Rl) = I,RlaRl = b, RlbRl = ai = Z oϕ D1 where ϕ is given by the presentation.

1 Figure 4.3: Wallpaper group W1 44

1 Figure 4.4: Example of wallpaper group W1 , including fundamental region Image credit: [8], Plate 11, Egyption No 8, Image 11

2 ∼ 3. W1 : G0 = D1 = {I,Rl} Lattice: Rectangular. Axis of reﬂection l is parallel to a basis vector. Without

loss of generality, let lka. The reﬂection Rl maps lattice points to lattice points and l is parallel to a but not on a, so a and b are perpendicular, and l bisects b. The symmetries and fundamental region of this group are shown in Figure 4.5.

2 −1 −1 −1 Presentation: ha, b, Rl| ab = ba, (Rl) = I,RlaRl = a, RlbRl = b i ∼ 2 = Z oϕ D1, where the deﬁnition of ϕ is given by the presentation.

l b a

2 Figure 4.5: Wallpaper group W1 45

3 ∼ 4. W1 : G0 = D1 = {I,Rl} Lattice: Rectangular. The reﬂection is given by a glide reﬂection, having an axis l lying on a basis vector. Without loss of generality, let l lie on a. Since

Rl maps lattice points to lattice points, and Rl lies on a, then a and b are perpendicular, and the lattice is rectangular. The reﬂection is given by a glide reﬂection. The symmetries and fundamental region are shown in Figure 4.6.

a a 2 a a −1 Presentation: ha, b, Rl, 2 | ab = ba, Rl, 2 = a, Rl, 2 a Rl, 2 = a, a a −1 −1 ∼ 2 Rl, 2 b Rl, 2 = b i = Z oϕ D1, where the action of ϕ is given by the presentation.

b l a

3 Figure 4.6: Wallpaper group W1

Note that the action of the point group on the translation subgroup is the same

2 3 for W1 and W1 , and so as abstract groups, these two are isomorphic. However,

3 2 because the reﬂection in W1 comes from a glide reﬂection, and W1 contains

2 3 only a reﬂection, when embedded in Isom(E), W1 6= W1 .

∼ ∼ 5. W2: G0 = C2 = {I,Sθ | θ = π} = Z2 Lattice: General. The 2-centers occur at lattice points, and the midpoint between any two lattice points. Because the rotation is a half-turn, there is no restriction on the angle contained by a and b, giving a general lattice. The 46

fundamental region is a parallelogram half the size of the parallelogram deﬁned by a and b, and is shown along with the symmetries of this group in Figure 4.7.

2 −1 −1 −1 −1 Presentation: ha, b, Sθ | (Sθ) = I, ab = ba, SθaSθ = a ,SθbSθ = b i ∼ 2 = Z oϕ C2 where the action of ϕ is given by the presentation.

Figure 4.7: Wallpaper group W2

1 ∼ 2 2 6. W2 : G0 = D2 = {I,Rl,Rm,RmRl | l ⊥ m} = Sθ,Rl | (Sθ) = (Rl) = I, −1 −1 RlSθRl = Sθ = Sθ Lattice: Rhombic. The two reﬂections have axes l and m where l bisects the angle formed by a and b, and m is the other diagonal of the rhombus deﬁned by a and b. The rotations are half-turns, and the 2-centers occur at lattice points and the midoints between any two lattice points. The fundamental region is one quarter of the rhombus given by the lattice, and is shown along with the symmetries of this group in Figure 4.8.

2 2 −1 Presentation: ha, b, Rl,Rm| ab = ba, (Rl) = (Rm) = I,RlaRl = b, −1 −1 −1 −1 −1 ∼ 2 RlbRl = a, RmaRm = b ,RmbRm = a i = Z oϕ D2 where the action of ϕ is given by the presentation. 47

a m l

1 Figure 4.8: Wallpaper group W2

2 ∼ 2 2 7. W2 : G0 = D2 = {I,Rl,Rm,RmRl} = Sθ,Rl | (Sθ) = (Rl) = I, −1 −1 RlSθRl = Sθ = Sθ Lattice: Rectangular. Axes of reﬂection l and m lay on the vectors a and b respectively. The rotations are half-turns, and the 2-centers occur at lattice points and the midoints between any two lattice points. The fundamental region is one quarter of the lattice rectangle deﬁned by a and b, and is shown along with the symmetries of this group in Figure 4.9.

2 2 −1 Presentation: ha, b, Rl,Rm | ab = ba, (Rl) = (Rm) = I,RlaRl = a, −1 −1 −1 −1 −1 ∼ 2 RlbRl = b ,RmaRm = a ,RmbRm = bi = Z oϕ D2 where the action of ϕ is given by by the presentation.

l a m

2 Figure 4.9: Wallpaper group W2 48

3 ∼ 2 2 8. W2 : G0 = D2 = {I,Rl,Rm,RmRl} = Sθ,Rl | (Sθ) = (Rl) = I, −1 −1 RlSθRl = Sθ 1 Lattice: Rectangular. The axis of reﬂection l lies parallel to a, of the way 4 along b. The 2-centers occur at lattice points and the midpoints between any two lattice points. The fundamental region is the quarter of the lattice rectangle deﬁned by a and b which is bounded by l. The symmetries of this group and its fundamental region are shown in Figure 4.10.

2 2 −1 Presentation: ha, b, Rl,Sθ | ab = ba, (Rl) = (Sθ) = I,RlaRl = a, −1 −1 −1 −1 −1 −1 ∼ 2 RlbRl = b ,SθaSθ = a ,SθbSθ = b i = Z oϕ D2 where the action of ϕ is given by the presentation.

b l

3 Figure 4.10: Wallpaper group W2

4 ∼ 2 2 9. W2 : G0 = D2 = {I,Rl,Rm,RmRl} = Sθ,Rl | (Sθ) = (Rl) = I, −1 −1 RlSθRl = Sθ Lattice: Rectangular. Lines l and m are both axes of glide reﬂections, l parallel

1 1 to a and lying 4 of the way along b, m parallel to b and lying 4 of the way along a. The 2-centers occur at lattice points and the midpoints of any two lattice points. Let P be the lattice point nearest the intersection of l and m,

a b 4 and note that Rl, 2 SP,θ = Rm, 2 . Then W2 contains both glide reﬂections, a and in fact the presentation could be written solely in terms of Rl, 2 and 49

b 2 Rm, 2 . Here, we express it in terms of Z and O (2). The fundamental region is one quarter of the lattice rectangle deﬁned by a and b, shown along with the symmetries of this group in Figure 4.11.

a a 2 2 Presentation: ha, b, Rl, 2 ,SP,θ, | ab = ba, Rl, 2 = a, (SP,θ) = I, a a −1 a a −1 −1 −1 −1 Rl, 2 a Rl, 2 = a, Rl, 2 b Rl, 2 = b ,SP,θaSP,θ = a , −1 −1 ∼ 2 SP,θbSP,θ = b i = Z oϕ D2 where the action of ϕ is given by the presentation.

b l

P a m

4 Figure 4.11: Wallpaper group W2

∼ 2 2π ∼ 10. W3: G0 = C3 = I,Sθ, (Sθ) | θ = 3 = Z3 2π Lattice: Hexagonal. Rotations are by , with 3-centers at lattice points and 3 the incenters of all the equilateral triangles determined by three noncollinear

lattice points. Let P be a 3-center. Then b (P ) = (SP,θ)(I, a)(P ), so b =

(I,SP,θa). Thus, the presentation of this group could be written solely in terms

2 of a and Sθ. Here, we give a presentation in terms of Z and O (2). The symmetries and fundamental region of this group are shown in Figure 4.12.

3 −1 −1 −1 Presentation: ha, b, Sθ | ab = ba, (Sθ) = I,SθaSθ = b, SθbSθ = (ab) i ∼ 2 = Z oϕ C3 where the action of ϕ is given by the presentation. 50

P a

Figure 4.12: Wallpaper group W3

1 ∼ 2 2 3 2 11. W3 : G0 = D3 = I,Sθ, (Sθ) ,Rl,SθRl, (Sθ) Rl = Sθ,Rl | (Sθ) = (Rl) = I, −1 −1 RlSθRl = Sθ 2π Lattice: Rhombic. Rotations are by , and 3-centers occur at lattice points 3 and the incenters of all equilateral triangles determined by three noncollinear lattice points. The axis of reﬂection l lays on the long diagonal of the rhombus,

and the fundamental region is half that of W3, shown along with the symmetries

of this group in Figure 4.13. The vector b = (I,Sθa), so the presentation for

2 this group could be written solely in term of a, Sθ and Rl. In terms of Z and O (2), the presentation is as follows.

3 2 −1 Presentation:ha, b, Sθ,Rl | ab = ba, (Sθ) = (Rl) = I,SθaSθ = b, −1 −1 −1 −1 −1 ∼ 2 SθbSθ = (ab) ,RlaRl = −b, RlbRl = (ab) i = Z oϕ D3 where the action of ϕ is given by the presentation.

1 Figure 4.13: Wallpaper group W3 51

2 ∼ 2 2 3 2 12. W3 : G0 = D3 = I,Sθ, (Sθ) ,Rl,SθRl, (Sθ) Rl = Sθ,Rl | (Sθ) = (Rl) = I, −1 −1 RlSθRl = Sθ 2π Lattice: Rhombic. Rotations are by , and 3-centers occur at lattice points 3 and the incenters of all equilateral triangles determined by three noncollinear lattice points. The axis of reﬂection l bisects the angle contained by a and b.

The fundamental region is half that of W3, and is shown along with the symmetries of this group in Figure 4.14.

3 2 −1 Presentation: ha, b, Sθ,Rl | ab = ba, (Sθ) = (Rl) = I,SθaSθ = ab, −1 −1 −1 −1 ∼ 2 SθbSθ = a ,RlaRl = b, RlbRl = a, i = Z oϕ D3 where the action of ϕ is given by the presentation.

a l

2 Figure 4.14: Wallpaper group W3

∼ 2 3 π ∼ 13. W4: G0 = C4 = I,Sθ, (Sθ) , (Sθ) | θ = 2 = Z4 π Lattice: Square. Rotations are either half-turns or rotations by . The 2-centers 2 occur at midpoints between lattice points which are at minimal distance from each other, while 4-centers occur at lattice points, and centers of lattice squares. The fundamental region is a square one quarter the size of the lattice square deﬁned by a and b, and is shown along with the symmetries of this group in Figure 4.15.

3 −1 −1 −1 Presentation: ha, b, Sθ | ab = ba, (Sθ) = I,SθaSθ = b, SθbSθ = a i ∼ 2 = Z oϕ C4 where the action of ϕ is given by the presentation. 52

Figure 4.15: Wallpaper group W4

1 ∼ 2 3 2 3 14. W4 : G0 = D4 = I,Sθ, (Sθ) , (Sθ) ,Rl,SθRl, (Sθ) Rl, (Sθ) Rl = 4 2 −1 −1 Sθ,Rl | (Sθ) = (Rl) = I,RlSθRl = Sθ π Lattice: Square. Rotations are either half-turns or rotations by . The 2-centers 2 occur at midpoints between lattice points which are at minimal distance from each other, and 4-centers occur at lattice points, and centers of lattice squares. Axis of reﬂection l lays on a diagonal of the lattice square. The fundamental

region is half that of W4, and is shown along with the symmetries of this group in Figure 4.16.

