Noneuclidean Tesselations and Their Groups
Pure and Applied Mathematics A Series of Monographs and Textbooks Editors Paul A. Smith and Samuel Ellenberg Columbia University, New York
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Noneuclidean Tesselations and Their Groups WILHELM MAGNUS Department of Mathematics Polytechnic Institute of New York Brooklyn, New York
ACADEMIC PRESS New York and London 1974 A Subsidiary of Harcaurt Brace Jwanovich, Publishera
COPYRIGHT 0 1974, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
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Library of Congress Cataloging in Publication Data Magnus, Wilhelm, Date Noneuclidean tesselations and their groups. (Pure and applied mathematics; a series of monographs and textbooks, v.61 ) Bibliography: p. 1. Tesselations (Mathematics) 2. Geometry, Noneuclidean. 3. Discontinuous groups. I. Title. 11. Series. QA3.P8 vol. [QA164] 510'.8s [511'.6] 73-18966 ISBN 0-12-465450-9
AMS (MOS) 1970 Subject Classifications: 20H10,5OC05 PRINTED IN THE UNITED STATES OF AMERICA
To the memory of Hanna Neumunn 1914x971
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Contents
Preface
ix
Abbreviations and Symbols CHAPTER I.
ELEMENTARY CONCEPTS AND FORMULAS 1.1 1.2 1.3 1.4 1.5 1.6 1.7
CHAPTER 11.
xiii
The Group G* of Homographic Substitution2 Action of G* on the Closed Complex Plane C Action of G* on Hyperbolic Three-Space Circle Groups as Groups of Motions of Hyperbolic Two-Space Notes on Elliptic and Spherical Geometry Illustrations. References and Historical Remarks Appendix: Hilbert’s Axioms of Geometry
1
2 10 19 37 41 43
DISCONTINUOUS GROUPS AND TRIANGLE TESSELATIONS 11.1 11.2 11.3 11.4 11.5 11.6 11.7
Introductory Remarks Discontinuous Groups and Fundamental Regions Triangle Groups, Local and Global Relations Euclidean, Spherical, and Elliptic Triangle Groups Hyperbolic Triangle Groups Some Subgroups of Hyperbolic Triangle Groups General Theorems. A Survey and References
52 56 65 68 81 90 95
CHAPTER 111. NUMBER THEORETICAL METHODS 111.1 The Modular Group 107 111.2 Subgroups and Quotient Groups of the Modular Group 112 111.3 Groups of Units of Ternary Quadratic and Binary Hermitian Forms 123 CHAPTER IV.
MISCELLANY IV.1 IV.2
Examples of Discontinuous Nonfuchsian Groups Fricke Characters vii
134 148
viii
Contents
CHAPTER V.
GROUPS THAT ARE DISCONTINUOUS IN HYPERBOLIC THREE-SPACE
V.l V.2
Index
Linear Groups over Imaginary Quadratic Number Fields Some Geometric Conatructions
151 153
FIGURES
157
REFERENCES
199 205
Preface The mathematical universe is inhabited not only by important species but atso by interesting individuals. C. L. Siege1
In the last decades of the eighteenth century, Georg Christoph Lichtenberg (1742-1799) , Professor of Physics in Goettingen and essayist, published a sequence of “Commentaries on Hogarth’s Engravings” which, a few years ago, were republished in English by I. and G. Herdan (1966). Lichtenberg’s commentaries go well beyond a description of the engravings. They tell a story about the persons appearing there and reflect on the usefulness and even the history of the equipment shown in the engravings, and also refer to many persons and events not shown. The present book intends to be a less comprehensive and purely mathematical version of Lichtenberg’s commentaries. The pictures that motivated the writing of the book are mainly those appearing in the mathematical works of Felix Klein and Robert Fricke. Most of them show tesselations of the noneuclidean plane. Comments on their group theoretical, geometric, and function-theoretical meaning are, of course, available in the more than two thousand pages published by Klein and Fricke on this subject. But these comments are not easily accessible. The present book tries to reach an audience of senior undergraduate or first-year graduate students and to serve as a useful companion volume for courses on geometry and group theory. It is an elementary book since it does not give any account of the theory of automorphic functions or of the large body of applications to special algebraic equations which had been assembled in the late nineteenth century and for which Volume 2 of the lectures on algebra by Fricke (1926) still seems to be the best source. Even on this level, lengthy and difficult proofs have been avoided whenever a good contemporary source was available as a reference. In spite of its classical material, the book is not merely a salvage operation for some nineteenth-century mathematics (although, in the opinion of the author, such an enterprise would be very much worthwhile). There are ix
x
Preface
numerous references to recent papers, showing how the old results stimulated new research. The theory of discontinuous groups is one of the best examples available to demonstrate the coherence of mathematics because the results and methods of many special disciplines enter into it. However, this fact also results in a fairly large number of prerequisites for the present book. Due to its elementary nature, the text does not require a deep knowledge of differential geometry, group theory, topology, or number theory. For differential geometry, a small fraction of the book by Stoker (1969) will suffice. Special references are given to the few results not to be found there. For group theory, special references are given only in exceptional cases. The concepts of a free group and of a presentation in terms of generators and relations, as well as the ReidemeisterSchreier method, are assumed to be known. As a general reference, “Combinatorial Group Theory” (Magnus, Karrass, and Solitar, 1966) will do more than suffice. For topology, Massey (1967) is more than adequate. On a few occasions, a little bit of algebraic number theory is used. Pollard (1950) will be fully adequate where no other reference is given. Various chapters in Siege1 (1972) will be quoted when appropriate. However, this text, which also offers an excellent introduction to the theory of automorphic functions, is utilized here only through its geometric theorems. The main characteristics of the book are the emphasis given to the special (and perfect) example rather than to the general theorem and the preference for the explicit formula wherever it appears to give information beyond abstract formulation of a result. The first feature is inherent in the motivating purpose of the book: to comment on the drawings. The second feature may contribute to the economics of mathematics in some cases. The author suspects that some tedious calculations which are then mentioned as “trivial” in research papers are actually the same calculations carried out for the hundredth time because nobody since the nineteenth century has dared to write the results out explicitly. The first chapter contains the geometric basis for the later parts. Noneuclidean geometry in the plane is developed by using the Poincare model of the upper halfplane. The fact that all axioms of Hilbert other than the parallel postulate are satisfied is verified explicitly. In an appendix, a translation of Hilbert’s axioms is given. The elliptic plane and hyperbolic threespace are described briefly. This chapter can be considered as a complete development of planar noneuclidean geometry. The second chapter deals with the triangle tesselations in planar euclidean, hyperbolic, and elliptic geometry and also with the triangle tesselations of the sphere.
Preface
xi
The third chapter deals with the modular group and other discontinuous groups defined by number-theoretical methods. The reports on B. H. Neumann’s nonparabolic subgroups of the modular group and on Fricke’s explicit presentation of unit groups of ternary quadratic forms cover material which seems to be little known even among experts in the field. Chapter IV contains a brief description of some nonfuchsian groups. It gives an idea of the incredible complications which can arise here. (The most pathological cases have been discovered only very recently or are merely known to exist although no explicit examples can be given.) The main feature of the fourth chapter is a complete and elementary proof of the pathological nature of the Jordan curve which carries the limit points of a particular nonfuchsian group. This result is widely known and frequently mentioned, but it seems that Fricke’s original proof is the only place where it appears in the literature. The fifth chapter deals with discrete groups of Mobius transformations which are discontinuous only in hyperbolic three-space. Most of it is very sketchy and contains mainly references. The only case dealt with in some detail concerns an example where the source is nearly inaccessible. The author is indebted to many colleagues for valuable comments and discussions. For extensive help, either in the form of calculations or through detailed criticism, special thanks are due to Bruce Chandler, Constance Davis, N. Purzitsky, Laurie Erens Spatz, and Carol and Marvin Tretkoff. The author also wishes to express his gratitude to the supreme experts who provided technical help: to Helen Samoraj who typed the whole manuscript with a critical eye for mistakes and inconsistencies and to Carl Bass who redrew the very difficult figures, and he would like to put on record his appreciation for unfailing patience and the understanding for his many special requests extended to him by the staff of Academic Press. Many of the drawings appearing in this work have been adapted from the versions appearing in the original publications, as noted beneath each figure. The courtesy of B. G. Teubner, Stuttgart, and Friedrich Vieweg und Sohn, Braunschweig, in granting permission to use these figures is gratefully acknowledged.
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Abbreviations and Symbols
Latin letters
Alternating group on n symbols Crossratio n-dimensional euclidean space Triangle group with angles r/Z, r / m , r / n in the elliptic plane Group of fractional linear mappings (Mobius transformations) with complex coefficients acting on a complex variable z G* Extension of G by the mapping z + Z I Unit matrix; In, unit matrix of degree n Klein’s model of noneuclidean (hyperbolic) geometry in two Kz, & or three dimensions (models in projective spaces) 0 Orthogonal group O* Orthogonal group extended by reflections Poincard’s model of, respectively, noneuclidean (hyperbolic) Pz, Pa two- and three-space, usually denoting the upper halfplane or halfspace, with the appropriate metric PSL(n, R ) Projective special linear group, arising from SL(n, R ) by factoring out its center Field of rational numbers The 2-sphere Special linear group of matrices of degree n with determinant + 1 and coefficients in the ring R tr M denotes the trace of the matrix M M t is the transpose of the matrix M Triangle group of orientation-preserving motions belonging to triangle with angles r/l, r / m , r / n Triangle group arising by reflecting the triangle with angles r/l, r / m , r / n in its sides A complex variable; z = z iy in the traditional notation
An CR En E*(Z, m, n ) G
+
xiii
xiv
Abbreviations and Symbols
Greek letters
r r* rm rl z n
all Q Q*
Restriction of G to selfmappings of upper halfplane; PSL(2, R) Restriction of G* to selfmappings of upper halfplane mth principal congruence subgroup of modular group r Modular group, PSL(2, 2) Symmetric group on n symbols Fundamental group of an orientable compact 2-manifold of genus g Restriction of G to selfmappings of the unit disk Restriction of G* to selfmappings of the unit disk
Special symbols
C
e
R Z
I
Q -
Field of complex numbers; the complex plane Complex plane closed b y a point at infinity Field of rational numbers Ring of integers x is the conjugate complex of I If H is a group, HI denotes its commutator subgroup. Angle Congruent
Noneuclidean Tesselations and Their Groups
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Elementary Concepts and Formulas
1.1 T h e Group G* of Homographic Substitutions We shall be concerned with the action of various groups on several spaces which may be merely sets but which will mostly be metric spaces. By “action,” we mean the following: T o every element g of the group, there is defined a transformation t of the space (i.e., a oneone mapping of the space onto itself) such that the mapping g + t defines a homomorphism of the group into a group of transformations of the space. All groups, spaces, and actions will be given in a very explicit form. The groups to be considered can be derived from a single one by forming subgroups and, in some cases, quotient groups of subgrou-ps.We shall call this comprehensive group G* and we proceed now with its definition. Let C be the field of complex numbers. The special linear group S L ( 2 , C) is defined as the group of all 2 X 2 matrices
The center of SL(2, C) consists of the matrices‘fl, where I denotes the 2 X 2 unit matrix. The quotient group of SL ( 2 , C ) with respect to its center is the projective special linear group P S L ( 2 , C ) , which we shall denote briefly by G . We may define an element g of G by a matrix of the form (1.1) with the understanding that this matrix and its negative will define the same g.
Elementary Concepts and Formulas
2
Now we construct the group G* as an extension of G as follows: We adjoin to G an element A* of order 2 such that G becomes a subgroup of index 2 in G*. This is done by subjecting A* to the relations
A*2
(1-2)
=
A*gA*-’
1,
=
9- ,
where
and where a bar over a complex number denotes its complex conjugate. Obviously, A* acts as an automorphism of G, and (1.2) and (1.3) define G* completely.
1.2
Action of G* on the Closed Complex Plane
e
We describe the complex plane as the range of a complex variable z = z
+ iy,
z, y
real
and we close the complex plane by adjoining one ideal point z = Q).The resulting closed plane (which, for the moyent, may be considered as nothing more than a set) will be denoted by C. The substitutions (1.4a) (1.4b)
2’
= 2,
e
define, respectively, transformati2ns t, and t* of by mapping z onto z’. We obtain the action of G* on C by mapping g --$ t, and A* + t*. The composition of two substitutions t and Z is given in the usual manner: If t maps z onto f(z) and t maps z onto f ( z ) , then ti maps z onto f ( .f(z)). From the theory of complex variables, we borrow the following elementary results [see Caratheodory (1954a)l.
(i) The mappings (l.?) preserve the absolute size of angles between differentiable curves in C. The mappings t, in (1.4a) (i.e., the transformations in G) are conformal, .and those in G* but not in G reverse the orieptaThe mappings t, are the only conformal transformations of C. tion of
e.
Z.2 Action of G* on the Closed Complex Plane
(ii)
Define a circle in
pz2
(1.5b)
3
6 as any curve defined by an equation
(1.5a) where p and a are real,
8
p
+ + jiz + u pz
=
0,
is arbitrarily complex, and P
2 0,
PLP
> pa.
Then the transformations of G* map circles onto circles. For p > 0, (1.5) defines a circle in the ordinary sense; its center is at - p / p , and its radius is given by ( p p - pa) l'2p-1
.
For p = 0, (1.5) defines a straight line; we may characterize the straight lines as the circles passing through z = 03. From elementary algebraic calculations, we obtain the following results. (iii) Lemma 1.1 Let z, (v = 1, 2, 3, 4) be four complex numbers a t least three of which are distinct. Define the crossratio CR(z,) by
Let 2,' be the value of z, under a substitution of (1.4a) of G. Then
CR(z,)
(1.7)
=
CR (2,') ,
If we choose in G* a substitution that is not in G, the crossratio of the z, goes into its conjugate complex value. Note: There are several definitions of the crossratio in the literature which arise from each other by a permutation of the z,. The definition given in (1.6) has the following property: If CR is considered as a function of zl,with fixed zz,23, 24, then CR assumes, respectively, the values 0, 1, co (in this order) if z1 assumes the values zz, 23, 24. The value of CR remains unchanged if the zy are permuted according to the permutations of their subscripts described by (1, 2 ) (3, 4), (1, 3) ( 2 , 4), and (1, 4)( 2 , 3 ) , i.e., according to the permutations of the four-group. If we put CR(z,) = y and apply the 24 permutations of the symmetric group to the zy, then y assumes only the six values
(1.8)
y,
7-1,
1 - y,
1 - Y-1
,
(1 -
Y(Y
- 1I-l.
The homographic substitutions of y which carry it into one of the expressions in (1.8) form a group isomorphic with the symmetric group on three symbols. This fact is the basis for the solution of the general equation of fourth degree. See, e.g., Van der Waerden (1948, p. 175 ff, Section 58).
4
E l e m n t a y Concepts and Formulas
(iv) Lemma 1.2 Given three distinct values z, ( p = 1, 2, 3), there exists exactly one circle C through these points. For any point z to be on C , it is necessary and sufficient that the crossratio
be real. This is an algebraic version of the theorem stating that in a circle the angle inscribed into an arc is constant. It also follows immediately from the next statement since it is obviously true if C is the real axis. (v) Let z, and w, be two triplets of distinct values of z, where we need not exclude the cases in which some or all of the w, are equal to some or all of the z,. Then there exists exactly one element t, E G of the type (1.4a) which maps z, into w, ( p = 1, 2, 3). It is given implicitly by
(1.9)
4:
2' --=--*
"1
- w1
w2 - w3 - wz w3 - 2'
z 21
- 21 2 2 - 23 - Zz 23 - Z
There exists exactly one mapping I,* in G* which maps every point of a circle C onto itself without being the identity mapping. I,* is an invoEution (i.e., I,*2 is the identity mapping) and it will be called the rejlection in the circle C. I,* never is an element of G. It is given implicitly by (1.10) where zl, zz,23 are any three distinct points on C. It should be observed that (1.9) is not in the form (1.4a). If we write (1.9) in the form
1.8 Action of
G* on the Closed Complex Plane
In the special case where z1 = 0,z2 = 1, 23 as follows: (1.12a)
fkY= w3(w2 - Wl),
(1.12b)
97
(1.12c)
=
w2 - w1,
Q2
= (Wl -
=
00,
these formulas simplify
Qfl
=
wI(w3 - WZ),
96
=
w3
w2)(w2 - w 3 )
5
- w2,
(w3
- Wl).