4 2 −1 Presentation: ha, b, Sθ,Rl | ab = ba, (Sθ) = (Rl) = I,SθaSθ = b, −1 −1 −1 −1 ∼ 2 SθbSθ = a, RlaRl = b, RlbRl = a i = Z oϕ D4 where the action of ϕ is given by the presentation.

l a

1 Figure 4.16: Wallpaper group W4 53

2 ∼ 2 3 2 3 15. W4 :G0 = D4 = I,Sθ, (Sθ) , (Sθ) ,Rl,SθRl, (Sθ) Rl, (Sθ) Rl = 4 2 −1 −1 Sθ,Rl | (Sθ) = (Rl) = I,RlSθRl = Sθ π Lattice: Square. Rotations are either half-turns or rotations by . The 2-centers 2 occur at midpoints between lattice points which are at minimal distance from each other, while 4-centers occur at lattice points, and centers of lattice squares. Axis of reﬂection l lies on 2-centers, bisecting a and b. The fundamental region

is half that of W4, and is shown along with the symmetries of this group in Figure 4.17.

4 2 −1 −1 Presentation: ha, b, Sθ,Rl | ab = ba, (Sθ) = (Rl) = I,SθaSθ = ab, SθbSθ = −1 −1 −1 −1 −1 ∼ 2 a ,RlaRl = b ,RlbRl = a i = Z oϕ D4 where the action of ϕ is given by the presentation.

a l

2 Figure 4.17: Wallpaper group W4

∼ n π ∼ 16. W6: G0 = C6 = I, (Sθ) | θ = 3 , 1 ≤ n ≤ 5 = Z6 2π π Lattice: Hexagonal. W contains half-turns, and rotations by and . The 2- 6 3 3 centers occur at midpoints between lattice points which are at minimal distance from each other, 3-centers occur at incenters of equilateral triangles determined by three noncollinear lattice points at minimal distance from each other, and 6-centers occur at lattice points. The symmetries and fundamental region of this group are shown in Figure 4.18. 54

6 −1 Presentation: ha, b, Sθ | ab = ba, (Sθ) = I,SθaSθ = b, −1 −1 ∼ 2 SθbSθ = a bi = Z oϕ C6 where the action of ϕ is given by the presentation.

Figure 4.18: Wallpaper group W6

1 ∼ n n π 17. W6 : G0 = D6 = I, (Sθ) ,Rl, (Sθ) Rl | θ = 3 , 1 ≤ n ≤ 5 = 6 2 −1 −1 Sθ,Rl | (Sθ) = (Rl) = I,RlSθRl = Sθ Lattice: Hexagonal. Axis of reﬂection l bisects the angle between a and b, lying on the long diagonal of the rhombus. The 2-centers occur at midpoints between lattice points which are at minimal distance from each other, 3-centers occur at incenters of equilateral triangles determined by three noncollinear lattice points at minimal distance from each other, and 6-centers occur at lattice points. The

fundamental region is half that of W6. It is shown along with the symmetries of this group in Figure 4.19, and an example wallpaper pattern in Figure 4.20.

6 2 −1 Presentation: ha, b, Sθ,Rl | ab = ba, (Sθ) = (Rl) = I,SθaSθ = b, −1 −1 −1 −1 ∼ 2 SθbSθ = a b, RlaRl = b, RlbRl = ai = Z oϕ D6 where the action of ϕ is given by the presentation. 55

a l

1 Figure 4.19: Wallpaper group W6

1 Figure 4.20: Example of wallpaper group W6 , including fundamental region Image credit: [8], Plate 30, Byzantine No 3, Image 19

4.4 Triangle Groups

The presentations in Section 4.3 are not unique in characterizing groups that tessellate the Euclidean plane. Instead of extending a point group by Z2, consider the group generated the reﬂections Rj,Rk,Rl in the sides j, k, l of a triangle having angles α, β, γ. Triangle groups and other reﬂection groups exist in the hyperbolic plane, and are discussed in Chapter 9, so it makes sense to understand the triangle groups 56

which populate the Euclidean plane. Due to the angle-sum restriction on Euclidean triangles, α + β + γ = π, there exist three Euclidean triangle groups, each having an orientation-preserving subgroup of index 2. As will be shown later, the angle-sum restriction for hyperbolic triangles, α+β+γ < π, allows an inﬁnite number of triangle groups.

Deﬁnition 4.10. An isometry group of type (α, β, γ) is the group generated by re- ﬂections in the sides of a triangle having angles α, β and γ. In all cases, α, β, γ ≥ 0.

A necessary, though not suﬃcient, condition for a group of type (α, β, γ) to be discrete kπ is that each angle of the triangle can be written with k and p relatively prime p π integers. A suﬃcient condition is that each angle of the triangle can be written p with p ∈ Z, 2 ≤ p ≤ ∞ ([1], p278).

Deﬁnition 4.11. A full (p,q,r) triangle group is the group generated by the reﬂec- π π π tions across the sides j, k, l of a triangle having angles , , , with 2 ≤ p, q, r ≤ ∞, p q r and p, q, r ∈ Z ([5], p280).

By the above discussion, a full (p, q, r) triangle group is discrete. In the Euclidean π π π case, the triplet (p, q, r) must also satisfy + + = π. p q r

To determine the presentation of a general (p, q, r) triangle group, let lines j and k π π π contain the angle , k and l contain the angle , and j and l contain the angle , p q r Figure 4.21. 57

j k

π p

π π q r l

Figure 4.21: Example of a (p, q, r) triangle

The rotation RjRk has order p, RkRl has order q, and RjRl has order r. Reﬂections have order 2. A full (p, q, r) triangle group G has presentation ([5], p281)

2 2 2 p q r hRj,Rk,Rl | Rj = Rk = Rl = (RjRk) = (RkRl) = (RjRl) = 1i. (4.1)

In the image of the fundamental triangle under the full (p, q, r) triangle group, pairs of reﬂected and unreﬂected triangles meet at each vertex. Since a full (p, q, r) triangle group is discrete, then an integer number of these pairs meet, and the sum of their angles is 2π, Figure 4.22.

Figure 4.22: Partial image of a fundamental triangle under the action of a full (p, q, r) triangle group

Deﬁnition 4.12. An ordinary (p, q, r) triangle group is the orientation preserving subgroup of a full (p, q, r) triangle group G generated by the rotations RjRk = S 2π , p

RkRl = S 2π and RjRl = S 2π ([5], p281). q r 58

2 Since (RjRk)(RkRl) = RjRkRl = RkRl, then the ordinary (p, q, r) triangle group is

generated by S 2π and S 2π , and has presentation p q

p q r hS 2π ,S 2π | (S 2π ) = (S 2π ) = (S 2π S 2π ) = 1i. (4.2) p q p q p q

The fundamental region of an ordinary (p, q, r) triangle group is the fundamental region of the full (p, q, r) triangle group union an adjacent reﬂection of that region, Figure 4.23. Thus, it is either an isosceles triangle or a quadrilateral.

Figure 4.23: Partial image of a fundamental quadrilateral under the action of an ordinary (p, q, r) triangle group

Under the action of the ordinary triangle group, the grey section of the fundamental region is mapped to other grey sections, and the white section is mapped to other white sections.

π π π Due to the angle sum restriction + + = π, there exist only three Euclidean p q r (p, q, r) triangles ([10], Theorem 2.5): (3, 3, 3) , (2, 4, 4) and (2, 3, 6). Each of these full (p, q, r) triangle groups is equivalent to a Euclidean wallpaper group.

1 Proposition 4.13. The full (3, 3, 3) triangle group is equivalent to W3 .

Proof. The full (3, 3, 3) triangle group has presentation

2 2 2 3 3 3 hRj,Rk,Rl | Rj = Rk = Rl = (RkRj) = (RlRk) = (RjRl) = Ii,

so RjRkRj = RkRjRk, RlRkRl = RkRlRk and RjRlRj = RlRjRl. 59

1 The wallpaper group W3 is previously described (page 50) by a presentation in terms

2 of Z and D3,

3 2 −1 ha, b, Sθ,Rl | ab = ba, (Sθ) = (Rl) = I,Sθa (Sθ) = b,

−1 −1 −1 −1 −1 Sθb (Sθ) = (ab) ,RlaRl = −b, RlbRl = (ab) i where the lattice is rhombic, and the axis of reﬂection l lies along the long diagonal of the rhombus, bounding one side of the fundamental region F , which is a (3, 3, 3) triangle. Deﬁne the other lines bounding F to be j and k. Then the corners of F are centers of rotations, each having order 3, Figure 4.24.

k l j b F a

Figure 4.24: Partial image of F under the action of the full (3, 3, 3) triangle group

1 The generators of W3 can be written in terms of reﬂections in j, k and l:

a = RlRjRlRk

b = RkRlRkRj

Sθ = RkRj

Rl = Rl.

1 In turn each reﬂection can be written in terms of the generators of W3 : 60

Rl = Rl

Rj = aSθRl

−1 Rk = b (Sθ) Rl.

Relations in each presentation hold when generators are written in terms of the generators of the other presentation. Thus, these two presentations generate the same group.

1 Proposition 4.14. The full (2, 4, 4) triangle group is equivalent to W4 .

Proof. The full (2, 4, 4) triangle group has presentation

2 2 2 4 4 2 hRj,Rk,Rl | Rj = Rk = Rl = (RkRj) = (RlRk) = (RlRj) = Ii,

so RjRl = RlRj, RkRlRkRl = RlRkRlRk and RkRjRkRj = RjRkRjRk.

1 The wallpaper group W4 is previously described (page 52) by a presentation in terms

2 of Z and D4,

4 2 −1 −1 −1 ha, b, Sθ,Rk | ab = ba, (Sθ) = (Rk) = I,SθaSθ = b, SθbSθ = a ,

−1 −1 RkaRk = b, RkbRk = ai, where the translation lattice is square. The axis of reﬂection k bisects the angle between the translation vectors a and b, bounding one side of the fundamental region F , which is a (2,4,4) triangle. Deﬁne the other lines bounding F to be j and l, Figure 4.25. 61

j F a k l

Figure 4.25: Partial image of F under the action of the full (2, 4, 4) triangle group

1 The generators of W4 can be written in terms of reﬂections in j, k and l:

a = RlRkRjRk

b = RkRlRkRj

Sθ = RkRj

Rk = Rk.

1 In turn, each reﬂection can be written in terms of the generators of W4 :

−1 Rj = (Sθ) Rk

Rk = Rk

Rl = a (Sθ) Rk

Relations in each presentation hold when generators are written in terms of the generators of the other presentation. Thus, these two presentations generate the same group.