In particular, we can map the points 0, 1, 0 0 , respectively, onto the points 1, i, - 1 (thereby mapping the real axis on the unit circle and z = i onto z’ = 0) by means of the substitution (1.13)
Z’ =
+ 1)z + +(i - 1) , +(1 + i ) z + +(i - 1)
-+(i
z=
+(i - 1)z’ - $(i - 1) - f ( i + 1)z’ - +(i + 1)
’
We add here a well-known result which will be needed later.
e
e
Lemma 1.3 The points of a circle in separate the remaining points of into two disjoint classes. Any two points of the same class can be connected by a segment of a straight line which does not intersect the circle. Any circular arc connecting two points of different classes intersects the circle.
In the case of an ordinary circle, the interior consists of that class of points of that contains the center of the circle, the other class being its exterior. In the case where the circle is a straight line, we shall speak of the two sides of a line. The mapping (1.13) maps the upper halfplane y > 0 of 6 onto the interior of the unit circle I z 1 = 1 in such a manner that z = i is mapped onto z’ = 0. Three distinct points P1,Pz, P3 always define a unique circle and, by their ordering, an orientation of the circle. This includes the case where one of the points is 00 and the circle is a straight line. Our statement (v) enables us to define the positive side of an oriented circle as follows: Let y > 0 denote the positive side of the real axis oriented by the succession 0, 1, 00 of three points. Then we can map these three points in a unique manner (1.9) on the points P1,Pz,Pa (in this order). The image of y > 0 is then P2,PS,which is oriented by called the positive side of the circle C through P1, this ordering of the points. In order to show that this definition of the P3and depends only on the orientation positive side is independent of P1,P2, defined by their ordering, it suffices to show that all selfmappings of the real axis onto itself which preserve its orientation map the upper halfplane y > 0 onto itself and vice versa. This can be derived from the formulas
Elementary Concepts and Formulas
6
(1.12), where w, is now the (real) coordinate of P, ( p = 1, 2, 3 ) . We see that 9' > 0 whenever all w, # m and the P, define the same orientation as 0, 1, 0 0 , that is, whenever
< wz < w3 or Q2 > 0 is equivalent
w2 < w3< w1
WI
or
w3< w1 < wz.
to the condition that a, p, y, 6 are real and +l. I n turn, these conditions are necessary and sufficient for a mapping to carry the real axis into itself and to map the upper halfplane onto itself. The cases where one of the w, = Q, have to be treated separately. We now turn around and define the positive orientation of a circle as the orientation that makes the interior of the circle its positiveside. I n anthropomorphic terms, the positive orientation of a circle is the one where the interior of the circle is on the left-hand side of a traveler who goes in the direction of the oriented circle. Now a6
- By
=
(vi) Conjugacy classes; traces; $xed points. As usual, we define the trace of a matrix
7
[;
(1.14)
+
(a6 - py = 1)
to be a 6. For the corresponding element t, of G, as defined by ( 1 . 4 4 , the trace is determined only up to its sign; we shall write
trt,
(1.15)
= a f 6
=
x
if we define t, by its matrix, but actually the trace of t, is the pair of numbers f(a 6). We have
+
Lemma 1.4 If t, is not the identity mapping, then t, is conjugate with a n element t,' if and only if
t r t,
=
f t r t,'
unless tr t, = f 2 and t,' is the identity mapping. I n particular, t, is conjugate with the element of G given by the substitution (1.16)
2' =
Xz/P
where (1.17)
if x
=
t r t, # f 2 . If
X =
x
$[x
= f2,
+ (x' - 4)''*]
t, is conjugate with a n element of G given by
I . 2 Action of G* on the Closed Complex Plane $
7
the substitution (1.18) of
2’ = 2
+
7.
The proof of Lemma 1.4 is based on the remark that the points 6 given by 1
11 = - [a - 6 2Y
(1.19)
1
- [a
{z
27
l1and pz
+ (x2- 4)’”],
- 6 - (x2 - 4)l’*]
(7 #
0)
are fixed points of the mapping x --f z’ defined b y the substitution (1.4a). I n the case where x2 = 4, we have t1 = l2= 6, where
t
(1.20) and in the case y (1.21)
=
= (a -
6)/2Y
0, we have
tl = w,
= P/(6 -a>
(2
If, in addition, a = 6 (which implies a (1.22)
( Y # 0)
{l
=
<2
=
t
(a # 6).
+ 6 = x = *2), (P
= co
we have
# 0).
If y = 6 = 0 and a = 6, we have, of course, the identity mapping for which every point is a fixed point. Using a coordinate transformation in 6 by introducing a new variable which assumes respectively the values 0 and w a t {I and t 2 , we obtain: Lemma 1.5 written as
The homographic substitution with matrix (1.14) can be
(1.23b)
2’
=
(1.23~)
2’
=
(1.23e)
X2(2
x’=z+P
- t2) - TI)
+t z +
PI
(t,# t z ; tl (11
z fz;
(tl
=
= w)
i-2 =
m)
=
t
# w)
(i-1 = t 2 =
t
= a),
t 2
where tl,tz,J’, X are defined, respectively, by (1.19), (1.20) , (1.17).
8
Elementary Comepts and Formdaas I n matrix form, the relation (1.23a) can be written as follows: Define T by
Then (1.24)
T
[T
:]
T-1 =
[$ :]
As a consequence of Lemma 1.5, we have
Lemma 1.6 Two elements # 1 of G, written as transformations of 2, will commute if and only if they have the same fixed points, unless both of them are of order 2. Proof: By conjugation, we can put one of the elements into one of the forms (1.23). For the corresponding conjugate of the other one to commute with the first is possible only if it is of the same form, differing merely in the value of X in the first three cases or in the values of y or p, respectively, in the fourth and fifth cases. The exception is X2 = -1.
Note: We have written out all formulas explicitly because, on later occasions, we may wish to know under which circumstances two elements of a subgroup H of G are conjugate within H . If H is defined arithmetically (e.g., by restricting a, p, y, 6 to a fixed algebraic number field), the ocl2becomes important. currence of square roots in the computation of X, As a consequence of Lemma 1.6, we have Lemma 1.7 Let A be an abelian subgroup of G which is not of exponent 2 and order 4. Then A is isomorphic with a subgroup of either the additive group of complex numbers or the multiplicative group of nonzero cqmplex numbers. Any maximal A is either of index 2 in its normalizer, in which case the elements f l of A have the same distinct fixed points, which are exchanged by conjugation u7it.h an element of the normalizer, or the quotient group of A in its normalizer is isomorphic with the multiplicative group of nonzero complex numbers. I n this case, all elements # 1 of A have a common single fixed point.
u ~ ~ ~ s , groups. We cl%ssify (vii) ClassiJicutzon of s u ~ s t ~ ~ one-parameter the homographic substitutions (i-e., the elements t, of G acting on C) as follows. If the matrix of a substitution is given by (1.14), and if X, {I, {Z are defined respectively by (1,15), (1.16), (1.17), (1.19), and if t, is not
I 2 Action of G* on the Closed Complex Plane
2
9
the unit element of G, then t, is called
+ 6)z = 4 (i.e., X = f l , l1 = lz = (a - 6)/2y); elliptic if a + 6 is real, (a + < -4; in this case # {z, I X 1 = 1, X Z f1; hyperbolic if a + 6 is real, (a + > 4;in this case ll l z , and X is real; Zoxodromic in all remaining cases. This means that ll # l2 and a + 6 and X
parabolic if (a
$
are not real, although X2 may be real and negative. With every to # 1 in G we associate a one-parameter group T,(s) (which is a subgroup of G) in the following manner. Let s be a real parameter, - m < s < 03. If t, is parabolic with fixed point co , the group T,(s) consists of all substitutions (1.25a)
2 ' = 2
If t, is parabolic with fixed point l # substitutions
+ sp. 01,
the group To(s)is given by the
(1.25b)
If t, is elliptic, we put ( 1.26a)
X = ei+,
4 real,
if t, is hyperbolic, we put ( 1.26b) X2 = e2u, u real, and if t , is loxodromic, we put X2 = e2Ufzir (0 < T < T). (1.26~) In all of these cases, X28 will then be defined uniquely by, respectively, (1.26d) e2i48 e2" e2(r+ir)a 1 9 1 and T,(s) is defined by (1.23a)-(1.23~) with XZ8 replacing X2. Let zo be any point in and let zo(s) (for - 00 < s < 03) be the image of zo under the substitutions of the one-parameter group associated with t,. Then the curve zo(s) is called the orbit of t, through zo. The orbits of the fixed points of t, are, of course, the fixed points themselves. The orbits of all other points are as follows:
el
In the case of a parabolic to: Circles through the fixed point l, excluding f itself, tangential at { to the ray from 0 to p in the case p = m and tangential to the parallel through l of the ray from 0 to l / y in the case where f # QO (i.e., y # 0 ) .
10
Elementary Concepts and Formulas
In the case of a hyperbolic t,: The arc of the circle through zo and the fixed points pl, pz of t,, excluding the fixed points themselves.
I n the case of a n eEEiptic t,: The full circle C through zo which is orthogonal t o all of the circles through the fixed points of t,. (More precisely, C is the projection of the orbit onto the complex plane. The orbit consists of the universal convering of C which is topologically the same as a n open straight line, wound around C infinitely many times. We shall, however, usually mean only C itself when talking about the orbit of a n elliptic t B . ) I n the case of loxodromic t,: The orbits are Eoxodromic curves, i.e., curves that intersect the circles through the fixed points of t, under a constant angle. I n the case where the fixed points are 0 and CQ , the loxodromic curves are logarithmic spirals ; in parametric form, the equation of the logarithmic spiral belonging to a given X and passing through a point zo is, of course, given by z = z(s) =
where
and
X2*
X%*
are defined in (1.17) and (1.26).
Illustrations: If an elliptic and a hyperbolic substitution have the same fixed points, their respective orbits are orthogonal trajectories of each other. Figure 8 shows both the orbits of an elliptic and a hyperbolic substitution with the same fixed points. Figure 9 shows the orbits of a parabolic substitution and their orthogonal trajectories, which are again orbits of a parabolic substitution with the same fixed point. Figure 10 shows the orbits and their orthogonal trajectories of a loxodromic substitution with fixed points a t z = CQ and z = 0. Figure 11 shows the same curves if both fixed points are finite.
1.3
Action of G* on Hyperbolic Three-Space
The group G* of homographic substitutions acts as a group of transformations on hyperbolic three-space. This is a riemannian manifold and, therefore, a metric space. We shall use two embeddings of this space into euclidean three-space. One of them has been used mainly by Felix Klein; the geodesics are segments of euclidean straight lines. We shall denote this model by K S .The other model, denoted by PB, was introduced by Henri Poincar6. The geodesic lines are euclidean semicircles perpendicular to a fixed plane in euclidean three-space.
G* on Hyperbolic Three-Space
I.3 Action of
11
We shall introduce both K 3 and ,P3 by using a well-known conformal mapping of the closed complex plane C onto the surface of the unit sphere S2, embedded in euclidean three-space. Let X , Y , Z be Cartesian coordinates in euclidean three-space. We can map the points of the two-sphere S 2 , given by
x2 + Y2 + 2 2 = 1 onto (which coincides with the plane Z the stereographic projection (1.27)
z = 2
0 in euclidean three-space) by
=
X + iY + iy = ____. 1-2
This mapping is one-one and known to be c?nformal, the north pole 2 = 1 corresponding to z = 0 0 . Every mapping of C onto itself under the action of a n element of G* induces a selfmapping of X 2 . This selfmapping can be extended to a projectivity of projective three-space which maps S2 and its interior onto itsel!, and all projectivities with this property are induced by the mappings of C onto itself contained in the group G* defined b y (1.4a) and (1.4b). The formulas describing this situation can be set up as follows. We introduce homogeneous coordinates tl, t2, t 3 , t 4 in euclidean three-space by putting
x
(1.28)
=
t&4,
Y
2
= t 4 4 ,
= ta/t4.
The points of projective three-space are then defined as the sets of quadruples (Xtl, Ah,
xt3,
x # 0,
A&),
t12
+ + + t22
t32
t42
> 0.
A point with t 4 # 0 can be identified with a point of euclidean three-space; the points for which t4 = 0 form the “infinite plane” which closes it. The homogeneous linear transformations with real coefficients and nonvanishing determinant define projectivities; if M and M‘ are the matrices of two such transformations, they define the same projectivity if and only if M = AM’, X real and X # 0. We are interested in those projectivities that map the unit sphere S2 and its interior onto itself. S2 is now given by (1.29)
t12
+ + t22
t32
- t42
=
0
and the matrix M of such a projectivity (after multiplication by a n appropriate constant) must satisfy the relation ( 1.30a)
MtLM
=
L,
12
Elementary Concepts and Formulas
:I,
where L is defined as 1 0 0 0 0 O1 0 0 1
(1.30b)
0 0 0 - 1 We find from (1.27) and (1.28) by assuming that (1.29) is satisfied
tl
= h(z
+E),
tz = --xi(z - z),
trr = h(z2
- l),
t4 =
h(zZ
+ l),
where X # 0. Writing (1.44 and (1.4b), respectively, as homogeneous substitutions (1.31a)
zl’= azl
+ pz2,
z2‘ = yzl
+ 6.a
(as
- 07
=
1)
and (1.31b)
21’ = E l ,
we can put
tl
= 2122
+
t3
= 21%
- 2222,
2122,
2;
=
22
- ZlZZ),
t2 = --i(Z122
(1.32) t4
= 2121
+
2*z2.
Clearly, the substitutions (1.31a) and (1.31b) induce linear substitutions t, (v = 1,. ., 4) in (1.32) with real coefficients. The resulting formulas are somewhat clumsy and we shall present them in a modified form by introducing
.
M of the
el = + ( t 3 + t4) = zlzl,
ez = +(tl
+ itz) = zlzz,
(1.33)
e3 = + ( t l - itz) = zlzz, e4 = +( -t3
+ t,)
= z2.z2.
Then (1.31a) and (1.31b) induce, respectively, the following substitutions on the 0”) the matrix of which is conjugate with M :
el’
=
+
ase1
+ npe3+ @Be4,
e2’ = a-iel + A e 2 + pqe3 + 8%
( 1.34a)
e3’ = me1 + h e 2 + rise, + gse,,
el
= Tiel
+ Tie, + 7% + sie,
and (1.34b)
el’ = el,
e2’ = e3,
Now we can formulate two lemmas.
e3’ = e2,
e4’ = e.,
Z.3 Action of G* on Hyperbolic Three-Space
13
Lemma 1.8 (Action of G* on K 3 ) The hyperbolic space K3, consisting of the points of the interior
x2 + Y2 + 2 2 < 1 of S2in euclidean three-space is a metric space in which a distance function A(x, y) between two points with coordinate vectors
x = x1, Yb 21,
y =
x2,
Yz, zz
is defined by
where (1.36)
K
> 0 and B(x, y)
=
1- XlX,
- YIY2 - 2122.
The geodesic lines are the segments of the euclidean straight lines within S2. The line element ds is given by
The projectivities M induced by (1.31s) and (1.31b) preserve distance. Any such projectivity can be induced by a combination of a substitution (1.31a) and (1.31b) or, if its determinant is +1, by a substitution (1.31a) alone. Lemma 1.9 (Noneuclidean geometry in K3) K3 is not only a metric space but a model of a three-dimensional noneuclidean geometry in the following sense : If we define points, straight lines, and planes to be identical with those points and parts of euclidean straight lines and planes inside the unit sphere, define order relations in the euclidean way, and use the distance function (1.35) to define congruence of line segments, triangles, and angles, then in this geometry all of Hilbert’s axioms (see the appendix) hold except for the parallel axiom, which states that for a line L and a point P outside of L, there exists exactly one line L’ through P in the plane determined by L and P such that L’ does not intersect L. The elements of G* act as rigid motions (including reflections) on K3,preserving distances and, therefore, mapping triangles onto congruent images. The measurement of angles can be derived from the formula (1.37) for the line element [see, e.g., Stoker (1969)l.