1 Proposition 4.15. The full (2, 3, 6) triangle group is equivalent to W6 . 62

Proof. The full (2, 3, 6) triangle group has presentation

2 2 2 6 3 2 hRj,Rk,Rl | Rj = Rk = Rl = (RkRj) = (RlRk) = (RlRj) = 1i

so RkRlRk = RlRkRl, RjRl = RlRj and RjRkRjRkRjRk = RkRjRkRjRkRj. The

1 2 wallpaper group W6 is previously described (page 55) in terms of Z and D6:

6 2 −1 ha, b, Sθ,Rk | ab = ba, (Sθ) = (Rk) = I,SθaSθ = b,

−1 −1 −1 −1 SθbSθ = a b, RkaRk = b, RkbRk = ai which has a hexagonal lattice. The axis of reﬂection k lies on the long axis and bisects the angle between translation vectors a and b, bounding one side of the fundamental region F , which is a (2, 3, 6) triangle. Deﬁne the other two lines bounding F to be j and l, Figure 4.26.

j F a k l

Figure 4.26: Partial image of F under the action of the full (2, 3, 6) triangle group

1 The generators of W6 can be written in terms of reﬂections in j, k and l: 63

a = RlRkRjRkRjRk

b = RlRkRlRjRkRj

Sθ = RkRj

Rk = Rk.

1 In turn, each reﬂection can be written in terms of the generators of W6 :

−1 Rj = b RkaSθ

Rk = Rk

−2 Rl = Rkb (Sθ)

Relations in each presentation hold when the generators are written in terms of the generators of the other presentation. Thus, the two presentations generate the same group.

Likewise, W3, W4 and W6 are equal to the ordinary (3, 3, 3), (2, 4, 4) and (2, 3, 6)

1 triangle groups respectively. If F is the fundamental region of W3 , note that the

1 fundamental region of W3 is F ∪ Rl (F ); if F is the fundamental region of W4 , then the fundamental region of W4 is F ∪ Rl (F ); and if F is the fundamental region of

1 W6 , then the fundamental region of W6 is F ∪ Rl (F ). 64

Chapter 5

Isometries in H

Hyperbolic geometry is the geometry based upon the ﬁrst four of Euclid’s axioms, along with a negation of the Euclidean parallel postulate, called the hyperbolic axiom.

Axiom 5.1 (Hyperbolic Axiom). In hyperbolic geometry there exists a line l and a point P not on l such that at least two distinct lines parallel to l pass through P ([4], p187).

The implications of this axiom result in a geometry distinct from Euclidean geometry.

This section contains a discussion of relevant implications of the hyperbolic axiom,

transformations of the hyperbolic plane H, and how these transformations may be understood in terms of the previously discussed transformations of E. 65

5.1 The Hyperbolic Plane

Euclid’s ﬁrst four axioms and the hyperbolic axiom (Axiom 5.1) lead to the more general Universal Hyperbolic Theorem.

Theorem 5.2 (Universal Hyperbolic Theorem). In hyperbolic geometry, for every line l and every point P not on l there pass through P at least two distinct parallels to l ([4], p188).

Using Euclid’s ﬁrst four aximos and the axiom of continuity, the following theorem may be proved.

Theorem 5.3 (Saccheri-Legendre Theorem). The sum of the degree measures of the three angles in any triangle is less than or equal to 180◦ ([4], p125).

Hilbert’s parallel postulate (Axiom 1.17) implies that the angle sum of a triangle is 180◦ ([4], p130). Therefore, since the hyperbolic axiom negates Hilbert’s parallel postulate, logically the angle sum of a triangle must not be 180◦ in hyperbolic geometry. Combining this with the Saccheri-Legendre theorem, then all triangles have an angle sum strictly less than 180◦. Since the existence of similar triangles is another equivalent of Hilbert’s parallel postulate, then in hyperbolic geometry, similar triangles do not exist. Thus, in hyperbolic geometry, if two triangles are similar, then they are congruent ([4], p190). In other words, any automorphism of the hyperbolic plane is an isometry. Thus the group of automorphisms of the hyperbolic plane is exactly the

group of isometries of the hyperbolic plane: Isom(H)=Aut(H). 66

5.2 Models of the Hyperbolic Plane

In the Poincar´edisc and the Poincar´eupper half-plane models, hyperbolic lines are given by arcs of circles or by particular Euclidean line segments or half lines. A major advantage of these two models is their preservation of angles.

Deﬁnition 5.4. When two hyperbolic lines intersect in either the Poincar´edisc or upper half-plane model, the magnitude of the angle formed is the number of degrees in the angle between the two rays tangent to the hyperbolic lines ([4], p233).

Deﬁnition 5.5. An ordinary point is any point in the hyperbolic plane H.

Deﬁnition 5.6. An ideal point is in the set of points “at inﬁnity” in each model.

In the Poincarédisc model, given a fixed Euclidean circle γ with center O, consider the ordinary points to be those interior to the circle, and γ itself to be the set of ideal points “at infinity.”

Definition 5.7. In the Poincarédisc model, Poincarélines are represented by open chords intersecting O, and by open arcs of those circles orthogonal to γ.

A circle δ is considered to be orthogonal to γ when the radii of δ and of γ through the point of intersection are perpendicular ([4], p232), Figure 5.1. 67

rδ

O rγ

Figure 5.1: Orthogonal circles

Deﬁnition 5.8. Two Poincar´elines intersect if they have a point in common.

Deﬁnition 5.9. If two Poincar´elines have no point in common, then they are parallel.

Unlike lines in E however, Poincar´elines can fail to intersect in two distinct ways. Consider k, l and m, Figure 5.2.

Λ Ψ

γ k m

O Φ l

Ω

Figure 5.2: Poincar´edisc model: divergently and asymptotically parallel lines 68

Since the Poincar´elines are open arcs or open chords, they do not intersect γ; but they do limit to ideal points in γ: k limits to Λ and Γ, l limits to Ω and Φ, and m limits to Φ and Ψ.

Deﬁnition 5.10. Two Poincar´elines l and m are divergently parallel if they do not intersect, and limit to distinct ideal points.

In Figure 5.2 k kd m and k kd l.

Deﬁnition 5.11. Two Poincar´elines l and m are asymptotically parallal if they do not intersect, and limit to a common ideal point.

In Figure 5.2, l and m do not intersect but limit to a common ideal point Φ, so

l ka m.

The disc is an excellent visual model, but the upper half-plane model is more suited for certain algebraic questions. The ordinary points make up the upper half-plane

itself, H = {z = a + bi ∈ C | b > 0}, with the ideal points of the disc mapping to R ∪ ∞. Let the Poincar´edisc be the unit disc in the complex plane, and consider the i + z isomorphism from the disc to the upper half-plane F : z → i ([4], p350). The i − z Poincar´edisc maps to H as in Figure 5.3. 69

F (i) = ∞

i H

0 F (0) = i −1 1 7−→

F (−1) = −1 F (1) = 1 R −i F (−i) = 0

Figure 5.3: Isomorphism between the Poincar´edisc and upper half plane models

Deﬁnition 5.12. A geodesic is the hyperbolic analogue of a Euclidean line, represented in the upper half-plane model by open semicircles

2 2 2 l = a + bi | b > 0, (a − a0) + b = r , a0 ∈ R

with centers a0 on the real axis, and by vertical open half lines

l = {a + bi | a is ﬁxed, b > 0} .

The upper half-plane model of the hyperbolic plane is then H∪R∪{∞}, equipped with |z − w| + |z − w| the metric d (z, w) = ln for z, w ∈ . Intuitively, the hyperbolic |z − w| − |z − w| H plane has “more space,”the farther one travels in the direction of the real axis. And since the real axis is the “line at inﬁnity,”it can never actually be reached . The visual upshot of this metric is that if two points having identical imaginary parts maintain a constant hyperbolic distance from each other while moving in the direction of the real axis, their Euclidean distance decreases, Figure 5.4. 70

i = z w = 1 + i

.5i = z1 w1 ≈ 0.13809 + .5i

Figure 5.4: Distance from z to w equals distance from z1 to w1

The Euclidean distance between points (0, 1) and (1, 1) is 1, as is the distance between (0,.5) and (1,.5), and if we plotted these two set of points, the distances would also

appear to be the same visually.√ However, under the hyperbolic metric, if z = i 3 + 1 and w = 1 + i, then d (z, w) = ln√ ≈ 1.31696. And when z and w are shifted 3 − 1 z → z1 = .5i, and w → w1 = a+.5i, then to maintain the hyperbolic distance between

z1 and w1 such that d (z1, w1) = d (z, w), we must have a ≈ 0.13809 (Figure 5.4). Thus

z1 and w1 appear visually to be closer together than z and w, although according to the hyperbolic metric, they are not. 71

Chapter 6

Classiﬁcation of Isometries in H

Nonidentity isometries of E are translation, rotation, reflection and glide, and all may be constructed by composing reflections. The isometries of H may be constructed from reflections in a similar manner: two reflections in intersecting geodesics gives a rotation about the point of intersection; two reflections in divergently parallel geodesics gives a translation with respect to the unique geodesic which is their common perpendicular; and a translation followed by reflection across the axis of translation gives a glide reflection. Additionally, two reflections in asymptotically parallel geodesics gives rise to a transformation which has no analogue in Isom(E), the parallel displacement. We begin by considering reflections.

Definition 6.1. A reflection in the Poincarédisc model is given by the Euclidean operation of inversion in circles.

Deﬁnition 6.2. (Figure 6.1) Let γ be a circle of radius r having center O. For any point P 6= O, the inverse P 0 of P with respect to γ is the unique point P 0 on ray −→ OP such that (OP )(OP 0) = r2, (where OP denotes the length of segment OP with 72

respect to a ﬁxed unit of measurement) ([4], p243).

O P P 0

Figure 6.1: Inverse P 0 of P in the circle γ

For example, given a circle γ having radius r = 4, and a point P such that OP = 2, then (2) OP 0 = 42 = 16, so OP 0 = 8.

The deﬁnition immediately gives the following three properties, Figure 6.2:

Proposition 6.3. ([4], p243)

(a) P = P 0 ⇔ P lies on the circle of inversion γ

(b) If P is inside γ then P 0 is outside γ, and if P is outside γ then P 0 is inside γ

(c)( P 0)0 = P

Since there exists the isomorphism F between the disc model and the upper half- plane, then in the Poincaréupper half-plane, reflection in a nonvertical geodesic is also given by inversion ([4], p351), while reflection on a vertical geodesic is given by Euclidean reflection. 73

l z

Figure 6.2: Reﬂection of z across the geodesic l is the inverse z0 of z in l

Deﬁnition 6.4. If the geodesics l and m intersect at point P , then RmRl ﬁxes P ,

and RmRl = SP,θ is a rotation about P by angle θ.

The orbit of a point z under rotation lies on a Euclidean circle with P the hyperbolic center of rotation. The point P is hyperbolically equidistant from all images of z, and so lies closer (by the Euclidean metric) to the real axis than the Euclidean center of the circle.

Deﬁnition 6.5. Given a center O and ratio k, the dilation of P with respect to O is −→ the point P 0 such that P 0 ∈ OP and OP 0 = k OP ([4], p250).

Deﬁnition 6.6. In the Poincar´eupper half-plane, translation with respect to any vertical geodesic takes place via Euclidean dilation.

Geometrically, this dilation is a radial movement with respect to O, where O is the point in R to which the vertical geodesic limits.

Deﬁnition 6.7. For every two divergently parallel geodesics l and m, there is a unique nonvertical geodesic t perpendicular to both ([4], p199). A translation having nonvertical axis t is constructed by composing the reﬂections Rm and Rl, either RmRl

or RlRm.