14
Elementary Concepts and Formulas
We shall not prove Lemma 1.9, which will be used only once, in the study of the Dehn-Gieseking group. However, we shall prove in the next section the analogue of Lemma 1.9 for two dimensions and shall explain in detail the meaning of the term “noneuclidean geometry.” Also, we shall derive the formulas of Lemma 1.8. However, we shall not prove the last assertion, according to which all matrices M satisfying (1.30) can be obtained from the substitutions (1.34a) and (1.34b). This result can be proved in an elementary manner [see Naimark (1964) 1.It also can be derived in an elegant manner by considering the sphere as a doubly ruled surface (like a one-sheeted hyperboloid) with complex straight lines. This approach is part of the process of algebraizing geometry which was carried out in the second part of the nineteenth century. For a brief description and reference to the papers of Cayley and Klein, see Fricke and Klein (1897, Chapter I, $12) or Hurwitz and Courant (1964). To derive the formulas of Lemma 1.8, we remark first that a point P in projective three-space is defined as a one-dimensional subspace of fourdimensional euclidean space E4 consisting of all multiples AT ( X real, X # 0) of a vector 7
= (ti,
tz, ta, h)
with components not all of which vanish. We call 7 (and any fixed multiple AT) a representative of P . Correspondingly, a straight line is defined as a two-dimensional subspace of E4, consisting of all points whose representatives are in the set Xi71
$-
h72,
where rl, 72 are two linearly independent vectors in E4, and X1 and X2 are real and do not vanish simultaneously. Let P I and P z be the points represented respectively by the vectors 7 1 and r2. Then the set of vectors 71
+ An,
X
real,
contains a representative vector for every point P on the line L joining PI and P2 if we admit the value X = co to obtain a representative of Pz. Let Q%be four points on L represented by the vectors r1 XZr2 (i = 1, 2, 3, 4). We can define the crossratio of the points Q2on L as crossratio y = CR (1,) of the X, if we can prove that y does not depend on the particular way in To do so, assume that ul, u2 which we have defined representatives of the Q2. are representatives of any two distinct points Pl‘, P2’, on L. Then every point on L has a representative of the form u1 pu2, and we have in particular as representatives of the &%(i = 1, 2, 3, 4)
+
+
UI
+
pZm=
+X
( ~ ~ ( 7 1
d
( P ~ 01% ,
real,
01%
# 0).
I.3 Action of
G* on Hyperbolic Three-Space
15
We also must have as representatives of Pll Pz 71
=
Pl(U1
+ rluz),
72
=
Pz(m
+ Yzuz)
(Pl,
Pz # 0 )
and therefore, for i = 1, 2, 3, 4, Q1
+
piuz =
OriC(P1
+
+
hiPdU1
(Ply1
+ ~iPzrz)uz].
Since u1 and uz are linearly independent, these relations give us Pi =
Ply1 P1
+ +
XiPZYZ XiPZ
and therefore C R ( h i ) = C R ( p i ) = y. We shall denote y also be C R ( Q i ) . We observe next that the application of a projectivity M maps the points Qi onto images Qi' such that CR = CR (Qi') since the Qi' have representahim, where tives ul (Q1)
+
~1
=
MT1,
~2
=
MT~.
Therefore, we can define a real function D(Pl, P z ) of PI and Pz which has the property that D(P1, Pz> = D(Pl', Pz')
if PI' and Pz' are respectively the images of P1 and Pz under a projectitviy which maps the interior of Sz onto itself. For this purpose, let F1 and Fz be the points in which the straight line L through P1 and Pz intersects Sz, and assume that P1 separates Fz and P, and also that P1 and Pz are in the interior of Sz.Let F1 and Fz be represented by the vectors 71
Then
+
fi7zl
71
+
fzT?.
is the function we want. It has the additional property that
(1.39)
W P l , PZ) .D(Pz, P3)
=
DV 1 , P3)
i f PSis a third point on the line L through P1,P2. Here we may not assume that B(P2, P3) 2 1, but we will still have D ( P z ,P3) > 0. We can compute D(P1, Pz)explicitly, and by passing from homogeneous coordinates to inhomogeneous ones (that is, to X , Y , 2 ), we obtain (1.35) if we define A by A(P1, Pz)
=
~ l o g D ( P 1Pz). ,
The invariance of A under the projectivities M induced by (1.31a) and (1.31b) implies the invariance of the line element ds given by (1.37) and
16
Elementary Concepts and Formulas
obtained from (1.35) by putting
x,= x, x,= x + d X , 2 1
=
2,
Yz
Y, = Y, 2 2
=
=
Y
+ dY,
z + dZ.
A direct proof of the invariance of ds under the action of the projectivities M would have to be based on the methods of the theory of Lie groups rather than on direct calculations involving the matrices M . Formula (1.39) implies the formula
+
A(P1, PZ) A ( P z , P3)
(1.40)
=
A(Pi, P3),
which shows that A has the additive property required for a distance function in euclidean or noneuclidean geometry. The geometric interpretation of the action of G* on KB which has been outlined in Lemma 1.9 leads to the consideration of various special types of projectivities. Those with a real fixed point-either inside or outside of S2-are of particular importance and have been used by Fricke and Klein in their development of the theory of discontinuous groups of hyperbolic motions. We shall need later only one special case, which may be described in the following
Lemma 1.10 The substitution (1.31a) induces a rotation of Sz if and only if it is unitary, i.e., if 6=E,
y =
-
-&
The proof is based on a simple calculation since, under a rotation, we must have 01’ 04’ = O1 04, and the point O2 = O3 = 0, O4 = O1 must be mapped onto itself. Of course, (1.31b) induces a reflection which maps Sz onto itself. The rotation induced by a unitary substitution with
+
a =
a1
+
+
ia2,
P
=
81
+
0112
$2,
+ + + a22
P12
022
=
1
is given by Euler’s formulas
+ - 2(a1a2 + Y’ 2 ( W Z - PlPZ>X + (a? + 2’ = + azP2)X + 2(a182 - Y +
X’
= (a?
- 0122 - P?
=
P22)X
PlPZ)
a22
2(a1P1
812
azP1)
P22)
+ 2(a2P2 - OLlPdZ, Y+ a,Pz)Z, + a2 - - PZ2)Z, Y
(alp
2(LyzP1
P12
where the axis of rotation (i.e., an eigenvector belonging to the eigenvalue 1) has the components pz, -PI, a2,and the angle w of rotation is given by a, = cos ( w / 2 ) .
I.S Action of G* on Hyperbolic Three-Space
17
Both Lemmas 1.8 and 1.10 are special cases of the general theory of representations of the Lorentz group and of the three-dimensional orthogonal group. For references see Naimark (1964) or Van der Waerden (1932). Lemma 1.10 is also well known to be a consequence of Hamilton's theory of quaternions. We shall use the formulas in Lemma 1.10 for the development of elliptic geometry in Section 1.5. There also exists a representation of G as a subgroup of the symplectic group (over the real numbers). For notations and definitions see Siege1 (1943). We state the result as Lemma 1.11 Let g be defined by (1.3) and let a =
a1
+
iaz,
P
=
P1
+
Y
$2,
= 71
+ iY2,
6
=
81
+ dz.
Then the matrices
a1
PI -02
P1
71 YZ
61
82
a1
[
-012
S(S> =
-Y2
a2
Y1
-82
P2
81,
provide a faithful representation of the multiplicative group of the matrices 9, and StJ*X = J*,
J*=[-i '1.
where the skewsymmetric matrix J* is given by 0 0 1
0 - 1 0 0
0 0 0 There exists a nonlinear representation of the group G* which arises if the interior of Sz (i.e., K 3 ) is mapped in an appropriate manner on the points of an open euclidean halfspace. The action of G* on the halfspace consists then in the conformal selfmappings (including those that change orientation) of the halfspace. We give the details in the following Lemma 1.12 (Action of G* on P 3 ) Let x, y, u be Cartesian coordinates in euclidean three-space and define r 2 0 by
r2 = x2
+ y2 + u2.
18
Elementary Concepts and Formulas
The open halfspace u > 0 becomes a metric space, denoted by Pa,if we define distance by introducing the line element ds as
ds2 =
dx2
+ dy2 + du2 U2
The geodesic lines are the open semicircles orthogonal to the plane u = 0 of euclidean space. K3 (as defined in Lemma 1.8) is mapped isometrically onto PI by the formulas (1.41) z
X 1-2’
y=-
= -
Y 1-2’
u=
(1 - x2 - y2 - Z2)1/2
1-2
Geodesics go into geodesics, the plane disks inside S2 go into hemispheres orthogonal to u = 0 (where a halfplane orthogonal to u = 0 is considered as a hemisphere of infinite radius), and the angles between geodesics in the metric of P3are equal to the euclidean angles. The selfmappings of K ) under the action of G* induce selfmappings of PIwhich, in their totality, are the conformal selfmappings of euclidean halfspace u > 0; the selfmappings that preserve orientation are those induced by the subgroup G of G*. Putting z = z i y , the substitution
+
induces the selfmapping of PI given by
,ej’ =
ym-2
yyr2
+ yBz + SEE + 6s + ySz + S ~ +Z ST, ’
and the substitution z’ (1.42b)
= r’2
2 induces the mapping
= r2,
2’
= 2,
2’ =
2.
In both cases, the boundary u = 0 of P3which is also defined by r2 = zZ is mapped upon itself under the action of G* on the closed complex z plane. Notes on the proof: Using the formulas (1.41) , the expression for the line element in P3 and the transformation formulas (1.42) follow immediately from Lemma 1.8 and from (1.34). In principle, one could
Z.4
Circle Groups as Groups of Motions
19
develop the remaining statements of Lemma 1.12 from here by using the methods of differential geometry and elementary calculations. One would then also arrive at a formula for the distance between two points p and q in P3 which can again be expressed as the logarithm of the crossratio of the complex numbers describing the coordinates of p , q and the points of intersection fi and f2 with the plane u = 0 of the geodesic joining p and q since these four points lie on a semicircle in a plane in which we can introduce complex numbers as coordinates. We shall give these formulas in Section 1.4, where w-e shall discuss the two-dimensional analogues Kz and Pz of the three-dimensional hyperbolic spaces K3 and P3. It should be mentioned that we could replace the euclidean halfspace used for the construction of P, by the interior of any sphere in euclidean three-space since we can map the halfspace conformally into a sphere in such a manner that spheres (including planes as limiting cases) go into spheres again. I n particular, the unit sphere can serve for the construction of both K3 and Pa. For the construction of K3 we could replace the unit sphere by any real surface of second degree with the same signature as the quadratic form defining the unit sphere by using an appropriate projectivity for mapping one onto the other. A proof of the fact that a selfmapping of euclidean three-space is conformal if and only if spheres (including planes as limiting cases) are mapped onto spheres may be found in Guggenheimer (1963). A simple and elegant way of writing (1.42a) and (1.42b) has been communicated to me by W. Abikoff. Using quaternions, with the traditional basis elements 1, i, j , k, and putting w = x i y + j u , we may write (1.42a) as
+
w’=
(aw
+ p) (yzo + 8)-1
(Because of a6 - py
1.4
=
=
(aw
+ p) (ti7 + I)Iyw +
61-2.
1, the coefficient of k in w’vanishes.)
Circle Groups as Groups of Motions of Hyperbolic Two-Space
A subgroup of G* whose elements, acting on the closed complex plane 6, map a given circle and its interior onto itself is called a circle group (a translation of the German word Hauptkreisgruppe). Here a straight line is again considered as a special case of a circle; the “interior” of the circle has to be replaced by one of the halfplanes bounded by the straight line. Siege1 (1971) uses the term “circle group” for discontinuous groups (see Chapter 11) only.
20
Elementary Concepts and Formulas
The subgroups of all elements of G* which, acting on a given circle in 2, map it and its interior onto itself is called a complete circle group. Since any circle in 2 and its interior can be mapped onto any other circle and its interior by an element of G* (and even of G) , all complete circle groups are conjugate in G*. We shall need only two complete circle groups, namely: The group r*,consisting of all elements of G* which map the upper halfplane y > 0 onto itself. It is generated by the group r of all substitutions az cz
(1.43)
+b +d
2’ = -
(ad - bc
=
1, a, b, c, d real)
and the reflection in the y axis given by (1.44)
z’ =
-z.
Of course, r = G n r* consists of the orientation-preserving mappings of y > 0 onto itself. We also need the following group: The group Q* of elements of G* which map the unit disk 22 5 1 onto itself is generated by the group Q of all substitutions (1.45) and the mapping (1.44), where Q = G n Q*. The connection between Q and r is given, according t o (1.13), by the relations
where (1.47a)
w =
(1.47b) a
+(a
+ ib
+ i b - ic + d ) ,
= w
- 4,
+ ( - a - i b - ic
4
=
d
+ ic = 4 + W.
+d),
DeJinition of the spaces P2 and KZ. We shall call the upper halfplane and, more generally, the disk fixed by any complete circle group in G* the space or model Pz of planar hyperbolic geometry. We shall explain the meaning of this definition a little later; the model Pz involves not merely a disk but also the action of the complete circle group (in particular or Q) on the disk. This action will allow us to describe the elements of the circle group as rigid motions of a noneuclidean geomety. It is advantageous to consider, simultaneously with P2, another model K2 of planar noneuclidean geometry which corresponds to the model K S introduced in Section 1.3. For this purpose, we introduce projective two-space as follows: Let Va be
1.4 Circle Groups as Groups of Motions
21
the linear vector space of three dimensions over the real numbers. We denote the components of an arbitrary vector by tl, tz, t 3 , and we define as points of projective two-space the one-dimensional subspaces of V3, with the exclusion of the vector (0, 0, 0). I n other words, a point consists of all vectors (At,, At2, A h ) , X # 0, X real, and tl, t2, tl fixed real numbers not all of which are zero. The two-dimensional subspaces of V 3 [with exclusion of the vector (0, 0, 0)] are called straight lines or simply lines. A line can be defined by two distinct points with coordinate vectors v and w (which, by definition, are linearly independent) ; the points on the line are then given by the onedimensional subspaces of the vector space containing all vectors ( 1.4Sa)
Alternatively, the same line can be defined by a linear form (1.4Sb)
L
= Cltl
+ cztz + c3t3
where the vectors with components tl, tz, t3 belong to the subspace defining the line if and only if L = 0. The ratios of the cr are then uniquely determined by the relations which express the fact that L = 0 for ( t l , tz, t3) = v and (tl, tz, t3) = w. If we use (1.4Sa) as the definition of a line, we call = X a projective coordinate on the line. The values X = 0 and X = 00 correspond, respectively, to the points defined by v and w. A mapping 3
(1.49)
t,’
=
C c,,t,
(V
=
1, 2, 3 ; c,,
real)
p=l
of V3 onto itself [with a nonsingular matrix C = ( G , ) ] is called a projectivity of V3. The matrices C and pC ( p real, p # 0) define the same mapping. We may therefore assume that the determinant det C of C equals +l. We have Lemma 1.13 A projectivity of V3is a one-one mapping of the points and also a one-one mapping of the set of lines of V3 onto itself. If X and p are any two projective coordinates on a line, and if Xi, pi ( i = 1, 2, 3, 4) are respectively the values of these coordinates for four fixed points on the line, then
(1.50) and we can therefore speak of the crossratio of four points o n a line. The value of this crossratio is an invariant under projectivities.