If w ∈ t is a point on the axis of translation, then T (w) ∈ t. Let the point z not lie on t. As a composition of reﬂections, T is isometric, so d (z, w) = d (T (z) ,T (w)). Thus 74

the orbit of a point z∈ / t under translation with axis t lies on a curve equidistant to t, and not on a geodesic, Figure 6.3.

z T (z)

t w T (w)

R Figure 6.3: Partial orbits of z and w under translation with respect to t

Deﬁnition 6.8. A parallel displacement about an ideal point Σ is the product of two reﬂections in geodesics l and m such that l and m are asymptotically parallel in the direction of Σ ([4], p326).

The parallel displacement has no analogue in Isom(E). Figure 6.4 shows a partial orbit of i under a parallel displacement about ideal point Σ = 2. The orbit of i lies on a Euclidean circle of Euclidean radius 2.5, tangent to the real axis at 2, Figure 6.4.

Figure 6.4: Partial orbit of i under parallel displacement about ideal point Σ = 2

Deﬁnition 6.9. A glide is the product of three reﬂections. Given divergently parallel

geodesics l and m, then they have a perpendicular bisector t. The composition RlRm 75

(or RmRl) is a translation having axis t, and RtRlRm (or RtRmRl) is a translation with respect to t followed by a reﬂection across t ([4], p338). 76

Chapter 7

Group Structure of Isom(H)

ax + b Definition 7.1. A linear fractional transformation is a transformation x → cx + d defined on a field K ([4],p353).

Direct isometries in the Poincar´eupper half-plane model are represented by all linear az + b fractional transformations, z 7→ such that a, b, c, d ∈ and ad − bc = 1 ([4], cz + d R az + b Prop 9.26). The opposite isometries are given by all transformations z 7→ such cz + d that a, b, c, d ∈ R and ad − bc = −1 ([4], p355).

Deﬁnition 7.2. Given the ﬁeld K, the projective line P1 (K) over K is K ∪ {∞} ([4], p352).

To describe these transformations in terms of matrices, note that each x ∈ K ∪ {∞} corresponds to an equivalence class of homogenous coordinates [x1, x2] with one or both of x1 and x2 not equal to 0, such that [x1, x2] ∼ [y1, y2] ⇔ [y1, y2] = [λx1, λx2] for λ a nonzero scalar. In particular, x ∈ K corresponds to [x, 1], and ∞ corresponds to [1, 0]. Letting K = C, and viewing these homogeneous coordinates as vectors, we see that they may be transformed by multiplication with 2 × 2 invertible matrices 77

having entries in R:

      a a x a x + a x  11 12  1  11 1 12 2     =   . a21 a22 x2 a21x1 + a22x2

The connection between the above matrix operation and a linear fractional transfor-   az + b z   mation z → of z ∈ C lies in taking the vector   associated with the ho- cz + d 1 mogenous coordinate [z, 1] corresponding to z ∈ C, multiplying it by a matrix whose entries reﬂect the elements a, b, c, d of the linear fractional transformation,

      a b z az + b           =   c d 1 cz + d

and then converting the homogenous coordinate [az + b, cz + d] back to the trans- az + b formed element of , . Thus any linear fractional transformation of a point C cz + d z ∈ C may be approached as matrix multiplication.

Deﬁnition 7.3. The general linear group of 2 × 2 matrices having entries in R is the group of invertible matrices

    a b    GL (2, R) =   : a, b, c, d ∈ R, ad − bc 6= 0 .  c d  78

  1 0   Let I =  , the identity matrix, and deﬁne the set 0 1

    r 0    H =   = rI : r ∈ R \{0} .  0 r 

Then H is a normal subgroup of GL (2, R) since for A, B ∈ H, A − B ∈ H, and for A ∈ H and B ∈ GL (2, R), AB = BA.

Deﬁnition 7.4. The quotient GL (2, R) /H is the projective general linear group     a b ak bk     P GL (2, R) of 2 × 2 real matrices, in which   ∼   for k ∈ R \{0}. c d ck dk

Deﬁnition 7.5. The subgroup

    a b    SL (2, R) =   ∈ GL (2, R): ad − bc = 1  c d  of GL (2, R) is the special linear group of 2 × 2 real matrices.

Consider the intersection H ∩ SL (2, R) = {I, −I}.

Deﬁnition 7.6. The quotient SL (2, R) / {±I} = PSL (2, R) is the projective   a b   special linear group of 2 × 2 real matrices having determinant 1, in which   ∼ c d   −a −b    . −c −d

Since a direct isometry in the Poincar´eupper half-plane may be represented by a 2 × 2 matrix A with det(A) = 1, then the group of direct isometries is a subgroup of 79

  a b   PSL (2, R). Conversely, each matrix A =   ∈ PSL (2, R) associates to a linear c d az + b fractional transformation z 7→ with ad − bc = 1. Thus, the group of direct cz + d isometries of the Poincar´eupper half-plane is isomorphic to PSL (2, R).

Opposite isometries of the Poincaréupper half-plane are represented by the trans- −az + b formations z 7→ for a, b, c, d ∈ with ad − bc = −1 ([4], p355). This −cz + d R   −a b   transformation corresponds to the matrix   having determinant −1, which −c d is an element of P GL (2, R). Recall that in the Poincaréupper half-plane, geodesics are represented by vertical lines or by half circles centered on the real axis, and that reflection in a nonvertical geodesic is given by the Euclidean operation of inversion in circles. If l is the vertical geodesic defined by a = k for k ∈ R, then z reflects in l by z 7→ −z + 2k ([4], p351). If k = 0, then l is the imaginary axis, and the reflection is simply z 7→ −z.

If l is not a vertical geodesic, then l is a Euclidean semicircle, having center k on the real axis. The reﬂection of an ordinary point z in l is the inverse of z in l, the point w such that |z||w| = r2 in the Euclidean metric.

Following Greenberg’s discussion ([4], pp351-355), let z be an ordinary point, and l an

arbitrary non-vertical geodesic having as its center the ideal point k ∈ R, and radius r. Then the inverse of z in l is a point w. Since any ideal point may be mapped to any other ideal point, we reduce to the simplest case, that of finding the inverse of a point z in a circle centered at the origin. To find the matrix which inverts z in l, first translate k 7→ 0, and define z0 = z − k. Then l0 is the geodesic having center 0 and radius r, and the inverse of z0 in l0 is w0 = w − k. The point w0 is determined by the 80

following system of equations:

|z0||w0| = r2 (7.1) w0 z0 = (7.2) |w0| |z0|

Equation 7.1 is the deﬁnition of the inverse of a point in a circle. Equation 7.2 requires that 0, z0, and w0 be collinear, by stating that the ray containing 0 and z0, and the ray r2 containing 0 and w0 intersect l at the same point. Solving for |w0|, we get |w0| = , |z0| and then z0 r2 z0 r2z0 r2z0 r2 w0 = |w0| = = = = . |z0| |z0| |z0| |z0|2 z0z0 z0

Then r2 r2 + kz − k2 kz + r2 − k2 w = w0 + k = + k = = . z − k z − k z − k 1 k Divide numerator and denominator by r, and then set c = , a = kc = and r r k2 b = r (1 − a2) = r 1 − . Then r2

k k2 z + r 1 − r r2 w = 1 k z − r r az + b = (7.3) cz − a

The previously discussed reﬂections across vertical geodesics deﬁned by the equation

a = k ∈ R may also be put in the form of Equation 7.3 by setting c = 0, a = −1, and (−1) z + 2k deﬁning b = 2k. Then z 7→ = −z + 2k. (0) z − (−1)

Since det(AB) =det(A)det(B), then for A, B ∈ P GL (2, R), with det(A) = det (B) = −1, the product AB has det(AB) = 1. So the composition of two odd isometries is an 81

even one, and elements of Isom(H) may be constructed by composing reﬂections.

Conversely, each even element of Isom(H) is the product of two matrices having   a b   determinant −1. Again following Greenberg’s argument ([4], p354), let A =  , c −a   a0 b0   and B =   Then the composition of A by B is c0 −a0

      a0 b0 a b x y       BA =     =   c0 −a0 c −a u v

where x, y, u, v ∈ R and

x = a0a + b0c

y = a0b − b0a

u = c0a − a0c

v = c0b + a0a

Solutions can be found satisfying the above matrix equation in the following four

cases, each of which correspond to elements of Isom(H). The matrices associated with each isometry type are discussed further in Section 7.2.

Case 1: u 6= 0. Then one solution is

 x − v   x − v    −v v − y −1 x y  u   u    A =   and B =   , so BA =   u v 0 1 u v

with xv − uy = 1. Depending on its trace (discussed in Section 7.2), the matrix 82

BA could be a translation, parallel displacement, or rotation.

Case 2: u = y = 0. We may assume that x > 0. Then one solution is

 1   √    0 √ 0 x x 0 A =  x and B =   , so BA =   . √   1   1  √ 0 0 x 0 x x

The matrix BA is of the form of the simplest translation in the Poincar´eupper half-plane model.

Case 3: u = 0, x = v = 1. Then one solution is

      −1 −y −1 0 1 y       A =   and B =   , so BA =   . 0 1 0 1 0 1

The matrix BA is of the form of a parallel displacement with respect to ∞.

Case 4: u = 0. Then one solution is

 y      1 x 0 x y  x      A =   and B =  1  , so BA =  1  . 0 −1 0 − 0 x x

In form, the matrix BA is the composition of a translation with respect to the imaginary axis and a parallel displacement with respect to ∞.

Thus, every element of PSL (2, R) is a product of two elements of P GL (2, R) having determinant −1. Geometrically, every even element in Isom(H) is a composition of two reﬂections.     a b a b 1     Recall that in P GL (2, R),   ∼ k  . Deﬁne k = p . Then c d c d |ad − bc| 83

The remaining discussion treats primarily on PSL (2, R), the orientation-preserving isometries of H.

7.1 Linear Fractional Transformations and Fixed

Points

Recall that in the Poincaréupper half-plane model, the hyperbolic plane is the union of the set of ordinary points, {z = a + bi | a, b ∈ R, b > 0}, and the set of ideal points, R ∪ {∞}. Linear fractional transformations may be classified by their fixed point set. az + b For a linear fractional transformation, the fixed point equation z = gives cz + d

cz2 + (d − a) z − b = 0,

q a − d ± (d − a)2 + 4bc having solutions . The nature of the ﬁxed points is then 2c determined by the discriminant

(d − a)2 + 4bc = d2 − 2ad + a2 + 4 (ad − 1)

= d2 + 2ad + a2 − 4

= (d + a)2 − 4.

If (d + a)2 − 4 > 0, then there are two real solutions, so the ﬁxed points are two ideal 84

points. If (d + a)2 − 4 = 0, then there is one real solution, so there is one ideal fixed point. If (d + a)2 −4 < 0, then there are two non-real complex solutions, one of which lies in the upper half-plane, so the transformation has one ordinary fixed point. Thus the fixed points are determined by the sum a + d, which is the trace of the matrix associated with each linear fractional transformation.

7.2 Matrices and Trace

  az + b a b   Each linear fractional transformation z 7→ associates to a matrix A =  , cz + d c d which has trace tr(A) = a + d. Since A ∈ PSL (2, R), then A ∼ −A, and tr(−A) = −tr (A), so tr(A) = ± (a + d) is not well-deﬁned. Thus, to discuss the trace of

an element of PSL (2, R) we must either use tr2 (A), or deﬁne the Trace, Tr(A) = |tr (A) | = |tr (−A) | ([9], p3).