22
Elementay Concepts and Formulas
The proof of Lemma 1.13 is based on formula (1.7) and standard arguments of linear algebra. See the proof of Lemma 1.8. We can embed euclidean two-space Ez into projective two-space in such a manner that euclidean points and lines ( = straight lines) coincide with those in projective two-space. For this purpose, we put
x = tl/t3,
(1.51)
Y
=
t2/t3
and consider X , Y as Cartesian coordinates in Ez. We cannot extend the metric of Ez to a metric of projective two-space. However, we can describe the rigid motions of EZ as those projectivities which map the line t3 = 0 (whose points would have infinite distance from all points of .E2) onto itself and which induce in the quotient space of Va defined by t 3 = 0 a transformation keeping t12 t 2 2 invariant. We shall now consider other subgroups of the group of projectivities of projective two-space which will turn out to be isomorphic to the full circle group r*. For this purpose, we define a real, nondegenerate conic as the set of points in projective two-space whose coordinates t, satisfy an equation
+
3
(1.52)
C
svrtvtr = 0
sV+ real),
( s , ~= s,;
u,p=l
where the determinant of the symmetric matrix
s = s(), is different from zero and where (1.52) has a t least one real solution other than the trivial one tl = t2 = t 3 = 0. [Since for any solution t, = T~ of (1.52) the values t , = AT, also provide a solution, (1.52) defines indeed a set of points in projective two-space.] A conic in the sense defined here coincides, of course, with the finite points (i.e., those with ta # 0) of a conic in euclidean two-space if we use the Cartesian coordinates (1.51). Let A,, A2, A3 be three homogeneous linear forms in tl, &, t3 (not identically zero) such that the lines A1 = 0 and A3 = 0 define tangents of the conic (1.52) and A2 = 0 defines the line joining the points of contact of these tangents with the conic. Equation (1.52) then assumes the form (possibly after multiplication of Al b y a nonvanishing constant) (1.53)
A1A3
-
112’ =
0.
We ask now for all projectivities which carry (1.53) into itself. Since the forms A, must be linearly independent (defining three lines not concurring in one points), we can use the A, as new coordinates (arising from the t, by a projectivity) . After a moderately involved and not very transparent
I.4 Circle Groups as Groups of Motions
23
calculation, we find that all projectivities mapping (1.53) into itself can be defined by the substitutions
+ 2abAz + b2A3, = acAl + (ad + bc) Az + bdA3 + 2cdAz + d2&,
A,' =
( 1 .54)
A,'
dA1
(a, b, c, d
real),
A3' = C2Al
which has the determinant (ad - bc)2. Wc may assume (ad - bC)2 = 1.
(1.55)
The conic (1.53) has an interior which is defined as the set of points from which no (real) tangent to the conic exists. The interior is given by A,& - Az2 > 0
and it is mapped onto itself by all substitutions (1.54). Now we are able to introduce the Definition of the hyperbolic space Kz. It consists of the interior of the conic (1.53) together with the group of selfmappings defined by (1.54) and (1.55). The selfmappings with ad - bc = +1 are called proper. They form a subgroup of index 2 with the particular substitution (1.56)
hi' = Al,
A,'
=
-A,,
h3' =
A3
as a coset representative. Now we can prove Lemma 1.14 The equations
(1.57)
x
=
Az/Aa,
X '
4-y2 =
A,/&
+
define a mapping of the upper half of the complex z = x i y plane onto the interior of the conic A1A3 - 82' = 0, which is one-one and has the following properties. (i) The semicircles orthogonal to the x axis are mapped onto the segments of straight lines within the conic. (ii) The selfmappings (1.43) and (1.44) of the upper halfplane induce, respectively, the projectivities (1.54) and (1.56) which map the interior of the conic (1.53) onto itself. (iii) The mapping (1.54) induced by the mapping (1.43) has exactly one real fixed point, which is located respectively inside the conic
on the conic
outside the conic
24
Elementary Concepts and Formulas
if and only if the inducing mapping (1.43) is elliptic
parabolic
hyperbolic
(iv) Let Fl, Fz be points on the real axis and let P1, P2 be points on the semicircle with F I F z as diameter. Assume that P1 is between Fz and Pz and that Pz is between Fl and P l . Let F1’,F2’, P1’,P2‘ be the corresponding points in projective two-space; they will lie on a straight line. Then
{ C R ( P i ,Fi, Pz, Fz) }’
=
CR(Pi’, FitjPz’, Fz’).
Note: This relation need not hold for four arbitrary points on a semicircle and their maps in project.ive two-space. Proof: The equation of a semicircle orthogonal to the 5 axis is of the form
+ y’) + Bs + C
A(?
(1.58)
=
0,
where A , B are not both zero. From (1.57), it follows that the points of the semicircle are mapped onto points of the straight line (1.59)
AA1
+ BAg + CA,
=
0.
This line will intersect the conic in two distinct points if and only if B2 > 4AC, which is also the condition for the circle (1.58) to have more than one real point. The statement (ii) of Lemma 1.14 is proved most easily by writing (1.43) in the homogeneous form (1.60)
21’
+ bzz,
= U Z ~
22’
+ dz2
= CZ~
(Z = z~/zZ)
and by putting (1.61)
A1
=
~121,
A2
+
= ~ ( Z ~ Z Z Z~ZZ),
A3
= 2222,
and computing the effect of the substitution (1.60) on the Av in (1.Gl). Since a, b, c, and d are real, one obtains (1.54). Statement (iii) can be derived from the remark that the points of projective two-space are mapped onto pairs of points of the complex z plane [according to (1.57)] in such a manner that points inside the conic are mapped onto conjugate complex points z and 2 of the z plane, whereas points outside or on the conic are mapped respectively onto two distinct or coinciding points on the real axis. This follows if (1.57) is written in the form (1.62)
x f i y =
Az f (Azz A3
A1A3)1‘2
I.4
Circle Groups as Groups of Motions
25
Since the fixed points of a mapping (1.43) of the z plane onto itself must be mapped onto the fixed points of the induced selfmapping of projective two-space, statement (iii) has been proved. Finally, we prove statement (iv) by observing that we can map any semicircle of the upper halfplane on the positive imaginary axis, without changing the crossratio of any four points on the semicircle, by applying a mapping of I’. The induced mapping of projective two-space also leaves crossratios unchanged. For the imaginary axis, we have respectively the z coordinates i y l , 0, iyz, co for Pl, F,, Pz, Fz, and the values of the projective coordinate AI/& of PI‘, F l f , Pz‘,Fz’ are respectively y?, 0, yz2, CO. The rest is a trivial calculation. For the investigation of the subgroups of the complete circle groups, it is very advantageous to use the language of noneuclidean, specifically hyperbolic, geometry. This is the geometry which arises from euclidean geometry if we replace the parallel postulate of euclidean geometry by the following postulate. Given a point P outside a line 1, there exist at least two lines through P which do not intersect 1. There is another alternative to the parallel postulate which says that any line through P intersects 1. This also leads to a noneuclidean, in this case elliptic, geometry which will be discussed briefly in the next section. I n his “Foundations of Geometry,” Hilbert ( 1930) introduces the concepts of “point,” “line,” “incident,” “in-between,” and “congruent” a s undefined terms, formulates a set of axioms in which these terms appear, and then proves the equivalence of this system of concepts and postulates with the body of knowledge contained in the algebraic version of euclidean geometry which is known under the name of analytic geometry. There would be no point in copying here Hilbert’s work for hyperbolic geometry. All of the mathematical objects introduced so far in this chapter are well defined in standard algebraic terms. What we need are two things: First, a language which appeals to our geometric intuition and utilizes our (euclidean) geometric experience. (A related example of such a language is provided by the use of geometric terms in spaces with more than three dimensions, including Hilbert space.) Second, we need a certain body of theorems. We shall be able to derive most of these rapidly; some of them could also serve as axioms of hyperbolic geometry but they are now provable assertions since the basic concepts listed above will appear as welldefined terms of ordinary analytic geometry. [In a sense, we are reverting here from Hilbert to Euclid, who attempted, not very successfully, to “define” the terms “point” (“that which has no parts”) and “line.”] We shall list now the basic concepts which we shall use, following Hilbert’s arrangement (see the appendix). Simple and nearly obvious results are
26
E ~ e ~ n Concepts t a ~ and Formulas
stated immediately after the definitions. More complex results will be stated later as lemmas. (i) Plane hyperbolic (noneuclidean) geometry. Definition of “point” and “line.” A hyperbolic point is a point in the upper halfplane of the complex x plane. A hyperbolic straight line (briefly denoted by “line”) is a n open semicircle in the upper halfplane. The endpoints of the semicircle (which are on the real axis) are called the ends of the line. (They are not points of the hyperbolic plane.) Recall that “semicircle” includes euclidean rays orthogonal to the real axis. (ii) Transfer principle. Every concept defined in the upper halfplane has its counterpart both in the unit disk of the complex plane [which is the map of the upper halfplane under the mapping (1.13) ] and in the interior of the conic (1.53), which is the map of the upper halfplane under the mapping (1.57). We shall not always prove that properties of the mathematical concepts defined in the upper halfplane are shared by their counterparts in the unit disk or inside the conic, because this is guaranteed by the very simple nature of the mappings used (they are one-one and continuous). According to the transfer principle, a hyperbolic line may also be viewed as the open arc of a circle inside the unit disk and orthogonal to it, or as an open segment of a euclidean straight line with endpoints on the conic, which we may choose to be an ellipse or even a circle. (iii) Incidence. A point is incident, or lies on a hyperbolic line if and only if this is true in the ordinary sense in which incidence is defined for points and semicircles in the upper halfplane. We have : Given two points Pl and Pz, there exists exactly one line 1 o n which both P1 and Pz lie. (Proof: Use the transfer principle. I n the conic, the hyperbolic line is part of a straight line in the euclidean sense, and the ellipse is a convex curve.) As a consequence we have: T w o distinct lines do not have more than one point in common. We also have: There exist three points which do not lie on a line. (iv) Order. Let A , 3,and C be three (distinct) points on a hyperbolic line 1. We shall say that C is between A and B if the same statement is true in the ordinary sense for the points A , B, and C on the euclidean semicircle representing E. The following postulates and theorems of Hilbert are obviously satisfied: If C is between A and B, then C is also between B and A . Given A and B, there exists a point D on 1 such that B is between A and D . Of three points on I , exactly one is between the two others. Given
1.4 Circle Groups as Groups of Motions
27
n points on a line 1, it is always possible to label them P, ( V = 1, 2,. . ., n) in such a manner that P,+1 is between P, and Pv+2for v = 1, 2 , . . ., n - 2. I n addition, Pasch’s axiom is satisfied. It states: Let A , B , and C be three points not on a line. Let a, b, and c respectively be the lines on which the pairs B , C and A , C and A , B lie and let 1 be a line which intersects a in a point between B and C and does not go through (is not incident on) any one of the points A , B , C. Then 1 intersects either b in a point between A and C or it intersects c in a point between A and B. The proof of the validity of Pasch’s axiom is reduced to the euclidean case if we use the interpretation of hyperbolic points and lines as points -and segments of euclidean straight lines within a n ellipse. (v) Definition of ray, side of a line, angle, interval, interior of a triangle. Let P be a point on a line 1. Two points Q and R on 1 which are distinct from P are said to lie on the same side of P if P is not between Q and R ; otherwise, they are on different sides of P . The points on the same side of P are called a ray with P as its initial point. P divides 1 into exactly two rays. Let 1 be a line and let P and Q be points not on 1. Then P and Q are said to be on the same side of 1 if the line I* incident on both P and Q has no point in common with 1, which is between P and Q (on I*). Otherwise, P and Q are said to lie on different sides of 1. A line has exactly two sides. A side of a ray is defined as a side of the line which contains the points of the ray. Let P be a point and let r and r’ be rays with P as initial point. Then all points of r‘ lie on the same side of r and vice versa. We call the sides thus defined, respectively, the positive sides of r and r’. The set of all points which are on the positive sides of both r and r‘ is called the proper angle of (or between) r and r’. ( I n Hilbert’s notation, it is simply called the angle. Hilbert does not admit angles greater than a straight angle.) The set of all points between two points P and Q on the line is called the interval with endpoints P and Q and is denoted by Let A , B, C be three points not on a line, and define the lines a, b, c as in the formulation of Pasch’s axiom. Define the positive sides of a, b, c, respectively, as those sides of these lines on which A , B, C lie. Then the points which lie on the positive sides of all three lines are called the interior of the triangle ABC. The angle of the rays with initial point A containing, respectively, B and C is denoted by QBAC. The interior of a triangle is convex, i.e., every interval whose endpoints are in the interior or on the sides of the triangle is completely in the interior of the triangle. All of these statements can, of course, be proved by using the euclidean meaning of point and line in the upper halfplane. However, they also follow easily from the previously stated properties of incidence and order. The concepts of polygon, closed polygon, and simple (i.e., non-self-
m.
28
Elementary Concepts and Forrnulas
intersecting) polygon are defined in the same manner as in euclidean geometry. We have
Lemma 1.15 Every closed simple polygon divides the plane (Le., the totality of all points) into two disjoint classes, called the interior and the exterior of the polygon, in such a manner that any two points can be connected by a simple polygon not intersecting the given one if and only if they belong t o the same class. The interior is defined as the set of points P such that every line 1 through P intersects the polygon. (Here one has to show that there exist lines which do not intersect the polygon.) We can, of course, prove Lemma 1.15 by applying the Jordan curve theorem t o any one of our three models of hyperbolic two-space. However, it is a remarkable fact that Lemma 1.15 can be proved by merely using Hilbert’s axiom of incidence and order or, alternatively, the facts collected under (iii) and (iv). I n the case of a triangle, this is an easy task; it is not even difficult to prove that there are lines which do not intersect the triangle. However, Dehn (1899) , in an unpublished manuscript, proved not only Lemma 1.15 in this elementary manner, but even the following result, which is not a consequence of the Jordan curve theorem.
Lemma 1.16 (Dehn) Every closed simply polygon has a t least three vertices P with the following property: If Q and R are the vertices adjacent to P, then the interval &R is either in the interior of the polygon or coincides with one of its sides (in which case the polygon is a triangle). Lemma 1.16 shows that a closed simple polygon with n vertices can be decomposed into the union of n - 2 triangles.
(vi) Congruence. Two hyperbolic configurations C1and Cz (for example, rays, intervals, angles, polygons) are said to be congruent if there exists an element of the group r* which maps the euclidean configuration in the upper halfplane corresponding to Cz onto the one corresponding to C1. (Of course, we have equivalent conditions for the unit disk and the interior of a conic.) It follows immediately that being congruent i s a rejlexive, symmetric, and transitive relation. Properties of incidence and order can easily be shown to remain unchanged under the mappings of I-*. This fact can be used to simplify some proofs since we may assume that a given ray and given side of it are in a prescribed position. We shall call r* (and, in the corresponding case of the unit disk, Q*) the group of (planar) hyperbolic (or, alternatively, noneuclidean) motions.
1.4 Circle Groups as Groups of Motions
29
This terminology is justified because of the strict analogy between I'* and the group of planar euclidean motions which will emerge from the following
Lemma 1.17 There exists exactly one element in r* which maps a ray r onto a given ray r' so that a given sidr of r is mapped onto a given side of r'. If r coincides with r', then the element,of r*which exchanges the sides of r leaves fixed all points of r and of the line 1 of which it is a part. We call this element the rejlectim in 1. An element 7 of the subgroup r of r* is uniquely determined by a fixed ray and its image under the action of 7. r is called the group of proper (noneuclidean or hyperbolic) motions. Lemma 1.17 can be derived from the remark that every element of I?* which maps a line 1 onto a line 1' must map the ends of 1 onto the ends of 'I and from the invariance of the crossratio [formula (1.6)]. We also see that all reflections are conjugate in r*. Lemma 1.17 establishes the validity of Hilbert's congruence axioms 111.1, 111.2, and 111.4. To establish the validity of 111.3 and 111.5, we need the following concepts [which are discussed in greater detail in the books of Caratheodory (1954a) and Siege1 (1971)l. (vii) Measures for intervals and angles. T o every interval and to every angle we can assign a positive real number, called the length of the interval and the size of the angle, such that two intervals are congruent if and only if they have the same length and two angles are congruent if and only if they have the same size. (Consequently, we shall frequently replace the word congruent by the word equal when talking about intervals and angles.) The size of the angle is defined as the size of the euclidean angle between the corresponding arcs of euclidean circles (including, possibly, a euclidean ray). The distance A(P1, P2) of two points P, with coordinates z, in the complex halfplane is defined in terms of the crossratio defined in Lemma 1.14, statement (iv) , by (1.63)
A(Pi,
P2)
=
log C R ( P 1 ,
F1, P2, F2).
+
If P, = x, iy, and x1 > xz,then respectively the real coordinates f1, of F 1 , Fz are given by (f2
+ fl)
- 22)
=
I 2 1 12 - I 2 2 12
(f2
- f l ) ( a - XZ)
=
{ (I 2 1
(21
(1.64) 12
-
I 22 12)Z
+ 4(sl - a)
(21 1x2
l2
- 22
121
I? )112
f2
30
Elementary Concepts and Formulas
and we find
since (2,
- f l ) ( 2 , - f2)
=
iYY( fl - f 2 ) .