Since for all A ∈ PSL (2, R), Tr(A) must be either less than 2, equal to 2, or greater than 2, then the trace fully classiﬁes elements of PSL (2, R), and corresponds to the number of ﬁxed points.

Remark. In some texts ([1],[9]), elements of PSL (2, R) are classiﬁed in the following manner:

• If Tr(A) > 2, then A is hyperbolic

• If Tr(A) = 2, then A is parabolic

• If Tr(A) < 2, then A is elliptic

Thus, “hyperbolic” now has two meanings. A “hyperbolic element” is an element of

PSL (2, R) having Tr > 2. A “hyperbolic translation” or “hyperbolic rotation” is the 85

analogue in Isom(H) of the Euclidean translation or rotation.

Translations (hyperbolic transformations). As noted previously, translation with respect to the imaginary axis is a Euclidean dilation with respect to 0. Consid- ering this geometrically, the ordinary points of the imaginary axis map back to the imaginary axis, and so as a geodesic, the imaginary axis remains invariant. Its ideal points 0 and ∞ remain ﬁxed, and the rest of the plane is moved radially towards or away from the ideal point 0. Thus translations have two ideal ﬁxed points. The linear fractional transformations which map 0 7→ 0 and ∞ 7→ ∞ are the transformations az + b 1 z 7→ such that c = b = 0, and to ensure that ad − bc = 1, set d = ([4], cz + d a p356). Then

az + 0 2 z 7→ 1 = a z, 0 (z) + a

where a = k ∈ R \{0}, and d (z, T (z)) = ln (k2). If k2 > 1, then ln(k2) > 0, and the translation is in the direction of ∞. If k2 < 1, then ln(k2) < 0, and the translation is in the direction of 0. If k2 = 1, then ln(k2) = 0, and this is the identity transformation.     k 0 k 0 The associated element of PSL (2, R) is T =  . Let T =   be nontrivial,  1   1  0 k 0 k 1 then |k| 6= 1, and Tr(T ) = |k + | > 2. Thus, translations T with respect to the k imaginary axis have Tr(T ) > 2.

Let l ∈ H be an arbitrary geodesic, having ideal points Ω and Σ, and without loss of generality, let Ω < Σ. To describe a translation with respect to l, transform the plane such that l is mapped isometrically to the imaginary axis, translate with respect to the imaginary axis, and then transform the plane back, mapping the imaginary axis back to l. Thus, we use the transformation P ∈ PSL (2, R) such that P (0) = Ω and 86

    Σ Ω 1 − Ω P (∞) = Σ: P =  Σ−Ω  with inverse P −1 =  Σ−Ω Σ−Ω  where P −1 (Ω) = 0  1    1 Σ−Ω −1 Σ and P −1 (Σ) = ∞. The conjugate of T by P is

      Σ Ω k 0 1 − Ω PTP −1 =  Σ−Ω     Σ−Ω Σ−Ω   1   1    1 Σ−Ω 0 k −1 Σ   k2Σ−Ω ΣΩ−k2ΣΩ =  k(Σ−Ω) k(Σ−Ω)   k2−1 Σ−k2Ω  k(Σ−Ω) k(Σ−Ω)

Trace is invariant under conjugation, so PTP −1 is itself a translation with axis l,

having ideal ﬁxed points Ω and Σ, and because P,P −1 ∈ PSL (2, R), then

d P −1 (z) ,T P −1 (z) = d PP −1 (z) ,PT P −1 (z)

= d z, P T P −1 (z) .

That is, the translation distance of P −1 (z) with respect to the imaginary axis is equal to the translation distance of z with respect to l.

The behavior of hyperbolic elements of PSL (2, R) diﬀers in several ways from the behavior of translations in Isom(E). Matrix multiplication is noncommutative in general, and given any two elements A, B ∈ PSL (2, R), AB = BA if and only if

Fix(A) = Fix (B) ([9], p35). Thus, two translations T1 and T2 commute if and only if they have a common axis. Additionally, a composition of translations may not

itself be a translation, so Isom(H) does not have a translation subgroup. Criteria for certain cases are presented in Section 7.3.

Parallel Displacements (parabolic transformations). The parallel displacement is an isometry whose ﬁxed point set consists of one ideal point. Algebraically, the 87

simplest parallel displacement is that of an ordinary point z about ∞. Recall that ∞         a b 1 a a         associates to the homogeneous coordinate [1, 0]. Then     =   and   c d 0 c c associates to ∞ if and only if c = 0. Then for any ordinary or ﬁnite ideal point x,       a b x ax + b           =  . 0 d 1 d

a b b a b Suppose that x is a fixed point. Then x + = x, so x = / 1 − = , d d d d d − a which is finite if and only if a 6= d. Since ∞ is the only fixed point, then a = d.

And since ad−bc = 1, then a = d = ±1, so the element of PSL (2, R) associated with     1 b −1 −b     parallel displacement is D =   ∼  , having Tr(D) = 2, and mapping 0 1 0 −1 z 7→ z + b, where b ∈ R is the Euclidean distance of the displacement of z in the real direction. If b = 0, then the transformation is the identity transformation. Remark. The hyperbolic distance between z = x+yi and D (z) = z +b = (x + b)+yi

depends on y. That is, let b, x, y, x0, y0 ∈ R, z = x + yi, z0 = x0 + y0i, and without loss of generality, assume y < y0. Then d (z, z + b) > d (z0, z0 + b).   1 1 0   In Figure 7.1, z = i, z = 2i, and b = 1, so D =  . 0 1 88

∞ √ 1 + 17 d (z0,D (z0)) = ln √ ≈ 0.4949329231 −1 + 17 2i = z0 D (z0) = 1 + 2i

i = z D (z) = 1 + i √ 1 + 5 d (z, D (z)) = ln √ ≈ 0.9624236501 −1 + 5 R Figure 7.1: d (z, D (z)) varies depending on the imaginary part of z

Note that if z and z0 diﬀer only in their real parts, then d (z, D (z)) = d (z0,D (z0)).

To construct the matrix of a parallel displacement with respect to a ﬁnite ideal point Ω, the strategy is similar to that used to construct an arbitrary translation, conjugat-   −Ω Ω + 1   ing by a transformation P ∈ PSL (2, R) such that P (∞) = Ω. P =   −1 1   1 −Ω − 1 −1   maps Ω 7→ ∞, and the inverse P =   maps ∞ 7→ Ω. Then 1 −Ω

      −Ω Ω + 1 1 b 1 −Ω − 1 −1       PDP =       −1 1 0 1 1 −Ω   1 − bΩ bΩ2   =   −b 1 + bΩ

Then Tr(D) = Tr (PDP −1) = 2, and parallel displacements are given by elements of

PSL (2, R) having Trace equal to 2 ([9], p23).

Rotations (elliptic transformations). The ﬁxed point set of a rotation consists 89

of one ordinary point. The simplest rotation algebraically is a rotation about i. Then ai + b the ﬁxed point equation i = gives b + c + i (a − d) = 0. Then b + c = a − d = 0, ci + d so b = −c, and a = d with ad − bc = 1. Deﬁne a = cos θ, b = − sin θ. Then a rotation   (cos θ) z − sin θ cos θ − sin θ   of z about i is given by z 7→ . The associated matrix   (sin θ) z + cos θ sin θ cos θ is the same matrix used to describe rotations by θ about the origin in the Euclidean

plane, so the group of rotations in Isom(H) is isomorphic to the group of rotations in Isom(E)([4], p357), and therefore to the special orthogonal group SO (2). There is a subtle diﬀerence though, as

        cos π − sin π z −1 0 z             =     sin π cos π 1 0 −1 1     −z z     =   ∼   , −1 1

which corresponds to z. What would be a half-turn in Isom(E) turns out to be a full rotation in Isom(H). Therefore, to describe a rotation by θ about i, and to achieve a one-to-one correspondence with rotations about the origin in Isom(E) ([4], p357), the matrix S ∈ PSL (2, R) is

 θ θ  cos − sin  2 2 S =  θ θ  . sin cos 2 2

θ If θ = nπ for n ∈ , then the above is the identity transformation. Since cos < 1 Z 2 θ for all 0 < θ < π, then Tr(S) = 2 cos < 2 for all nonidentity rotations. To 2

construct the matrix of a rotation about an arbitrary ordinary point z0 = x + yi, we conjugate by a transformation P ∈ PSL (2, R) such that P (i) = z0. The matrix 90

√ x   1 x  y √ √ −√  y  −1  y y  P =  1  maps i 7→ z0, and P =   maps z0 7→ i. Then  0 √  √ y 0 y

  √ x  θ θ   1 x  y √ cos − sin √ −√ −1  y   2 2  y y  PSP =  1   θ θ     0 √  sin cos √ y 2 2 0 y  θ x θ θ x2 θ  cos + sin −y sin − sin =  2 y 2 2 y 2 .  1 θ θ x θ   sin cos − sin  y 2 2 y 2

θ Then Tr(S) = Tr (PSP −1) = 2 cos < 2, and rotations are given by elements of 2 PSL (2, R) having Trace less than 2.

7.3 Composing Translations and Parallel Displace-

ments

Let T1,T2 be two translations in Isom(H). Unless they have their ﬁxed points in common, and therefore have a common axis, then at the very least hT1,T2i contains inﬁnitely many translations with respect unique axes ([9], pp39-40). It is also possible

for hT1,T2i to contain parallel displacements and rotations. Proposition 7.7 and Examples 7.8 and 7.9 state conditions which determine the isometry that results from the composition T2T1 of certain translations, or P2P1 of certain parallel displacements.

Recall that for A ∈ PSL (2, R), tr2 (A) identiﬁes the isometry type.

2 Proposition 7.7. Let T1 and T2 be nontrivial translations by ﬁnite distances ln (j ) and ln (k2) with respect to orthogonal axes, or with respect to asymptotically parallel 91

axes. Then j, k ∈ R \{0, 1}. If the axes are orthogonal, then the composition

T2T1 is a translation for all j and k. If the axes are asymptotically parallel, then the

composition T2T1 is a parallel displacement if and only if the displacement parameters 1 j and k are related by j = ± ; otherwise, the composition is a translation. k

2 Proof. Let T1,T2 ∈ PSL (2, R) be nontrivial translations by ﬁnite distances ln (j ) and ln (k2). Then j, k ∈ R \{0, 1}.

First consider the case where the axes of T1 and T2 are orthogonal. Since any pair of orthogonal geodesics may be mapped to any other, we reduce to the case where

the axis of T1 is the positive imaginary axis, and the axis of T2 is the geodesic having

ideal points −1 and 1. The associated elements of PSL (2, R) are

    k2+1 k2−1 j 0 2k 2k T1 =   T2 =    1   k2−1 k2+1  0 j 2k 2k

and the composition T2T1 is

 2  j(k +1) k2−1  2k 2jk  T2T1 = 2  j(k −1) k2+1  2k 2jk

j (k2 + 1) k2 + 12 with tr2 (T T ) − 4 = + − 4. 2 1 2k 2jk

j (k2 + 1) k2 + 12 Case 1: + − 4 < 0. This never occurs, and so the composition is 2k 2jk never a rotation.

j (k2 + 1) k2 + 12 Case 2: + − 4 = 0. This occurs when the parameters j and k 2k 2jk q q ±2k ± −1 (k2 − 1)2 ±2k ∓ −1 (k2 − 1)2 are related by j = or by j = , in k2 + 1 k2 + 1 92

which case j is nonreal complex. But j, k ∈ R, so the composition is never a parallel displacement.

j (k2 + 1) k2 + 12 Case 3: + − 4 > 0. Since Cases 1 and 2 are impossible, then Case 2k 2jk 3 is the only option, and the composition of translations in orthogonal axes is

a translation for all j, k ∈ R \{0}.