The equation of the euclidean circle orthogonal to the real axis on which z1 and z2 lie is given by
(x - fl) (x - f2> If
x1 = 2 2
and y1
> y2, we have f1
(1.66)
=
+ Y2
x1 and f 2
CR(P1, Fi, P 2 , F2)
=
=
0.
= a,and
y1/~2.
The euclidean semicircle orthogonal to the real axis on which P1 and Pz lie is in this case the ray x = xl,y > 0. The fact that the congruence of intervals and angles is equivalent, respectively, to the equality of the lengths of the intervals and the sizes of the angles follows from the statements a t the beginning of Section 1.2. The following results imply the validity of Hilbert's axioms 111.3 and 111.5. Lemma 1.18a Let PI, P2, PI be three points on a noneuclidean line and assume that P3 is between P1and P,. Then ( 1.67a)
A(P1,
P2)
=
A(Pi,
P3)
4- A ( p 3 , P2).
Lemma 1.18b Let A, B, C and A', B', C' be respectively the vertices of two triangles T and TI. Assume that A(A, B)
=
A(A', B'),
A(Al C)
=
A(A', C')
and that the angles QBAC and QB'A'C' have the same size (are congruent). Then we also have (1.67b)
QABC
=
QA'B'C',
and also that A ( B l C) gruent.
=
A(B', C'), i.e., the intervals BC and B'C' are con-
QACB
=
Proof: We have to prove (1.67a) only if Pl, P2,P3are on the y axis since we can map every noneuclidean line onto the y axis without changing distances. If y1 > y3 > yz and 2, = iy, ( Y = 1, 2, 3), we find A(P1,
P2)
=
lOg(Yi/Y,), A(P3,
P2)
= lOg('?43/y2),
A ( P i , P3) = log(Yi/Yo),
1.4 Cireb Groups as Groups of Motions
31
which proves (1.67a). However, we can also use (1.65). I n this case, the proof of (1.67a) reduces to the remark that in a right triangle, the product of the legs equals the product of the altitude and the base, both products being equal to the area of the rectangle whose sides are the legs. To prove (1.67b), we use a transformation g of r*which maps A onto A’, the ray A B onto the ray A’B’, and which maps the side of the ray A B onto which (I lies onto that side of the ray A’B‘ on which C‘ lies. Then g is uniquely determined (Lemma 1.17), and, since g preserves distances and angles, B is mapped onto B’ and C is mapped onto C’. Therefore, corresponding sides and angles in the triangles ABC and A‘B‘C‘ are equal in pairs. (viii) Parallel postulate. We discussed this before. Obviously, there exist infinitely many noneuclidean lines through a point P which do not intersect a given line 1. (ix) Archimedean axiom and axiom of completeness. The archimedean axiom states: Let r be a ray with initial point Po,and let P, ( v = 1, 2 , . . .) be points on r such that P , is between P,-l and P,+l and all intervals P., PPcl( V = 0, 1,. . .) are congruent. Let I be any interval on r with Poas a n endpoint. Then there exists a natural integer n such that the other endpoint of I is between Po and P,. Since.Lemma 1.18 [specifically (1.67a)l shows that the length of an interval is greater than the length of any of its proper parts, and since the lengths of intervals are positive real numbers in our model of hyperbolic geometry, the archimedean axiom reduces to the corresponding property of real numbers. The completeness axiom has been used very little by Hilbert; its most important consequence is that every real positive number is the length of an interval. This implies, for instance, that every interval has a midpoint, a fact which could also be proved by using considerably weaker postulates. (x) Area and line element. The development of a theory of polygonal area without the use of the archimedean axiom or, in a more familiar terminology, without the use of the concepts of continuity and limit, is perhaps the most admirable achievement of Euclid’s “Elements.” (It,too, has been perfected by Hilbert.) In hyperbolic geometry, it is still an elementary proposition to show that the sum of the angles of a triangle is less than T . If we map in our model (the upper halfplane) one side of the triangle onto the y axis, this result is equivalent to an elementary statement about circles. Let a,0,y be the angles of a hyperbolic triangle. We call a-a-p--y
32
Elementary Concepts and Formulas
the deficiency of the triangle, and we see immediately: If a triangle i s decomposed into the union of smaller triangles, the sum of the deficimcies of the parts i s the deficiency of the whole triangle. Therefore, the deficiency of a triangle has one of the basic properties which we must postulate for the concept of area. But we cannot copy Euclid's method of developing it, and the most efficient way to show that the area of a triangle can be expressed in terms of its deficiency is provided by the methods of differential geometry: Pz and K 2 are metric spaces with a riemannian line element, and area is expressed in terms of a double integral. We have [see Siege1 (1971) for a proof] : Lemma 1.19 The distance formula (1.63) implies that the line element ds is given in the upper halfplane by
ds2 =
( 1.68a)
dx2
+ dY2 Y2
and in the unit disk by ds2 = 4
(1.68b)
+
dx2 dy2 (1 - 2 2 - y2)2'
Accordingly, the area of a polygon is given in the upper halfplane and in the unit disk respectively by the double integrals
/JY,
d x dy 4 1 1 (1 - 2 2 - y 2 ) 2 .
(1.69)
The calculations to prove Lemma 1.19 are straightforward, with the exception of those proving (1.68a). By taking the logarithm on both sides in (1.65) and putting 21 = z dz, 22 = Z , we find
+
ds
=
log (1
+ ")- - log (1 + z fi
(2
( fl - f 2 ) dz -fl) ( z - f2)
-f2
+ higher terms in dz.
The right-hand side must be real (observe that fl, f2 depend on d z and on z ) , and by multiplying it by its conjugate complex value, we find ds2 =
dz dz ($1 - fi)'
I
- fi
12
I z - l2 . f2
I.4
Circle Groups as Groups of Motions
33
But the factor of dz dZ on the right-hand side is y-2 since the crossratio in (1.65) equals 1 for z1 = zz = z. To obtain (1.68b), we use, of course, the mapping ( 1.13) . For the distance of a point x from the origin, we find from (1.68b) (1.70) Let zl,z2 be any two points in the unit disk. We can use a n element of Q to map z1 onto 0 and z2 onto a point on the real axis with coordinate p (0 < p < 1).If this mapping is given by [see (1.45)]
we find
and therefore 1 + P
(1.71b)
log 1 - P
the hyperbolic distance of the points zl,z2. The proof that the area of a triangle with angles a, p, y equals its deficiency T - a - p - y is now a consequence of the Gauss-Bonnet formula. It can also be proved directly in a n explicit manner. Siege1 (1971) provides such a proof together with a proof of the following results. 21s
I n hyperbolic geometry, the triangle inequality for distance i s true. T h e locus 1 of all points f o r which the distances f r o m two given points P1,Pz are equal i s a hyperbolic line through the midpoint of the interval PIPz and orthogonal to it. T h e diference of the distances A(Q, P,) - A(Q, P z ) of the point Q from the points P1,Pzi s positive o n that side of 1 o n which Pz lies and negative o n the other. For the proof of a few results in hyperbolic geometry, it is advantageous to use a special case of the model Kz. Using the notations and definitions following equation (1.52) , the information needed may be stated as follows. Lemma 1.20 Let C be the conic defined by t32
-
t12 - t22
=
0
34
Elementary Concepts and Formulas
where
x
h/& are the euclidean coordinates corresponding to the homogeneous projective coordinates t,. The interior of C is then the interior of the euclidean unit circle. If we put A, =
t3
=
- ti,
Y
tl/t31
A3
=
t3
=
+ tl,
A2
=
t2,
a mapping of the upper halfplane onto the interior of C is, according t o (1.57), given by Y l - x ( 1 -x2- y2)1 / 2 x=x2+y2=1+Xr Y = l + X X =
1 - 2 2 - y2 1 x 2 + y2l
+
Y =
1
+
22 x2+
y2'
It maps the point z = i onto the center of C and the y axis onto the interval - 1 < X < 1 of the X axis. For the line element we find
which agrees with (1.37) for 2 = 0 and K = 4. The (noneuclidean) distance between two points is again expressible in terms of a crossratio of the values of the projective coordinates of four points on a straight line, according to (iv) of Lemma 1.14. The result is a special case ( 2 = 0, K = +) of (1.35). The proof is implicit in the proof of Lemma 1.8. (xi) Some theorems in hyperbolic geometry. We need, in addition to the results already proved in this section, a certain body of theorems valid in hyperbolic geometry. For instance, we have not yet proved that a triangle y < T, and that all with given angles a!, p, y actually exists if a! p such triangles are congruent. This will be done later, in the section on triangle groups. Here, we mention only an extension of the concept of a triangle: We shall agree to consider noncompact triangles which have one or more angles which are equal to zero. Some of the sides of such a triangle will be rays with a common end. I n the upper halfplane, these rays will be either rays on lines parallel to the y axis or arcs of semicircles which have a common tangent at their point of intersection with the real axis. Figure 36 provides an abundance of examples for such triangles. We have:
+ +
Lemma 1.21 All triangles in which the three angles are equal to zero are congruent and their area equals T ,
1.4 Circle Groups as Groups of Motions
35
Proof: The triangles in question have vertices on the real axis (including the point a t z = co ). Any triplet of such vertices can be mapped into any other triplet by an element of I?. [See (v) of Section 1.2.1 The area of the particular triangle bounded by I z I = 1 and the rays x = -1, y > 0 and x = +l, y > 0 can be computed directly from (1.69). A hyperbolic (noneuclidean) circle is defined as the locus of all points which have a given hyperbolic distance p from a fixed point Zo,the center of the circle. We have
Lemma 1.22 Noneuclidean circles are the orthogonal trajectories of the noneuclidean rays through the center 2 0 . They are also euclidean circles. However, the euclidean and noneuclidean centers may be different points. Conversely, every euclidean circle within the upper halfplane (or, if we use the unit disk as a model, every euclidean circle within the unit disk) is also a noneuclidean circle. Proof: Map Zoonto the center of the unit disk and let Pobe a point on the positive real axis with the prescribed noneuclidean distance p from 2,. The rays through Zoare the euclidean radii of the unit disk. The hyperbolic motions in s2 which map one radius onto another and keep Zofixed are the euclidean rotations with Zoas center. Therefore, the noneuclidean circle is just the orbit of Po for the elliptic substitutions that keep Zofixed and is a euclidean circle with the origin as its center. It is orthogonal to all radii, and this property remains unchanged under homographic substitutions, in particular under the mapping which carries the unit disk onto the upper halfplane and under the transformations of both s2 and r. We also see that the noneuclidean circle is completely determined by one point Po and by two of the rays with the same initial point on which it is orthogonal. Consider now an arbitrary euclidean circle Co entirely in the upper halfplane. Draw a euclidean line I parallel to the y axis through its center. This is also a noneuclidean line orthogonal to Co. From its intersection xo with the real axis, draw the tangents to Co. (By assumption, 2 0 is outside Co.) Construct the euclidean circle C1, with xo as a center and the euclidean length toof the tangent as the radius. C1is also orthogonal to Co.It intersects 1in a point 20. Then Co is the orthogonal trajectory of all noneuclidean rays through 20, which is its noneuclidean center. We note here that the rays through Zo are, in turn, all of the euclidean circles orthogonal both to Coand the real axis. They are precisely the circles intersecting in Zoand in 2 0 , the point conjugate complex to 2 0 (which, of
36
Elementary Concepts and Formulas
course, lies in the lower halfplane). Since we can map any arbitrary circle onto the real axis by using an element of G, we have
Lemma 1.23 Let C1 and Cz be two nonintersecting euclidean circles. The circles orthogonal to both C1and Cz all intersect in the same two points P and P . Except for P and P , exactly one of these circles goes through any given point in the complex plane. Let C1‘ and C; be two euclidean circles which intersect in two points. By mapping these onto z = 0 and z = m (using a homographic substitution), C1’ and Cz’ become straight lines intersecting a t z = 0. Through any point P other t’hanz = 0 and z = 00, there goes exactly one circle orthogonal both to C1’ a,nd Cz‘. Assuming now that C1’ and Cz‘ are orthogonal to each other, and mapping C1’onto the real axis and Cz’ onto a prescribed noneuclidean line, we find Lemma 1.24 Given a hyperbolic line 1 and a point P outside of 1, there exists exactly one noneuclidean line k through P and orthogonal to 1. Lemma 1.23 allows us to prove another result in noneuclidean geometry. In contradistinction to Lemma 1.24, it has no euclidean counterpart. We have
Lemma 1.25 Let 1, and 1, be two noneuclidean lines which have no finite point and no endpoint in common. There exists exactly one noneuclidean line I* orthogonal to both 11 and 1,. Proof: I n the upper halfplane, 1, and 1, are parts of euclidean circles orthogonal to the real axis and without common point. The euclidean circles orthogonal to both 1, and l2 are exactly the circles which go through two fixed points of the real axis, according to Lemma 1.23. One of these is also orthogonal to the real axis.
Next, we shall prove a result which has no noneuclidean significance but which we shall need later and which is related to the previous lemmas.
Lemma 1.26 Let C1,Cz, C1 be three (euclidean) circles which either are pairwise disjoint or which are such that any two of them have a t most one point (and then, of course, a tangent in this point) in common. We include the case where three circles have one point (and one tangent in this point) in common. Then there exists a t least one circle C* orthogonal to C1, CZ,and Co.
1.5 Notes on Elliptic and Spherical Geometry
37
Proof: If two of the circles touch, we can map them onto parallel lines, and the lemma becomes obvious. Assume then that the three circles are disjoint in pairs. We map Ca onto the real axis. The circles orthogonal to the maps C1' and Cz' of C1 and CZall go through two common points PI and P z . The centers of these orthogonal circles lie therefore on the (euclidean) line 1 orthogonal to the line PlPz and bisecting the interval PIP2. Let Q be the point of intersection of 1 with the real axis. The (euclidean) circle with center in Q and passing through P I and Pz is orthogonal to the real axis and to C1' and Cz'. ~
Finally, we shall make use of the model Kz in its special form described in Lemma 1.20. We claim: Lemma 1.27 A closed, simple noneuclidean polygon is convex if and only if none of its interior angles exceeds r .
This result is easy to prove for simple, closed euclidean polygons. Suppose now we have such a noneuclidean polygon, represented in the upper halfplane. We map it onto its counterpart in the model Kz. The sides of the noneuclidean polygon are now represented by the sides of a euclidean polygon, but the angles become distorted. However, we know that a straight angle goes into a straight angle and that the angles are distorted in a continuous manner. Therefore, an angle not exceeding ?r appears as an angle between euclidean lines which is also less than T . Therefore, the map of the noneuclidean polygon is a convex euclidean polygon, and convexity of a polygon must be preserved under the mapping since the definition of convexity involves only properties of incidence and order. Note: The mapping of the upper halfplane onto the interior of a conic becomes discontinuous for angles a t points of the boundary since, in the upper halfplane, two noneuclidean lines may meet at a zero angle, whereas their maps in the conic may meet under any (euclidean) angle less than T . However, since angles in the conic with vertices on the boundary cannot exceed r , Lemma 1.27 remains valid even if some of the vertices of the polygon are improper, i.e., a t infinite distance.