Now let T1 and T2 have axes which are asymptotically parallel. Since any three ideal points may be mapped to any other three ideal points, then any two asymptotically parallel geodesics may be mapped to any other two asymptotically parallel geodesics.

Therefore, we reduce to the case where the axis of T1 is the positive imaginary axis,

and the axis of T2 is the geodesic having ideal points 0 and 1. The associated elements

of PSL (2, R) are

    j 0 k 0 T1 =   T2 =    1   k2−1 1  0 j k k

and their composition T2T1 is

  jk 0   T2T1 = 2  j(k −1) 1  k jk

1 2 with tr2 (T T ) − 4 = jk + − 4. 2 1 jk

1 2 Case 1: jk + − 4 < 0. This never occurs, so the ﬁxed point set is never a single jk nonreal complex point. Thus, the composition is never a rotation.

1 2 1 Case 2: jk + − 4 = 0. This occurs when j and k are related by j = ± . In this jk k case, there is one real ﬁxed point, so the composition is a parallel displacement. 93

1 2 Case 3: jk + − 4 > 0. This occurs when j and k are related in any way other jk than that stated in Case 2. There are two real ﬁxed points, and the composition is a translation. The two real ﬁxed points determine the geodesic which is the axis of translation.

The following example examines one case where the axes of translations are divergently parallel.

2 Example 7.8. Let T1 and T2 be nontrivial translations by ﬁnite distances ln (j )

and ln (k2), j, k ∈ R \{0, 1}, with respect to axes which are divergently parallel. We

consider the case where the axis of T1 is the positive imaginary axis, and the axis of

T2 is the geodesic having ideal points 1 and 2. The associated elements of PSL (2, R) are     2k2−1 −2k2+2 j 0 k k T1 =   T2 =    1   k2−1 −k2+2  0 j k k

and their composition T2T1 is

 2  j(2k −1) −2k2+2  k jk  T2T1 = 2  j(k −1) −k2+2  k jk

j (2k2 − 1) −k2 + 22 with tr2 (T T ) − 4 = + − 4. Depending on j and k, this 2 1 k jk statement may be less than, equal to or greater than zero, and so translations in this pair of divergently parallel axes may result in any of the three even isometries in

Isom(H): rotation, parallel displacement or translation.

j (2k2 − 1) −k2 + 22 Case 1: + − 4 < 0, then the composition is a rotation. This k jk 94

1 3 occurs when j and k are related by |j| = √ and |k| > √ , or by 2 2 2 n o |j| ∈/ ±1, ± √1 and 2 s √ s √ 2j4 − 3j2 + 2 − 2 2j (j2 − 1) 2j4 − 3j2 + 2 + 2 2j (j2 − 1) < |k| < (2j2 − 1)2 (2j2 − 1)2

with j, k 6= 0.

j (2k2 − 1) −k2 + 22 Case 2: + − 4 = 0, then the composition is a parallel displace- k jk √ ±k ± 2 (k2 − 1) ment. This occurs when j and k are related by j = . 2k2 − 1

j (2k2 − 1) −k2 + 22 Case 3: + − 4 > 0, then the composition is a translation. This k jk 1 3 occurs when j and k are related by |j| = √ and 0 < |k| √ , or by 2 2 2 s √ 1 2j4 − 3j2 + 2 ± 2 2j (j2 − 1) |j|= 6 √ and |k|= 6 2 (2j2 − 1)2

with j, k 6= 0.

Composition of parallel displacements can also result in any of the three even isometries of the hyperbolic plane. In the following example, the displacement parameters of the two transformations are equal, and the isometry type of the composition varies as the ﬁxed ideal points Σ and Ω vary.

Example 7.9. Let D1,D2 be nontrivial parallel displacements with respect to ideal points Σ and Ω respectively. Let the displacement parameter be the same in both

transformations, say j ∈ R \{0}. The associated elements of PSL (2, R) are

    1 − jΣ jΣ2 1 − jΩ jΩ2     D1 =   D2 =   −j 1 + jΣ −j 1 + jΩ 95

and their composition is

  1 + j2Ω (Σ − Ω) − j (Σ + Ω) j (Σ2 + Ω2 + jΣΩ (Ω − Σ))   D2D1 =   j (−2 + j (Σ − Ω)) 1 + j2Σ (Ω − Σ) + j (Σ + Ω)

2 2 having trace tr(D2D1) = 2 − j (Σ − Ω) . This trace varies as Σ and Ω vary.

2 Case 1: 2 − j2 (Σ − Ω)2 − 4 < 0, then the composition is a rotation. This occurs 2 when Σ ∈ and Σ < |Ω| < Σ + . R j

2 Case 2: 2 − j2 (Σ − Ω)2 − 4 = 0, then the composition is again a parallel displacement. This occurs when Σ and Ω are related by the linear equations Σ = Ω or 2 Σ = Ω ± . j

2 Case 3: 2 − j2 (Σ − Ω)2 − 4 > 0, then the composition is a translation. This occurs 2 when Σ ∈ and |Ω| > Σ + . R j

7.4 Discrete Subgroups of Isom(H)

The following deﬁnition is restated for the hyperbolic plane.

Deﬁnition 7.10. A subgroup G ≤ Isom (H) is discrete if, for each arbitrary pair of points z, w ∈ H, there exist neighborhoods U of z and V of w such that the following set is ﬁnite: {g ∈ G|g (U) ∩ V 6= ∅}.

Definition 7.11. Given a group G that acts upon H, if for all compact subsets K ⊂ H, g (K) ∩ K 6= ∅ for only finitely many g ∈ G, then G acts properly discontinuously on H. That is, a group G acts properly discontinuously on H if the orbit of any point z ∈ H under G is locally finite ([9], p27). 96

The cyclic group hRli generated by a reﬂection in a geodesic l is discrete, as is hDli,

the group generated by a glide reflection having axis l. Thus like Isom(E), Isom(H) has orientation-reversing subgroups of both finite and infinite order.

Deﬁnition 7.12. A Fuchsian group is a discrete subgroup of PSL (2, R), and so contains only the identity, and orientation-preserving isometries of H ([9], p26).

If a group of translations or parallel displacements is cyclic, then it is discrete, and therefore Fuchsian; and if a group of rotations has ﬁnite order, then it is Fuchsian     * k 0 + * 1 k + ([9], p29). Thus, groups conjugate to   or to   , generated by a  1    0 k 0 1 translation or parallel displacement respectively, are discrete groups of inﬁnite order. Groups conjugate to   * θ θ + cos 2 − sin 2   ⊂ PSL (2, R)  θ θ  sin 2 cos 2

or to     * θ θ + cos 2 − sin 2 −1 0   ,   ⊂ P GL (2, R) ,  θ θ    sin 2 cos 2 0 1

θ where is a rational fraction of π, are discrete of ﬁnite order. 2

7.5 Finite Subgroups of Isom(H)

Theorem 7.13. The ﬁnite subgroups of Isom(H) are cyclic and dihedral groups, Cn and Dn with n ≥ 1, as they were for Isom(E) ([4], p366).

For completeness, the proof is presented here, but the only substantive change is to add that, in addition to a translation, a ﬁnite discrete group cannot contain a parallel 97 displacement.

Proof. It will be shown that a ﬁnite subgroup G of Isom(H) contains no nonidentity translations, parallel displacements or glide reﬂections, and that it must be either a cyclic group Cn, or a dihedral group Dn.

Let l and m be two non-intersecting geodesics. If l kd m, deﬁne a translation T =

RlRm. If l ka m, define a parallel displacement P = RlRm. Define the geodesic k as the reflection of m across l, and we may write T = RkRl (respectively P =

2 2 RkRl). Then T = (RkRl)(RlRm) = RkRm (resp. P = (RkRl)(RlRm) = RkRm).

2 2 Deﬁne j as the reﬂection of l across k. Then RjRl = T (resp. RjRl = P ), and

3 3 T = (RjRl)(RlRm) = RjRm with j kd m (resp. P = (RjRl)(RlRm) = RjRm

n with j ka m). By induction on this algorithm, for any n ∈ N, T = RjRm 6= I (resp.

n −n −n P = RjRm 6= I). A similar argument shows that T 6= I (resp. P 6= I). Suppose the group containing T (resp. P ) were ﬁnite and T m = T n (resp. P m = P n) with m 6= n. Then T n−m = I (resp. P n−m = I) for n 6= 0 which contradicts the above conclusion. Therefore, a ﬁnite subgroup of Isom(E) can contain no translations (resp. parallel displacements) ([4], p366).

Let l, m ⊥ t, so l k m (either divergently or asymptotically), and define a glide reflection B = RtRlRm. Define the geodesic k as the reflection of m across l, then

2 k k m and B = RkRlRt. Then B = (RkRlRt)(RtRlRm) = RkRm, a translation. If n is even, then Bn is a translation, and we reduce to the previous case. If n is odd, then Bn is a glide reﬂection, therefore an odd isometry, and so not the identity.

Therefore, no finite subgroup of Isom(E) may contain a glide reflection, so G contains only rotations SP,θ and reflections Rl. Since both I and rotations are even, and the product of even isometries and the inverse of an even isometry are even, let us begin with the group containing only even isometries. 98

Let G contain only rotations and I. Then SP,θ ∈ G with 0 ≤ θ ≤ 2π. Suppose ←→ SQ,φ ∈ G such that P 6= Q, and θ, φ 6= 2kπ for k ∈ Z. Let t = PQ. Then there exist

geodesics p through P and q through Q such that SP,θ = RpRt and SQ,φ = RtRq. G also contains the composition

−1 −1 SP,θSQ,φSP,θSQ,φ = RtRpRqRtRpRtRtRq

= RtRpRqRtRpRq

2 = (RtRpRq)

Since p, q and t do not all intersect at the same point, then RtRpRq is a glide ([4], p340), the square of which is a translation. But G contains no translations, so P = Q. Thus, all nonidentity rotations in G have center P , and Fix(G) = {P }.

Since G is discrete, then there exists SP,ψ ∈ G where ψ is the minimal rotation. Let

SP,φ ∈ G. If there exists k ∈ Z such that 0 < φ − kψ < ψ, then this contradicts the k minimality of ψ. Therefore, φ = kψ, and SP,φ = (SP,ψ) for each SP,φ ∈ G, for some

k ∈ Z. Thus G = Cn, the cyclic group having n elements.

Suppose a ﬁnite discrete group G contains reﬂections in geodesics l and m with l 6= m.