1.5 Notes on Elliptic and Spherical Geometry The parallel postulate, in Hilbert's version, has two alternatives. One of them leads to hyperbolic geometry which has been developed in the previous section. The other alternative states that every line 'I through a point P
38
Elementary Concepts and Formulas
outside of a line 1 intersects 1 in a (exactly one) point. I n other words, any two lines in the plane intersect. Although Euclid did not formulate any axioms of order, the proof of Propositions 16 and 27 of the first book of the “Elements” shows clearly that the axioms of order are incompatible with the assumption that any two lines meet. Specifically, it will then be impossible to define two sides of a line or to say that of three points of a line, exactly one is between the other two. I n fact, the straight line must now have a finite total length. To see that, we construct on a line 1 a t two points P , and P2the lines 1, and 1, orthogonal to 1. Then, if the lines 11 and 12 meet in a point Q, the triangle PlP2Qand its image reflected in the line 1 are congruent and the total length of a line is twice the distance pl&, where Q can be reached from P1 by going on either one of the intervals PI& starting on different sides of 1 at P I . I n spite of the difficulties arising from the necessity to reformulate and to weaken the axioms of order, it is rather easy to construct a geometry th a t is very close to euclidean geometry and in which any two lines intersect. This geometry, which we shall develop now, is called elliptic geometry. We have seen that we can complete the euclidean plane t o a space which we called projective two-space by introducing homogeneous coordinates. They were denoted by tl, t z , ts on p. 21, and the quotients t l / t 3 and t 2 / t 3 were the Cartesian coordinates of a euclidean point whenever t a # 0. Lines (meaning: straight lines) were defined by a (homogeneous) linear relation between the t,, and the fact that any two lines intersect becomes a n elementary result of linear algebra. In fact, we have the same algebraic machinery available as in euclidean geometry except that now we are unable to define congruence and distance. On the other hand, spherical geometry provides a well-studied and simple model of a geometry where we can define lines (the great circles), where we can define a group of rigid motions (the rotations of the sphere together with the reflections in the planes of great circles) which allows us to introduce the concept of congruence, and where we have a natural definition of length together with the fact th a t the length of a line is finite. The only difficulty here is that two great circles (representing now the straight lines) intersect in two points instead of one. These points are endpoints of a diameter of the sphere. Therefore, all we have t o do to establish elliptic geometry is to map the pairs of diametrically opposite points of the sphere onto the points of projective two-space in such a manner that the great circles on the sphere are mapped onto the straight lines of projective two-space. This will then provide us with formulas for distance and for rigid motions (i.e., congruence-defining selfmappings) in projective two-space. We have developed already the formulas needed; we present the result as
1.6 Notes on Elliptic and Spherical Geometry 39
Lemma 1.28
Let X , Y , Z be a triplet of real numbers such that
xz+
(1.72)
Y2+22= 1.
Then each triplet represents a point on the unit sphere. At the same time, the triplets
X , Y ,Z
(1.73)
- X , - Y , -2
and
represent one point in projective two-space [since any set of triplets Atl, At2, At3, X # 0, represents a point, the condition (1.72) leaves exactly two possibilities if X = i t l , Y = Ah, 2 = i t s ] . A homogeneous linear equation
AX
+ BY + CZ = 0
+ BZ+ C2# 0 )
(A2
defines both a great circle on the sphere and a line in projective two-space. The linear substitutions defined in Lemma 1.10, together with
(1.74)
X’
=
x,
Y’
=
-Y,
2’
=
2,
generate a group which is both the group of proper and improper (orientation-reversing) rigid motions of the unit sphere and a group of projectivities.
If we consider projective two-space as a completion of the euclidean plane, by defining (1.75) u = x/z, v = Y/Z as planar Cartesian coordinates, the line element ds on the sphere, defined by
(1.76) ds2 = dX2
+ dY2 + dZ2
=
dX2
+ dY2 - ( Y dX - x dY)2 1 - x2 - Y2
7
expresses itself in terms of u and v as
(1.77)
ds2 =
dU2
+ dv2 + dv - v (1 + + (U
u 2
dU)’ 1
v2)2
defining an elliptic metric in the euclidean plane. The assertions of Lemma 1.28 are proved by straightforward calculations. To derive (1.76) , one needs, of course, that
+
+
X dX Y dY Z dZ = 0. The derivation of (1.77) is then tedious but not difficult. The total length of the u axis (v = 0), and, therefore, of every straight line, is given by
r+-
du
40
Elementary Concepts and Formulas
A formula for the distance of two points PI,P2,analogous to (1.35) can be derived from spherical geometry. It involves now a n arccosine function instead of a logarithm, and the distance can also be expressed in terms of the crossratio of the values of the projective coordinates of the points P1,P2 and the coordinates of the points of intersection of the line joining P1,PZ with the conic x 2
+ Y2 + 22
=
0
(which has no real points). However, we shall not need these results; they are stated explicitly by Fricke and Klein (1897 pp. 41-44). Summarizing, we have Lemma 1.29 Let O* be the extended orthogonal group, i.e., the group generated by the proper rotations 0 defined in Lemma 1.10 and by the reflection (1.74). Then O* has two representations as a group of selfmappings of a plane. The first representation is given by the group of elliptic motions, that is, by the projectivities induced by
(1.78) in the projective plane with coordinates X , Y , Z according to the formulas of Lemma 1.10. Since the projectivity (1.79)
X’
=
-x,
Y’
=
-Y,
Z ’ = -2
defines the identical selfmapping of the elliptic (projective) plane, this group is isomorphic with 0. The mapping (1.79) is an improper (orientation-reversing) selfmapping of the sphere. Under the stereographic projection of the sphere onto the complex plane, (1.79) defines the mapping of
e onto itself.
2’
= -l/Z
e,
The other representation of O* is the group of selfmappings (1.78) of extended by the orientation-reversing mapping z‘ = 2. This representation is faithful.
Lemma 1.29 implies that there exists a motion of the elliptic plane other than the identity which leaves all points of a given line fixed, which is an involution (i.e., of order 2) and which is derivable from the identical mapping by a continuous change of the parameters a,@ which describe the elliptic motions. For later purposes (Section 11.3) , we need another implication of Lemma 1.29. The stereographic projection of the sphere onto the closed complex
1.6 Illustrations. References and Historical Remarks
41
e
plane provides us with a planar model of spherical geometry. The great circles (which are the intersections of the sphere with a plane through its center) represent the analogue of straight lines on the sphere. Because of the formulas z =z
(1.80)
+iy
=
x + iY ~
(X2
1-2
+ Y2 +
2 2
=
l),
which describe stereographic projection, and their inversion (1.81) X
=
r2+
r2 - 1 z=r2 + 1
y = - 2Y
22 ~
1’
r2+
1’
(r2 = x2
+ y2 = z Z ) ,
we find that the great circles on the sphere are mapped on the circles
2Az
(1.82)
+ 2By + C ( x 2+ y2 - 1) = 0 ( A , B, C real; A 2
e. The line element on the sphere is given by
in
ds2 = d X 2
+ d Y 2 + dZ2,
XdX
+ B2 + C2> 0)
+ Y d Y + Z d Z = 0.
We find from (1.81)
r dr
=
x dx
+ y dy,
and therefore obtain
ds2 = 4
+
dx2 dy2 (x2 Y 2
+ +
as the expression for the spherical line element in
1.6
e.
Illustrations. References and Historical Remarks
Illustrations
At the end of Section 1.2, we commented on Figures 8-11, which show orbits of substitutions of various types in the complex plane, together with their orthogonal trajectories, which are again orbits of substitutions with the same fixed points.
42
Elementary Concepts and Formdas
I n circle groups, loxodromic substitutions cannot occur since all circle groups are conjugate with subgroups of r*in which the substitutions have real entries and therefore real traces. Similarly, the unitary substitutions in Lemma 1.29 that represent motions of the elliptic plane or rotations of the sphere are all elliptic since a a is real and 1 a I 5 1. Figure 12 is a sketch of the mapping (1.57) of the upper halfplane onto the interior of a conic, which is here drawn as a n ellipse. The grating of parallel lines in the halfplane represents the orbits of the parabolic substitub. The horizontal lines belong to the case of a real b ; for the tions x' = x vertical lines, b would have to be purely imaginary; in this case, the substitution is still parabolic but does not belong to I'*. Only the vertical lines of the grating are noneulcidean lines; they are mapped onto the straight lines concurring in a point of the ellipse. The horizontal lines of the grating are mapped onto ellipses which are tangential to the conic which is the image of the real axis, i.e., of the boundary of the halfplane. I n Figure 14, the right-hand part of Figure 12 has been completed. Every point of the upper halfplane corresponds to a point inside the strongly drawn ellipse, which represents the conic whose interior is the model K2 of the hyperbolic plane. The exterior of the ellipse does not belong to the model K2. However, the projectivities that keep the ellipse fixed act on the exterior also, and the orbits of the various points under a parabolic substitution are also drawn; they are the ellipses outside the conic (i.e., the strongly drawn ellipse). Figure 13a and 13b show, respectively, the orbits of a hyperbolic and of an elliptic substitution presented as projectivities mapping a conic (the strongly drawn ellipse) and its interior onto itself. The straight lines (or, rather, their segments inside the heavily drawn ellipse) represent noneuclidean straight lines; one may consider each one as the map of the previous one under a fixed noneuclidean motion. They all meet in one point, but only in the case of an elliptic substitution (Fig. 13b), which represents a noneuclidean rotation is this point in the interior of the heavily drawn ellipse. The heavily drawn straight line is called the axis of the substitution, and the point of concurrence of all the weakly drawn straight lines is called its pole. Note that the axis is mapped onto itself by the substitution; in the case of a hyperbolic substitution, part of it represents a noneuclidean straight line, which shows that a hyperbolic substitution behaves along one straight line like a euclidean translation.
+
+
References and Remarks
The uniform language which can be applied to the discussion of noneuclidean motions once they are viewed as projectivities provides a certain
I.7
Appendix: Hilbert’s Axioms of Geometry
43
advantage of the model K z over the model Pz. In the publications of Felix Klein and his school, this language is used widely, although by no means exclusively. The advantages of the model Pz are twofold: Angles, which play a dominant role in noneuclidean geometry, are recognizable in the model, and, above all, the applications of noneuclidean geometry to the theory of R i e m a p surfaces make the presentation of noneuclidean motions as mappings of C onto itself the only feasible one. The interaction of linear algebra (including the theory of quadratic forms) and geometry had been developed by many authors before Felix Klein. His most important contribution to the synthesis of geometry and algebra is the introduction of group theoretical concepts and ideas. The older history of the subject, with proper credit and references to Cayley, Beltrami, Klein, Poincar6, and others is nicely sketched in the “Introduction” of the work by Fricke and Klein (1897, pp. 1-59). For later developments, the standard text is the book on “Geometric Algebra” by Artin (1957). It also gives a few references to other sources and to comprehensive bibliographies. A special chapter, important because of its applications and its interrelations with other fields, is the theory of the real symplectic groups of degree n (which, for n = 1, is the group r of Section 1.4) and its action on an analytic space of n ( n 1)/ 2 complex dimensions. It is fully represented in a paper by Siege1 (1943) with the title “Symplectic Geometry.” A modern presentation of projective geometry and projective metrics is the book by Busemann and Kelley (1953). Of texts on noneuclidean geometry, we mention Coxeter (1957) and, on a more elementary level, Meschkowski (1964).
+
1.7
Appendix: Hilbert’s Axioms of Geometry
Preamble
We start with three distinct classes of abstract objects. The objects of the first class we shall call points and we shall denote them by A , B, C , . . . ; the objects of the second class we shall call “lines” or “straight lines” and we shall denote them by a, b, c, . . . ; the objects of the third class we shall call planes and we shall denote them by a, p, y,. ... The points are also called elements of linear geometry; points and lines are called elements of planar geometry; and points, lines, and planes are called elements of solid geometry or elements of space.
44
Elementay Concepts and Formulas
Between these objects we introduce relations which will be denoted by terms like “lying on or being incident with,” “in between,” “congruent,” ‘lparallel,ll “continuous.” The exact and mathematically complete description of these relations is given by the axioms of geometry. We shall subdivide the axioms into five classes. The axioms of each class express certain interconnected facts appearing in our geometric intuition. We shall denote these five classes of axioms as follows:
I. 1-8. Axioms of Incidence 11. 1-4. Axioms of Order 111. 1-5. Axioms of Congruence IV. Axiom of Parallels V. 1-2. Axioms of Continuity
Comments
Although both our basic objects and the relations between them arise from our geometric intuition, we consider both these objects and these relations as abstract entities which are defined exclusively by the fact that they satisfy the axioms. It can be shown that highly ungeometric objects (namely sets of real numbers or sets of pairs or triplets of real numbers) can be substituted for the objects and set theoretic or algebraic relations can be substituted for the intuitively geometric relations in such a manner that the axioms are satisfied. Other substitutions, in many cases of a very different nature, can be carried out if only a part of the set of axioms is assumed t o be valid.
1. Axioms of Incidence
1.1 Given two (distinct) points A , B, there exists a line a incident with both of these points.
1.2 Given two points A , B, there does not exist more than one line a incident on both A and B. Instead of “incident” we shall also say: Passing through. We shall also say that A , B lie on a or that a j o i n s A and B.
1.3 On every line there lie a t least three points. There exist at least three (distinct) points which do not lie on one line.
f.7 Appendix: €filbert’s Axioms of Geomlry
45
1.4 Given any three (distinct) points A , B , C which are not incident on (do not lie on) one and the same line, there exists a plane a that is incident on each one of the three points A , B, C. Given any plane a,there always exists (at least) one point A incident on a. We shall also say: A is a point of a, A lies on a, etc.
1.5 Let A , B, C be three (distinct) points which do not lie on one and the same line. Then there does not exist more than one plane a incident on A , B, C .
1.6 Let a be a line on which there lie two distinct points A , B . Let a be a plane incident on A and B. Then all points on a lie on a. I n this case we shall also say: a lies in a, or a is incident on a, etc. 1.7 If two distinct planes CY, /3 are both incident on (have in common) a point A , then they have a t least one more point B in common. 1.8 There exist a t least four distinct points which do not lie in the same
plane.
II. Axioms of Order The axioms of order define the concept “in between” and thereby enable us to introduce a n ordering of the points of a line and concepts like “side of a line in a plane” or “side of a plane,” etc. The term “in between,” which refers to a relation between the points on a line, is defined by the following axioms. 11.1 If a point B lies between a point A and a point C , then A , B, C are three distinct points on a line a, and B also lies between C and A . 11.2 Given two (distinct) points A and 6, there exists at least one point B on the line AC (i.e., on the line determined uniquely by A and C according t o 1.1 and 1.2) such that C is between A and B.
11.3 Given any three distinct points on a line a, at most one of them will lie between the two others. 11.4 (Pasch’s axiom) Let A , B, C be three distinct points which do not lie on one line. Let a be a line in the plane ABC (uniquely determined by 1.4 and 1.5) which is not incident on any one of the points A , B, C . Assume
46
Elementary Concepts and Formulas
that a is incident on a point X between A and B (and, therefore, on the line A B ) , then a is also incident either with a point Y between C and B or with a point Z between A and C. Intuitively, this means: If a line enters the interior (not yet defined but definable) of a triangle, then it has to leave the triangle. It can be proved that the line cannot have both a point X between B and C and a point Z between A and C. The axioms of order allow us to define a ray on a line and the sides of a line in the plane. Let A be a point on the line a. We shall say that two points B, C ( A ,B, C are distinct) lie on the same ray or on the same side of A if A is not between B and C. We can prove that then the points on a distinct from A are divided into two disjoint classes which are called the sides of A on a or the rays through A . Similarly, we shall say that two distinct points B, C lie on the same side of the line a if they are not incident with a and if the interval BC does not contain a point of a. Here the interval BC is defined as the set of all points X on the (uniquely determined) line b which goes through B and C such that X is between B and C . Pasch’s axiom then allows us to show that a line has exactly two sides. This means that the points outside the line a can be divided into two disjoint classes by our definition of the side of a line.
ZZZ.
Axioms of Congruence
The intervals (defined by pairs of points, see the preceding comments) enter into certain relations which are characterized by the terms ‘(congruent” or “equal.” These relations will allow us to define the concept of motion. 111.1 Let A , B be two (distinct) points on a line a and let A’ be a point on a line a’, which may or may not be identical with a. Then it is possible to find on a‘, on a given side of A’, a point B’ such that the interval A‘B’ is congruent (or equal) with the interval AB. We shall express this by writing AB = A’B’. This axiom postulates that it is possible to lay of a n interval. That it is possible in exactly one manner will be proved later. Since an interval A B is defined by the unordered pair of its endpoints A , B, the formulas
A B = A’B’,
A3
3
B’A’,
are equivalent with each other.