If l k m, then RlRm is a translation, so it must be that l and m intersect at some

point P . The rotation RlRm ﬁxes P . Suppose G contains a reﬂection in another

geodesic k 6= l 6= m. If k k l then RkRl is a translation, and similarly for if k k m. If k is skew to l and to m, then k intersects each of them. Suppose k intersects l at

some point Q. Then the rotation RkRl leaves Q fixed. By the previous argument, the rotations of G have a common fixed point, so P = Q, and all axes of reflection intersect at a common point, which is also the center of rotation.

Let G be a ﬁnite discrete group which contains at least one reﬂection. The identity 99

transformation I ∈ G, and products and inverses of even isometries are even, so the

even isometries of G form a subgroup. By the previous argument, this is Cn, having order n, for some n ∈ N. Then P is a fixed point of G, so P ∈ l for any geodesic l such that the reflection Rl ∈ G. Let G contain a reflection Rl. The products

2 n SP,θRl, (SP,θ) Rl, ..., (SP,θ) Rl form n distinct odd isometries in G. These odd isometries are reflections; otherwise, they would be glide reflections, which contradicts the finiteness of G. Thus in addition to n rotations, G contains n reflections, in which

case G = Dn, the dihedral group having 2n elements ([11], p66, [4], pp367-8). 100

Chapter 8

Frieze Groups in H

Given a frieze group G, its translation subgroup is the inﬁnite cyclic group hT i, where T is the minimal translation. The orbit hT i (z) of a point z under the translation subgroup hT i is hT i (z) = {T n (z) ∈ P GL (2, R) | n ∈ Z}. As in Isom(E), we may transform any geodesic into any other, and translate any point to any other. There- fore, we reduce to considering the frieze groups containing translations with respect to the imaginary axis, generating rotations about i, and reﬂections in the imaginary axis and the geoodesic orthogonal to the imaginary axis at i, having ideal points −1 and 1.

Because of discreteness, there exists a minimal translation having distance a such that any translation in the group is equivalent to the minimal translation being applied an integer number of times. Then the point group G0 of each frieze group G consists of elements of PSL (2, R) which ﬁx an ordinary point, where each element is the hyperbolic analogue of the Euclidean transformation. As in Isom(E), the point groups are cyclic or dihedral groups. 101

Given the imaginary axis as the axis of translation, the generating transformations of the frieze groups in the Poincar´eupper half-plane are:   1 0   • I =  : Identity transformation 0 1

  −1 0   • RIm =  : Reﬂection across the imaginary axis 0 1

  0 −1   • Ri =  : Reﬂection across the geodesic orthogonal to the imaginary −1 0 axis at i, having ideal points −1 and 1   0 −1   • Sπ =  : Half-turn (rotation by π about i) 1 0

  k 0 • T =  : Translation with respect to the imaginary axis  1  0 k

 √  − k 0   • DIm =  : Glide reﬂection across the imaginary axis 0 √1 k

Each frieze group in Isom(E) is isomorphic to the the group generated by analogous elements of PSL (2, R).

∼ • G = Hop: hT i = Z.

G0 = {I}. Hop is the group generated by a single translation, and so contains only translations by integer multiples of the smallest translation distance, ln (k2), Figure 8.1. 102

0 1

Figure 8.1: Hop

2 ∼ • G = Jump: hT,RIm | RIm = I,RImTRIm = T i = Z × Z2

∼ G0 = hRImi = Z2. Jump translates an object by integer multiples of ln (k2), and reﬂects the object across the imaginary axis, which is also the axis of translation, Figure 8.2. These two actions commute. Jump is abelian

n m with a direct product structure, and every element has the form T (RIm) for

m, n ∈ Z. The imaginary axis is the axis of symmetry, and Jump has no point of symmetry ([11], p78). 103

0 1

Figure 8.2: Jump

2 −1 ∼ • G = Sidle: hT,Ri | Ri = I,RiTRi = T i = Z o Z2

∼ 2 G0 = hRii = Z2. Sidle translates an object by integer multiplies of ln (k ), and reﬂects the object across the geodesic orthogonal to the imaginary axis at i,

−1 m n Figure 8.3. Since RiT = T Ri, then each element has the form (Ri) T ([11], p80). These two actions do not commute: the translation may be acted upon by the reﬂection in the y-axis, depending on the order of composition. Sidle is a semidirect product. Sidle has no point of symmetry, but does have an axis of symmetry orthogonal to the imaginary axis. 104

0 1

Figure 8.3: Sidle

2 ∼ • G = Step: hDIm | DIm = T i = Z

∼ G0 = hRImi = Z2. The transformations in Step result in the object being translated by integer multiples of ln (k2) on one side of the imaginary axis, and the object reflected across the imaginary axis and also translated by integer ln (k2) multiples of ln (k2), but offset by a distance of , Figure 8.4. The reflection 2 2 in the imaginary axis is from a glide reflection where (DIm) is a translation

with respect to the imaginary axis. Since DIm generates T , we need state only

DIm as a generator. Step has no axis nor point of symmetry, but the imaginary axis is preserved by the glide reﬂection ([11], p80). 105

0 1

Figure 8.4: Step

2 −1 −1 ∼ • G = Spinhop: hT,Sπ | Sπ = I,SπTSπ = T i = Z o Z2

G0 = hSπi. A half-turn is both the only rotation, and the only even isometry other than translation which preserves the imaginary axis. Spinhop translates the object by integer multiples of ln (k2), and rotates the object by π, Figure 8.5. The centers of rotation are half-translation points.

0 1

Figure 8.5: Spinhop

2 2 −1 −1 • G = Spinjump: hT,RIm,Ri | RIm = Ri = I,RImTRIm = T,RiTRi = T i 106

∼ = (Z o Z2) × Z2

G0 = hRIm,Rii. In addition to translating, Spinjump reﬂects the object across both the imaginary axis and the geodesic orthogonal to the imaginary axis through i, and rotates it by a half-turn with centers of rotation at half-

translation points, since RImRi = RiRIm = Sπ, Figure 8.6. The translation T

and the reﬂection Ri in the geodesic orthogonal to the imaginary axis at i do

not commute, giving the semidirect product Z o Z2. However RIm commutes

with both T and Ri and with their composition, giving the direct product

(Z o Z2) × Z2. Spinjump has axes of symmetry in the imaginary axis and in geodesics orthogonal to the imaginary axis at half-translation points, and points of symmetry at half-translation points ([11], p80).

0 1

Figure 8.6: Spinjump

2 2 −1 ∼ • G = Spinsidle: hDIm,Sπ | Sπ = [DImSπ] = I,SπDImSπ = DImi = Z o Z2

Centers of half-turns are at half-translation points, and the composition

DImSπ gives a reﬂection in a geodesic perpendicular to the imaginary axis at a quarter- or three-quarter-translation point, depending on whether the center of rotation is a translation point or a half-translation point, Figure 8.7. Half-turns 107 and glide reﬂections do not commute. Spinsidle is a semidirect product. It has a point of symmetry, and an axis of symmetry orthogonal to the imaginary axis ([11], p81).

0 1

Figure 8.7: Spinsidle 108

Chapter 9

Analogues of Wallpaper Groups in

Recall that each Euclidean wallpaper group may be written as an extension of its point group by Z2, and under the action of the group, a fundamental region tessellates E. While PSL (2, R) does contain subgroups isomorphic to Z, no discrete subgroup of PSL (2, R) is isomorphic to Z×Z ([9], p36), and so P GL (2, R) contains no subgroup which is structurally equivalent to any Euclidean wallpaper group. The relations in

Isom(E) between the translation elements of a wallpaper group do not hold for their analogues in PSL (2, R), and the hyperbolic analogues of Euclidean generators of a wallpaper group generate a wildly diﬀerent group of isometries in the hyperbolic plane, which may or may not tessellate, and may or may not be discrete.

Isom(H) contains triangle groups, inﬁnitely many in fact. And while six of the

1 1 1 Euclidean wallpaper groups, W3 ,W3,W4 ,W4,W6 , and W6 are also full or ordinary (p, q, r) triangle groups, these presentations do not successfully transfer directly to the hyperbolic plane, as no hyperbolic triangle has an angle sum of π. 109

Thus, instead of attempting to “transplant” Euclidean wallpaper groups into the hyperbolic plane, we end with a discussion of groups which do tessellate the hyperbolic plane.

9.1 Dirichlet Region

The following deﬁnition is restated for the hyperbolic plane.

Deﬁnition 9.1. A closed region F ⊂ H, being the closure of a nonempty open set ◦ F ⊂ H, is a fundamental region for a group G of isometries if

S 1. g∈G g (F ) = H

◦ ◦ 2. F ∩ g(F ) = ∅ for all g ∈ G with g 6= I.

◦ The set ∂F = F −F is the boundary of F . Under the action of G, F tiles or tessellates the plane ([9], p49).

◦ Another characterization of F is as a set containing exactly one point from each orbit of a chosen point z under the action of the group G ([1], §9.1), and so the fundamental region varies with the choice of z. Regardless of the choice of z however, the area ◦ of F is invariant ([1], Theorem 9.1.3). Since the boundary of F has area 0, then ◦ area (F ) = area (F ), and so is invariant as well.

The Dirichlet region (Deﬁnition 9.2) is one possible fundamental region for a Fuchsian group, and in fact, Dirichlet regions can be constructed for every Fuchsian group ([9], Theorem 3.2.2). Thus, inﬁnitely many orientation preserving groups of isometries tessellate the hyperbolic plane. 110

To begin the discussion of Dirichlet regions, we ﬁrst state the following lemma.

Lemma 1. Let G ≤ PSL (2, R) act properly discontinuously on H, and z ∈ H be ﬁxed by some element of G. Then there is a neighborhood W of z such that no other point of W is ﬁxed by an element of G \{I} ([9], Lemma 2.2.5).

Deﬁnition 9.2. Let G ≤ PSL (2, R) be discrete, and let the point w ∈ H such that for all A ∈ G \{I}, w∈ / Fix (A). The Dirichlet region for G centered at w is deﬁned as

Dw (G) = {z ∈ H | d (z, w) ≤ d (z, A (w)) , ∀A ∈ G} .

Equivalently ([9], §3.2),

Dw (G) = {z ∈ H | d (z, w) ≤ d (A (z) , w) , ∀A ∈ G} .

For a given A ∈ PSL (2, R) and w ∈ H with w∈ / Fix (A), the region {z ∈ H | d (z, w) ≤ d (z, A (w))} is the half-plane containing all points closer to w than to A (w). The boundary of this half-plane is the geodesic which is the perpendicular bisector of the geodesic segment connecting w and A (w), all of whose points z satisfy d (z, w) = d (z, A (w)) ([9], Lemma 3.2.1). The boundary of the Dirichlet region is made of sections of geodesics. For a discrete subgroup G ≤ PSL (2, R), with w ∈ H

not ﬁxed by A ∈ G, the Dirichlet region Dw (G) of G centered at w is constructed by taking the intersection of the half-planes

\ A (w) = {z ∈ H | d (z, w) ≤ d (z, A (w))} . A∈G\{I}

For example, in Figure 9.1, let A1 and A2 be isometries in G ≤ PSL (2, R), and w a point not ﬁxed by any element of G. Then δ deﬁnes the half-plane containing points 111

closer to w than to A1 (w), and γ deﬁnes the half-plane containing points closer to

w than to A2 (w). The shaded region is the intersection of these two half-planes and so contains the Dirichlet region of G. The Dirichlet region itself is the region deﬁned by the intersection of all half-planes deﬁned in this manner by images of w under G.