BA = A’B’,
BA = B’A’
1.7
Appendix: Hilbert’s Axioms of Geometry
41
111.2 If an interval A’B’ and an interval A”B” are both congruent with the interval AB, then A’B’ and A“B“ are also congruent with each other. We can show that A B = A B since we can find an interval A’B’ not identical but congruent with AB. Then 111.2 applied to A B = A’B’ and A B = A‘B‘ implies A B 3 AB. Similarly we can show that A B = A’B’ implies A‘B’ = AB. We anticipated this fact already in the formulation of 111.2. We could have written merely that A’B’ is congruent with A””’. 111.3 Let A B and BC be two intervals without common points (the endpoints do not belong to the interval) on a line a. Let A’B‘ and B‘C‘ be two intervals without common points on a line a‘. Then the congruences
A B = A’B’,
BC = B’C’
imply AC = A’C’. This axiom expresses the additive property of the “laying off” process of intervals. The next axiom (111.4) involves the concept of an angle. To define it, let a be any plane, let 0 be a point in a , and let h, k be two rays starting at 0 and belonging to distinct lines through 0. (Recall that a ray starting at 0 is defined as one side of a line through 0.) We call the unordered pair of rays h, k an angle and denote it by Q ( h , k) or Q (k,h) . We define the interior of the angle Q ( h, k) as follows: Let H be a point on h and let K be a point on k. (Then 0, H , K are three distinct points.) Denote as the positive side of the line of which k is a part that side on which H lies, and similarly denote as the positive side of the line on which h lies that side on which K lies. Then the interior of Q (h, k) consists of those points which lie on the positive side of both h and k. It can be shown that an interval A B is entirely in the interior of Q (h, k) if A and B are in the interior. Note that our definition excludes angles which (in the usual language) are straight or larger than a straight angle. We call 0 the vertex of the angle 0:( h , k) and we shall call h and k the legs of the angle. Suppose we have an angle with vertex B on whose legs there lie respectively the points A and C. Then we denote this angle also b y the symbol QABC or, briefly, by QB. Sometimes, we shall also denote angles by lowercase Greek letters. Angles enter into certain relations which again will be described by using the terms ‘(congruent” or “equal.”
111.4 Let the angle Q (h, lc) be in a plane a and let a‘ be a line in plane a’. Assume that a definite side of a’ in a‘ has been given. Let h’ be a ray on a‘ which starts a t the point 0‘ of a‘, Then there exists in a’ one and only one
48
Elementary Concepts and Formulas
ray k’ such that 0:(hJk) is congruent or equal to the angle Q (h‘, k’) where all points of the interior of Q (h’, k‘) lie on the given side of a’. We shall express this by writing Q (h, k)
= Q (h’, k’).
Every angle is congruent to itself, that is, we have always Q ( h , k) = Q (h, k). We express the contents of 111.4 briefly by saying that every angle can be laid off in any given plane on a given side of a ray in exactly one manner. We did not consider oriented intervals and we shall not consider oriented angles either. This means that the formulas
4 ( h , k, 0: k’) 0:(k,h ) = Q (h’, k’), (h’7
J
Q (h,
= 0:(k’, 4 (k,h ) = 0:(k’,h’) ”17
all express the same fact. 111.5 Let ABC and A’B’C’ be two triangles. Then, if
A B = A‘B’,
AC = A’C’,
QBAC = QB’A’C‘,
we also have the congruence
QABC = QA‘B‘C‘. Note: By using the term “triangle,” we imply that A , B , C are distinct points which do not lie on one line. According to 1.4 and 1.5, they determine uniquely a plane. The corresponding statements hold for A‘, B’, C’. IV. Axiom
of
Parallels
IV (Euclid’s axiom) Let a be any line and let A be a point which does not lie on a. Let a be the plane in which a and A lie (and which is uniquely determined by these data). Then there exists in cx a t most one line a‘ that goes through A and which does not intersect a. From the previous axioms, we can derive the fact that at least one line a’ exists. Axiom IV postulates that a’ is uniquely determined by a and A . We shall call a‘ the parallel of a through A .
V . Axioms of Continuity V.l (Axiom of measuring or Archimedean axiom) Let A B and CD be any two intervals. Then it is possible to find on the line A B a certain number of points A1, A 2 , .. ., A , such that the intervals AA1, A1A2, A z A 3 , .. .,
Exercises
49
A M A , are congruent with the interval CD and such that B lies between A and A,. V.2 (Axiom of linear completeness) The points on a line form a system which cannot be extended any more if we maintain the validity of the axioms of linear order, the first congruence axiom, and the Archimedean axiom (that is, the axioms 1.1, 1.2, 11, 111.1, and V . l ) . This means that it shall be impossible to adjoin to this sytem of points on a line a further points such that in the system arising from this adjunction the axioms mentioned are still valid.
Comment. Hilbert points out that Axiom V.2 does not make sense without Axiom V . l . The meaning of V.2 is the following: Take all previous axioms together, it is possible to show that the geometry obtained is isomorphic with the coordinate geometry arising from linear algebra over an ordered (commutative) field. Without the Archimedean axiom, we can always embed such a field in a bigger one, obtaining thereby more possibilities for the coordinates of our points. However, the Archimedean axiom prohibits such an embedding once we have used the field of real numbers for the coordinates of our points. For a more detailed analysis, see Hilbert (1930).
Exercises
Show that the crossratio CR of zl,zz,z3, 24 [formula (l.S)] does not change if a , . .., 24 are replaced by their images under a Mobius transformation, by proving the following proposition : Any Mobius transformation (with complex coefficients) can be obtained by combining the Mobius transformation z’ = - l / x finitely many times with Mobius transformations of the type z‘ = z c, where the complex number c may have different values if this type is applied several times. (In other words: z‘ = - l/z’ and the continuously many LLtransvections” z’ = z c generate G.)
+
+
In the note after Lemma 1.1, it is mentioned that the crossratio CR of zl,z2, 23, 24 assumes only six different values under the action of the 24 permutations of zl,.. ., 24. Show that there also exists a homogeneous polynomial of second degree in zI,. . ., 24 with the same property. Given two disjoint circles C1and Cz, show that there always exists a hyperbolic Mobius transformation mapping C1onto Cz [Hint: Show this in the case where C1is the real axis, using the formulas (1.12).]
50
Elementary Concepts and Formulas
4. Use the solution of Exercise 1 to show that Mobius transformations map circles onto circles.
5. Use the solution of Exercise 1 to show that Mobius transformations are conformal mappings. 6. Show: Two disjoint circles can always be mapped, by a Mobius transformation, onto two concentric circles.
7. Consider the set of all circles which have two distinct points PI and Pz in common. Show that their common orthogonal trajectories are circles. 8. Assume that a Mobius transformation T which maps the upper halfplane and, therefore, the real axis onto itself has the property that it maps three points P1,Pz, PI with rational coordinates on the real axis onto points PI’, Pz’,P3’ which again have rational coordinates. Show: T will map any point with a rational coordinate onto a point with a rational coordinate, but the coefficients of the matrix for T need not be rational (if, as usual, we postulate that the determinant is +1). Note: to is considered to be a rational number. 9. Show that the matrices S(g) in Lemma 1.11 have determinant +1
and that there are symplectic matrices other than those of the form of S k ) . 10. Let CI, Cz, CB, Cq be four circles. Assume that the center of any one of them is outside of the disks bounded by the three others and that the pairs of circles (cl, c2)
1
(c21 c3)1
(c31
c4>
1
(c4,
cl> 1
(czl
c4)
have exactly one point in common and that there do not exist any other points which are on two distinct circles. Show: There cannot exist a circle which is orthogonal to all four of the circles C1, Cz, C3, C4. [Hint: Map Cz and C4 onto parallel lines.] 11. Formulas (1.34a) describe the action of a Mobius transformation on the space K3. Show: A Mobius transformation which is not the identical mapping has a fixed point in Ks if and only if it is elliptic. I n this case, the fixed points fill a straight line. (Note: I n the parabolic case, there exists a fixed point, but it lies on the euclidean sphere bounding K 3 and, therefore, is not a point of K 3 . ) [Hint: Show first that we can replace the transformation by any conjugate one to prove the proposition.]
12. Formulas (1.42a), after Lemma 1.12, describe the action of a Mobius transformation on the space Pa. Show : Only elliptic transformations
Exercises
51
can have fixed points, and these then fill a geodesic line. [Hint: Same as in Exercise 11.] 13. Let L1 and Lz be two nonintersecting straight lines in the noneuclidean hyperbolic) plane. Assume that L1 and Lz do not even have a point on the euclidean boundary of the noneuclidean plane in common. Let P1 and PZbe, respectively, points on Ll and LZand let D be their (noneuclidean) distance. Show: D assumes a minimum for a particular position of both PI and 2‘2, and the line joining PI and Pz in this particular position is orthogonal to both L1 and Lz. [Hint: Show first that we may assume that L1 and L2 are represented by concentric semicircles orthogonal to the real axis and then use formula (1.68a) of Lemma 1.19 (Section I.4).] 14. After the proof of Lemma 1.14, there starts a discussion of plane hyperbolic geometry on the basis of Hilbert’s axioms (Section 1.7). Using these, prove the statement in (v) (immediately preceding Lemma 1.15) that a line has exactly two sides. 15
Using the statements in (iv) and (v) preceding Lemma 1.15, Section 1.4, prove with the help of Hilbert’s axioms: Given a triangle and the three lines L1,LZ, LI on which the sides of the triangle lie, there exists in the plane no point that is on the negative side of all three of these lines.
Chapter
I1 Discontinuous Groups and Triangle Tesselations
11.1 Introductory Remarks
We define a tesselation of the euclidean plane as a covering of the plane, without gaps and overlappings, by polygons all of which have the same size and shape or, in other words, are congruent to each other. Each individual polygon will be called a tile; the simplest, and probably the earliest, type of a tesselation is the one which uses a square as a tile and in which the squares are arranged in the same manner as on a chess board. The concept of congruent tiles implies the existence of a group of selfmappings of the euclidean plane (namely the group of rigid motions), which allows us to define polygons of the same shape and size in different parts of the plane. Obviously, the concept of a tesselation can be generalized to any space for which a group of selfmappings with the properties of rigid motions can be defined, as, for instance, n-dimensional euclidean space or the hyperbolic and elliptic spaces of dimensions three and two introduced in Chapter I. In fact, weaker requirements than those characterizing a group of motions suffice to define tesselations, as is shown by the example of the geometry associated with the symplectic group (Siegel, 1943). In the case of the chessboard tesselation of the euclidean plane with squares, the tesselation is closely associated with a subgroup T of the group
II.1 Introductory Remarks
53
of euclidean motions in such a manner that each square represents exactly one element of the subgroup. T consists of the translations t,,, defined by (2.1)
t,,,,,:
z’=z+n+mi
( n , m = O , f l , f 2 ,...)
and each square is the map of the square so,odefined by 0 5 x 5 1, 0 5 y 5 1 under the action of t,,,. The group T is uniquely determined by the chessboard tesselation if we admit only translations for the mapping of one tile onto another. However, the tile (namely the square) is not determined a t all by the group T . An example is given in Figure la. The big squares form a chessboard pattern. However, the union of a pair of two of the smaller squares also forms a tile which, together with its replicas under the action of T , fills the plane without gaps and overlappings. It is intuitively clear that the two tiles (big square and union of two smaller squares) must have the same area. Indeed, this statement is simply the Pythagorean theorem, and it has been conjectured that it was discovered by a contemplation of tiling designs. Figure l b indicates the proof of the Pythagorean theorem resulting from our two tesselations. From an axiomatic point of view, this proof is a particularly economical one since it shows that the two smaller squares can be dissected into a bounded number of pieces which then can be put together in such a manner that they just fill the big square. From this, it follows that the Pythagorean theorem can be proved without using the archimedean axiom. Euclid’s proof does not establish this fact so clearly because Euclid derives the Pythagorean theorem by showing that the square over a leg of a right triangle has the same area as the rectangle whose sides are the hypotenuse and one of its segments produced by the altitude of the triangle. Since the larger side of this rectangle may be greater than any given (fixed) multiple of the smaller side, it is not possible to dissect it into a bounded number of triangles which then can be used to fill the square over the leg. We remark in passing that Euclid’s proof of the Pythagorean theorem is also independent of the archimedean axiom. But this is not so obvious. Given a tesselation, then certainly a t least one element of the group of rigid motions belongs to every tile, namely a motion that maps a fixed tile onto the given one. (If the tile itself has symmetries, i.e., if there exist rigid motions mapping it onto itself, there exist, of course, several motions that can be assigned to a given one. I n the case of the chessboard tesselation, the number of symmetries of the tile is eight.) It may be impossible to select the rigid motions assigned to the tiles of a tesselation in such a manner that they form a subgroup of the group of rigid motions. An example can be constructed as follows.
54
Discontinuous Groups and Triangle Tesselations
We divide the plane into parallel strips of width 1. Each strip we divide into squares of side length 1, but we stagger our strips so that the left lower corners of our tiles have the coordinates z , , , ~defined by (2.2)
z , , , ~ = pm
+ n + mi
where po = 0 and otherwise the substitutions (2.3)
(n,m = 0, f l , f 2 , . . .), pm are
2' =
z
arbitrary fixed real numbers. The
+ zn,m
will map the square so,o with its left lower vertex at z = 0 onto the square s,,,,,,with its left lower vertex a t z , , , ~ .The translations (2.3) will not, howpm - PZ+%is ever, form a group unless for all l, m = 0, f l , f2,..., pE a n integer. It is not very difficult to show that even if we use not only translations but a combination of translations and rigid selfmappings of so.o for the purpose of mapping so,oonto s , , , ~ , the pn can be selected in such a manner that the mappings carrying so,ointo s , * ~will not form a group. Another example of a tesselation in which the motions carrying a fixed tile into the other tiles do not form a group can be obtained from the chessboard tesselation as follows: We draw in each one of the squares one of its two diagonals. We thus obtain a tesselation with an isosceles right triangle as a tile. By choosing either one of the two diagonals of the squares in a random manner, we can obtain the desired effect. The example of a tesselation with staggered squares arising from a given one under the action of the translations (2.3) is connected with a conjecture formulated by Minkowski which was proved half a century later by HajBs (1941). Although the translations (2.3) do not form a group, they nevertheless contain a subgroup generated by a translation in the direction of one of the coordinate axes. I n other words: The tesselation contains strips of squares in which each square has a neighbor with which it has a full side in common. Minkowski conjectured and Haj6s (1941) proved (using a new theorem on finite abelian groups) the following result. Let C be the hypercube defined by 0 5 xy 5 1 ( Y = 1 , 2 , . ., n) in euclidean n-space, let x be the vector with components xy,and let vy ( V = 1 , . . .)n ) be a n y $xed set of n linearly independent vectors. Assume that the action of the translations
+
.
x +x
+ v,
n
v
=
C myvy
( m y = 0, f l , f 2 , .
. .)
V-I
applied to C produces a tesselation of the space. Then at least one of the vectors v has the property that all but one of its components are zero, the remaining component having the value f1. I n geometric language : A tesselation of euclidean
11.1 Introductory Remarks
55
n-space by n-dimensional cubes in which the centers of the cubes form a lattice necessarily contains a column of cubes. In euclidean two-space, it is easy to see that this theorem is true for any tesselation with squares, and the corresponding theorem has been proved by Keller (1937) if the number of dimensions does not exceed six. So far, our remarks have merely shown the great complexity of the relationship between an arbitrary tesselation and the group or groups of rigid motions which may be associated with it. We shall not continue to operate on such a broad basis, but we shall impose strong restrictions on the tesselations which we wish to consider both in this and in the next chapter as well as in Chapter V. We state:
I.
Restriction on Tesselated Spaces
We shall (in this chapter and in Chapters I11 and V) consider only the euclidean and the elliptic plane, the sphere, the hyperbolic plane, and the hyperbolic three-space as objects of tesselations. The groups of rigid motions which define congruence of the tiles will respectively be the standard group of planar euclidean motions, the groups mentioned in Lemma 1.29, the groups I'* and a* defined in Section 1.4 (acting, respectively, on the upper halfplane and the unit disk), and the group G* acting on Paaccording to Lemma 1.12 or acting on K3 according to Lemma 1.9.
2.
Restriction of Tesselations t o the Canonical Type
A tesselation will be called canonical if it satisfies the following conditions: (i) The tiles are congruent polygons (in two-dimensional spaces) or polytopes (in three-dimensional space). (ii) Two tiles which are not identical have in common either no point, or exactly one point, which then is a vertex of both tiles, or a line segment, which then is a side (in two dimensions) or an edge (in three dimensions) of both tiles, or, finally, in the three-dimensional case, a polygonal part of a plane which is a face of both tiles.