A1 (w) A2 (w)

w δ

Figure 9.1: Intersection of half-planes to determine a Dirichlet region

As an intersection of half-planes, the Dirichlet region is a hyperbolically convex region, and a connected fundamental region for G ([9], Theorem 3.2.2). Three is the minimum number of sides that such a fundamental polygon may have, in both the Euclidean and hyperbolic cases. A nice example of a Dirichlet region is that of the Modular group, which is itself an ordinary triangle group.

Deﬁnition 9.3. The Modular group is the group M = PSL (2, Z) ≤ PSL (2, R), whose matrices have integer entries, and are generated by a parallel displacement     1 1 0 −1     about ∞, P =  , and a half-turn about i, H =  . 0 1 1 0

Example 9.4. To ﬁnd a nicely symmetrical Dirichlet region of the Modular group,

deﬁne the ordinary point w = ki, with k ∈ Z, k > 1. Then w is ﬁxed by no

transformation A ∈ M, and Dw (M) = {z ∈ H | d (z, w) ≤ d (z, A (w))} is a (3, 3, ∞)

−1 triangle, having vertices v1, v2 and ∞. Note that v1 = P (v2), and v2 = P (v1), Figure 9.2. 112

w = ki

Dw (M)

π i π 3 3

v1 v2

1 0 2 1

Figure 9.2: Tesselation of the upper half-plane by the Modular group

The Modular group is the index two orientation preserving subgroup of the group G =

P GL (2, Z), which contains a reﬂection Rl such that Dw (M) = Dw (G)∪Rl (Dw (G)). Deﬁne l to be the angle bisector of the zero angle at ∞, which perpendicularly bisects

−1 the segment from P (i) to P (i) at i. Then Dw (M) can be characterized either as a (3, 3, ∞) triangle, or as a quadrilateral of the same shape whose fourth vertex is the midpoint of its base, i ([10], II.6, Ex.1). M is the orientation preserving subgroup of the full (2, 3, ∞) triangle group. Under the action of M, the fundamental region

Dw (G) ∪ Rl (Dw (G)) tessellates as in Figure 9.2 ([10], Figure 15 and [9], Figure

9). Under the action of P GL (2, Z), the fundamental region Dw (G) tessellates as in Figure 9.3 ([10], Figure 15). 113

Rl (Dw (G))

Dw (G) l

π i 2 π 3

1 0 2 1

Figure 9.3: Tesselation of the upper half-plane by P GL (2, Z)

For any full triangle group, the Dirichlet region is the intersection of three half-planes. The Dirichlet region of any ordinary triangle group is the intersection of either three or four half-planes, depending on whether the fundamental region is a triangle or a quadrilateral.

9.2 Triangle Groups

Similarly to the Euclidean case, a hyperbolic full triangle group in P GL (2, R) is a

group of type (α, β, γ), and is generated by the reﬂections Rj,Rk,Rl in the sides j, k, l of a triangle. It has an index 2, orientation preserving subgroup in PSL (2, R), its associated ordinary triangle group. Taking into account that for hyperbolic triangles π π π + + < π, the general presentations for hyperbolic full and ordinary triangle p q r groups are as for the Euclidean full and ordinary triangle groups, (4.1) and (4.2), 114

Section 4.4.

Recall that a necessary, though not suﬃcient, condition for discreteness of a group mπ of type (α, β, γ) is that each angle of the triangle can be written as with m, p p relatively prime integers. A suﬃcient, though not necessary, condition for discreteness π is that each angle of the triangle can be written with p ∈ , 2 ≤ p ≤ ∞. The p Z connection between these two conditions is that a group of type (α, β, γ) having angles mπ of the form is discrete if and only if it is also a group of type (α, β, γ) having p π angles of the form ([1], p278). Example 9.5 demonstrates one case, and also shows p that the generating triangle need not be a fundamental region of the group.

Example 9.5. Let T1 and T2 be the two triangles in the Figure 9.4: the corresponding π π groups are G = hR ,R ,R i (of type (0, , )), and G = hR ,R ,R i (of type 1 h k j 2 3 2 h k l 2π (0, 0, )). Then R R = R R gives that R = R−1R R , so R ∈ G , and G ⊂ G ; 3 j k k l l k j k l 1 2 1 −1 and that Rj = RkRlRk , so Rj ∈ G2 and G1 ⊂ G2. Thus G1 = G2. Note that while

the groups generated are equal, T1 and T2 diﬀer both in side-length and in area. In particular, 2π π π area(T ) = π − = 2(π − − ) = 2area(T ). 2 3 2 3 1

Since any two fundamental regions of a group have equal area, then even though it

is the generating triangle, T2 cannot be a fundamental region of G2 ([1], p277). 115

0 0 j k l

Rj Rk Rl

T1 T2 π 2 π v 3 2π 1 3 v2 h Rh 0

v3 1 0 2 1

Figure 9.4: Triangles T1 and T2 generate the same group.

Consider now an arbitrary full (p, q, r) triangle group G. G is discrete, and the triangle π π π π having angles , , is a fundamental region F for G. Let the angle occur at p q r p π π vertex P , occur at vertex Q, and occur at vertex R. Then 2p triangles meet q r at P , 2q triangles meet at Q, and 2r triangles meet at R. The associated ordinary

(p, q, r) triangle group is generated by the rotations S 2π ,S 2π ,S 2π having orders P, p Q, q R, r p, q and r respectively.

Reﬂect F 2p times across its sides intersecting at P . If q = 2 or r = 2, then the union of the resulting copies of F is a polygon having p sides which is both equiangular and equilateral. Without loss of generality, let r = 2. Then the polygon has p angles of 2π , so an n-gon is formed with n = p. If on the other hand r = q ≥ 3, then the q 2π regular polygon has 2p angles of , and the n-gon formed has n = 2p. q

If the (p, q, r) triangle is neither isosceles nor right, then the polygon formed in the above manner need be neither equiangular nor equilateral. And yet because its symmetry group is a subgroup of a full (p, q, r) triangle group, it still tiles the plane. 116

Thus, while the Euclidean plane may be tiled only by rhombuses, regular hexagons, and three speciﬁc triangles an inﬁnite set of convex polygons have symmetry groups which tile the hyperbolic plane.

9.3 Conclusions

An infinite number of discrete subgroups of Isom(H) tessellate H, and none of them are structurally identical to any Euclidean wallpaper group. Isom(H) is structurally distinct from Isom(E). In Isom(H), no isometry type forms a subgroup isomorphic to Z2, as translations do in Isom(E); and further, Isom(H) contains an isometry not ex- istent in Isom(E), the parallel displacement. The frieze groups exist in both Isom(H) and Isom(E). The isometry groups which tile H regularly with a convex polygonal fundamental region are infinite in number, as a Dirichlet region can be constructed for every Fuchsian group. Any regular convex polygon can be understood as a union of congruent triangles, and so a discrete isometry group generated by reflections in the sides of a regular polygon is in fact a subgroup of a triangle group.

All of the above comes from my predecessors in this ﬁeld of study. The original results in this thesis come from investigation into precisely when and how translations and parallel displacements are not closed in Isom(H). The isometry type of a composition of two nontrivial translations is determined by whether the axes of translation cross and at what angle, and the distances translated with respect to each axis; and the isometry type of a composition of two nontrivial parallel displacements is determined by the ideal points about which the displacements take place, as well as the distances displaced. The partial answer to this is presented in Section 7.3. Unless someone else has addressed the general case, then it remains unanswered. 117

9.4 Questions and Future Work

We began this research with far more questions than knowledge of isometry groups and the peculiarities of the hyperbolic plane, and many of these questions have already been discussed in the literature. Some questions that have not been answered in the course of this thesis are:

• What are the structures of the groups generated by the hyperbolic analogues of the generators of the Euclidean wallpaper groups?

• How is the structure of the groups proposed in the previous question aﬀected as the translation distance and axes of the translation are varied?

• Given the composition of two hyperbolic translations having divergently parallel axes, what relationship between the translation distances and the ideal points of the axes determines the isometry type of the composition? 118

Appendix: Semidirect Product of Groups

In the direct product H ×K of two groups H and K, group elements are ordered pairs (h, k) with h ∈ H and k ∈ K, and the group operation is deﬁned componentwise:

(h1, k1)(h2, k2) = (h1h2, k1k2). Both H and K are normal subgroups of the group G = H × K.

The semidirect product of two groups is a generalization of this idea, where each group element can be understood as an ordered pair (h, k) with h ∈ H and k ∈ K, but only one of H and K need be a normal subgroup of the constructed group, and the group operation is deﬁned diﬀerently.

Suppose G contains two subgroups H and K such that

(a) H/G, but K need not be

(b) H ∩ K = 1, and

Then every element in HK can be written uniquely as a product hk for h ∈ H and 119

k ∈ K. Given two elements h1k1, h2k2 ∈ HK, their product is

−1 (h1k1)(h2k2) = h1k1h2 k1 k1 k2

−1 = h1 k1h2k1 k1k2 (9.1)

= h3k3

−1 where h3 = h1 k1h2k1 ∈ H and k3 = k1k2 ∈ K.

Since H/G is a normal subgroup, K acts on H by conjugation: khk−1 ∈ H. This

−1 gives a group homomorphism ϕ : K → Aut (H) by ϕ (k) = ϕk, where ϕk (h) = khk , and so Equation 9.1 becomes

(h1k1)(h2k2) = h1ϕk1 (h2) k1k2

Now, instead of assuming the existence of a group G having subgroups H and K satisfying conditions (a) through (c), begin with the two groups H and K, and a homomorphism ϕ : K → Aut (H), and deﬁne G in terms of them.

Theorem 9.6. ([3], Theorem 5.10) Let H and K be groups and ϕ : K → Aut (H) a group homomorphism. Let ∗ denote the (left) action of K on H determined by ϕ.

Let G = H oϕ K be the set of ordered pairs (h, k) with h ∈ H and k ∈ K, and deﬁne the following multiplication on G:

(h1, k1)(h2, k2) = (h1k1 ∗ h2, k1k2) .

(1) This multiplication makes G in to a group.

(2) The sets {(h, 1) | h ∈ H} and {(1, k) | k ∈ K} are subgroups of G and the maps 120

h 7→ (h, 1) for h ∈ H and k 7→ (1, k) for k ∈ K are isomorphisms of these subgroups with the groups H and K respectively:

H ∼= {(h, 1) | h ∈ H} and K ∼= {(1, k) | k ∈ K} .

Identifying H and K with their isomorphic copies in G described in (2), we have

(3) H/G

(4) H ∩ K = 1

(5) G = HK:(h, k) = (h, 1) (1, k).

−1 Note that in G, (1, k)(h, 1) (1, k ) = (ϕk (h) , 1).

Corollary. There is a short exact sequence which splits,

π ιH 1 → H → G K → 1 ιK

where ιH : H → G by ιH (h) = (h, 1), ιK : K → G by ιK (k) = (1, k), and π : G → K by π (h, k) = k. Remark. Given a short exact sequence that splits,

π ιH 1 → H → G K → 1 s

∼ it can be shown that G = H oϕ K where ϕ : K → Aut (H) is given by ϕk (h) = s (k) hs (k)−1 ∈ H. 121

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