(iii) Let S be any side (in the two-dimensional case) or face (in the threedimensional case) of a particular fixed tile To. Then there exists exactly one side (or face) S' of Totogether with a rigid motion M (S) such that S
56
Discontinuous Groups and Triangle Tesselations
and S’ are congruent, and M ( S ) maps S onto S‘ and T oonto a tile T o ( S ) of the tesselation which has exactly the points of S‘ in common with To. Also, the inverse M-l( S ) of M (Sj maps S’ onto S and the tile Toonto a tile To(S‘) which has exactly the points of S in common with To.We admit the possibility that S coincides with S‘. I n this case A l ( S ) is the reflection of T o in the line on which S lies. (iv) Let S , ( p = 1 , . . ., r j be the sides (or faces) of the tile To.Then the motions M (8,)generate a group D such that every tile of the tesselation is the image of To under the action of exactly one element of D. D is called the group of the canonical tesselation. Of the tesselations of the euclidean plane with square tiles which have been discussed so far, only the chessboard tesselation can be considered as a canonical one; we can makc it canonical by defining its group as the group of translations (2.1). However, we could achieve the same result by defining its group as the one generated by the reflections in the four sides of one of the squares. The definition of a canonical tesselation includes the d e j l n i t i m of its group. Our definition of “canonical” is more specialized than thc onc given by Fricke and Klein (1897, pp. 182-190), but agrees with the usage by Siege1 (1971). The usual definition of a canonical tesselation starts with tho definition of a discontinuous group D for which Tois a canonical fundamental region. We shall define these concepts in the next section.
11.2 Discontinuous Groups and Fundamental Regions I n this section, we shall define first the terms “discontinuous group” and “fundamental region” in a rather general manner which is applicable in many cases. After that, we shall impose additional restrictions on these concepts which are relevant only for thc special cases studied in this monograph. Let S be a topological space and let H be a group of one-one topological mappings of S onto itself. Let P be a point in S, and let H ( P ) be the orbit of P under H , i.e., the set of all images of P under the action of H . A point Q will be called a limit point of P if in every open set containing Q there are infinitely many points belonging to H ( P ). Here we count the images of a point P under the action of the group elements hl # h2 of H as diffcrent points of H ( P ) . This implies that any fixed point P of any mapping h H is automatically a limit point of P if h is of infinite order.
II .2 Discontinuous Groups and Fundamental Regions
57
We shall say that the group H is discontinuous in a part S* of S if S* is an open set which is mapped upon itself by all elements of H and if, for any point P C S*, no limit point of P is in S*. Two points P and P' in S are said to be equivalent under H if there exists an h H such that P' is the image of P under the action of h. We shall P' or simply P P' if H is selfunderstood. write P
-
-
II
Assume that H is discontinuous in S*. We shall call a domain D ( H ) a fundamental region of H if it has the following properties: D ( H ) is the union of an open set D' in S* and those of its boundary points which are not limit points of H . Every point in S* is equivalent to a point in D ( H ), and no point in the open set D' is equivalent to a different point in D' or to a point on the boundary of D'. This definition implies that the fixed points of the elements h # 1 of H are not in D'. Therefore, the existence of a fundamental region implies that even the fixed points of the elements h # 1 of finite order are not dense in S*. All groups H that Mte consider have elements defined by 2 X 2 matrices of the type (l.l),or a t least H contains a subgroup of index two whose elements are defined by such matrices; the latter case arises if H contains elements belonging to G* but not to G [as defined by (1.2)]. The matrices M ( h ) corresponding to these elements of H themselves form a group d l ( H ) under matrix multiplication. We shall say that such a matrix group contains infinitesimal substitutions if there exists a sequence of distinct group elements hn ( n = 1, 2 , . . .) with corresponding matrices
such that (2.5)
lim an = lim Pn n-m
n-m
=
0,
lim an = lim 6, n-m
=
fl.
n-m
(This term has nothing to do with the terminology used formerly in the theory of Lie groups.) We shall say that H is discrete if M ( H ) contains no infinitesimal substitutions. We shall prove: Lemma 2.1 A subgroup H of G = PSL(2, C) which acts as a discontinuous group on a part S* of 6 or on hyperbolic three-space has countably many elements. Proof: There exists an open set u containing a n arbitrarily selected point P S* such that only finitely many images of P are in u ;otherwise, P
58
Discontinuous Groups and Triangle Tesselations
would be a limit point of H and could not be in S*.Similarly, to each image h ( P ) under the action of h E H there exists a n open set U h containing h ( P ) and only finitely many other images of P. We note that it does not matter whether we use euclidean or noneuclidean metric to define the topology in S* since a e!clidean open set in S* is also a noneuclidean one and vice versa. Both in C and in hyperbolic three-space, we can introduce euclidean coordinates, and we can define “rational points” as points with rational euclidean coordinates. Choosing a rational point in each Uh and assigning to it the finitely many images of P in Uh, the countability of these images follows from the countability of rational points. It is rather obvious that H cannot be discontinuous either in a part of or in hyperbolic three-space if the associated matrix group M ( H ) is not discrete.
Theorem 2.1 T h e subgroup of H of PSL(2, C ) = G acts as a discontinuous group o n hyperbolic three-space P3 (Section 1.3) if and only i f M ( H ) i s discrete. In this case, there exist onlyfinitely m a n y elements h of H which leave a given point Q E P3fixed, and there exists a n open set containing Q in which there lie n o $xed points of a n y elements of H except those which leave Q $xed. It should be noted that h H acting on P3 has either no fixed points in P3 or else keeps a (noneuclidean) line pointwise fixed. Therefore, the fixed points of h cannot be isolated in three-space. After the remarks preceding Theorem 2.1, we can prove the theorem by showing that the existence of a limit point of H in P3 would imply the existence of a n infinitesimal substitution in M ( H ), and that the same would be true if infinitely many elements of H would have fixed points in an open set containing Q E Pa. To prove that a discrete group H is discontinuous, we use the formulas (1.42a). We define u and u‘ respectively by ~2
=
r2
- 22,
~ ‘ = 2 r‘2
- 2’2’
(u > 0,
u’> O ) ,
and we find
as the result of the action of z’ prove :
= (az
+ @)/(-yz + 6)
on P3. We shall
Lemma 2.2 Let h, ( n = 1, 2, 3,. . .) be an infinite sequence of distinct elements of H . Let u,r, z be coordinates of a point p , in P3 and let u,, r, be
I I . 2 Discontinuous Groups and Fundamental Regions
59
respectively the u- and r-coordinates of the images of p o under the action of k,. Then, if u* = lirn u,, r* = lirn r, n-m
n-m
exist and u* f 0 and r* # 0, the group H is not discrete. (Note that the definition of a limit point given a t the beginning of this section implies the existence of the limits u* and r*.) Proof of Lemma 2.2: According to (1.42a) and (2.6),
+
+
P ) / (rnz 6 , ) . Since u > 0, if h, is given by the substitution z, = (a,z the existence of u* # 0 implies that the 7 , I are bounded and, in turn, the boundedness of the I Y, I implies that the 1 6, I arc also bounded. Once this has been established, it follows from the boundedness of the values rn2 that the I an 1 and I 0, I are also bounded. Now we can find a subsequence n k ( k = 1, 2, 3,. , .) of the sequence of integers n such that simultaneously the limits
I
lirn anr,
lim fink,
lim
k - ca
k+ m
k-m
ynx,
lim 6,, k-m
exist. We may assume nk = k since we may omit unnecessary h, from our original sequence. Then we can show by an elementary calculation that the entries of the matrices
converge toward the unit matrix. The matrices Mk are those defining the elements hk-'hk+l of H . By assumption, none of them is the unit matrix. Therefore, H contains an infinitesimal substitution and is not discrete. We have proved now that, if p o is not a fixed point of any element # 1 of a discrete group H , then the maps of p , under the action of H do not enter a certain neighborhood of p,. We have also shown that po can be the fixed point of only finitely many elements of H . We still have to prove that a sufficiently small neighborhood of a point p* in Pa can contain fixed points of only finitely many elements of a discrete group H ; here p* may or may not be a fixed point of an element of H (if it is a fixed point, we know already that it belongs to a finite number of elements of H only). To complete the proof of Theorem 2.1, let us assume that H contains a sequence of distinct elements h, such that h, has a fixed point with coordi-
60
Discontinuous
Groups and
Triangle Tesselations
nates un,r,, zn in P3 and such that the limits lim un = u*
(2.8)
> 0,
lim rn = r*
n+m
> 0,
lim zn
n+m
= z*
n+m
exist. We find from (2.7) (2.9)
YnTnUn'
+I
YnZn
+
6n
1'
=
1,
rn'
= anGUn'
+I
anzn
+
Pn
1.'
Since (2.8) imposes upper and lower positive bounds on u, and r, and an upper bound on I zn 1, it follows again that the sequences I an 1, 1 On I, I y n I, and 16, 1 are bounded. As before, this implies that H cannot be discrete. We note here that a subgroup H of G may be discrete and, therefore, discontinuous in Pa without being discontinuous in any part of the complex plane. The simplest example for such an occurrence is provided by the group PSL( 2, Z (i)) of fractional linear substitutions (2.10)
z'=
az + O -
YZ
+6
[a6 - Py = 1 ;
a,P, Y,6
E Z(i)l.
+
Here Z ( i ) denotes the ring of gaussian integers a bi, where a and b are ordinary integers. This group was investigated first by Picard (1884) and has been named Picard's group by Fricke and Klein (1897) (who actually studied a finite extension of this group). We shall report on Picard's group in Chapter V. Here we shall merely prove:
T h e group PSL ( 2 , Z (i)) i s discrete and discontinuous in P3, but in the complex plane C,the point z = 0 can be mapped onto every point z = rl rzi, where rl, r2 are arbitrary rational numbers. Theorem 2.2
+
That P S L ( 2 , Z ( i )) is discrete follows immediately from the definition. Therefore, the group is discontinuous in Pa, according to Theorem 2.1. On the other hand, the last statement of Theorem 2.2 shows that the group cannot be discontinuous in any part of the complex plane since every open set contains a point whose images are everywhere dense in the plane. T o prove the statement, we write
r1
+ r2i = P / 6
where p, 6 are in Z ( i )and are coprime. This is possible because in Z (i) we can define a division process with the properties of Euclid's algorithm. Therefore, we have unique factorization into prime numbers (rather than prime ideals), up to factors which are units and therefore powers of i, and we also have the result that for any pair of coprime numbers P, 6, there exist two other elements a,y of Z (i) such that a6 - By = 1. The a,P, y, 6 thus constructed give us the mapping (2.10) which maps 0 onto P/S.
II.2 Discontinuous Groups and Fundamental Regions
61
It is a difficult problem to decide whether a given discrete subgroup of G is discontinuous in any part of the complex plane C.We shall encounter a few examples of groups which are discontinuous in a part of C with a rather complicated boundary (Chapter IV) , but most of the groups studied will be groups of motions in euclidean, spherical, elliptic, or hyperbolic two-space, with a few examples of discontinuous groups of motions acting on hyperbolic three-space (Chapter V) . For two-dimensional groups of motions, we have the analogue of Theorem 2.1, which may be stated as follows. Theorem 2.3 have either
Let H be a subgroup of G such that all elements h # 1 of H
(i) a common, single fixed point (euclidean case) , or (ii) a common pair of distinct fixed points, or have the property that all of them (iii) m a p the interior of a fixed circle onto itself (hyperbolic noneuclidean case).
T h e n H i s discontinuous in the complex plane outside of the fixed points or inside the fixed circle i f and only i f H is discrete. A subgroup H of G which represents a group of (two-dimensional) spherical or elliptic noneuclidean motions i s discontinuous i f and only i f i t i s Jinite and therefore discrete. We shall prove statement (i) of Theorem 2.3 by describing all discontinuous groups of euclidean two-dimensional motions in great detail (Theorem 2.4). Statement (ii) is nearly trivial: By putting thc common fixed points of the elements of H a t the points z = 0 and z = C Q , we find that all elements of H can be defined by substitutions z’ = Xz, where the values of X form a group under multiplication. Therefore, there either exists a sequence of distinct values X = X, ( n = 1, 2, 3,. . . ) such that limn-.- A, = 1, in which case H is neither discrete nor discontinuous anywhere, or the total supply of values of X consists of the powers Xon ( n E Z) of a fixed Xo # 1. If 1 Xo I = 1, Xo must be a root of unity if H is to be discrete; in this case, H is of finite order and a fundamental region is given by the sector between two rays through z = 0 which form a n angle of size 2?r/m,where m = I H I is the order of H . If Xo is real and positive, then H is discontinuous and the annulus between the circles I z I = 1 and 1 z I = XO
62
Discontinuous Groups and Triangle Tesselations
forms a fundamental region. Finally, if Xo = pei+
($,p
real;
p
# 1,
I#I
# 2nr,
n E Z),
H is again discontinuous and a fundamental region is defined as the region between the two logarithmic spirals z =
eit(logp+i+)
and
z
= X&(lOw+i+)
(- a
m),
that is, by the region defined as the set of points z for which =
e(it+o)(logp+i+)
(0 5 0
< 1; - co < t <
a).
Statement (iii) of Theorem 2.3 can be proved in exactly the same manner as Theorem 2.1 if we represent the group r of orientation-preserving hyperbolic”motions by the substitutions
We shall not go through the proof, the more so since Siege1 (1971) gives not only the details of it but also proves the existence of a fundamental region (which, however, does not have the properties of a canonical fundamental region). Examples of discontinuous subgroups of I’ will be given in Section 11.6. Clearly, statement (iii) is valid for any discrete group fixing a circle since it is valid for the discrete subgroups of I’. These discontinuous groups are also known as ‘yfuehsian groups.” They form a large and important class of groups; some general theorems valid for them will be stated without proofs in Section 11.7. I n the case of a group H of spherical or elliptic motions, the matrices
+
representing them have entries for which E = 6, B = -y, and CYE @B = 1. Therefore, all entries are uniformly bounded in absolute value and if H is infinite, we can find a sequence h, of distinct elements in H with entries a,,P,, y,, and 6, in the corresponding matrices such that lirn a, = lim 6, = 1 and lirn8, = lim y, = 0. Since all elements define elliptic substitutions, H contains an infinitesimal elliptic substitution and therefore cannot be discontinuous anywhere. Therefore a discrete group H must be finite, and finite groups are trivially discontinuous. Instead of proving statement (i) of Theorem 2.3, we prove the stronger Theorem 2.4 A discrete group of orientation-preserving, planar euclidean motions i s either a cyclic group of finite order or a subgroup of the group
I I . 2 Discontinuous Groups and Fundamental Regions
63
generated by the rotation (2.11)
2’
-f
= E(Z
-f),
where E is a root of unity of order 1, 2, 3, 4 , or 6, and the translations (2.12)
2’
=z
+
2’
w1,
= z
+
w2,
where w1 # 0 and w2/w1 i s real if and only if w2 = 0, and where also ewl and em2 are of the form nul muz,where n, m are integers. All of these groups are discontinuous in the whole plane C.
+
Proof: The group E of proper euclidean motions is given by the mappings
(2.13)
z’
= EZ
+6
(I e I = 1 ;
All of them have the fixed point z euclidean plane) and the b e d point (2.14)
r#J =
6
= co
arbitrary complex). (which does not belong to the
6 / ( 1 - E)
if e # 1. If e = 1, they are translations. The group E is metabelian; its commutator subgroup E‘ consists of translations and the quotient group E/E‘ is isomorphic to the multiplicative group of all E which appear in any element of E. If we denote the substitution (2.13) by h ( e , 6 ) , we have the formulas T)
=
h ( l , 6 ( 1 - q ) - ~ ( -1 e ) )
h(e, 6 ) h ( l ,7)h-1(el 6 )
=
h(1, ET).
(2.15) h(e, 6)h(q,T ) h - l ( e , S)h-‘(q, (2.16)
Assume now that H is a subgroup of E which is discrete in the euclidean plane. We distinguish two cases. 1. H doesnot contain any translations. In this case [according to (2.15)], the commutator subgroup H’ is of order 1, H is abelian, and, according t o Lemma 1.6, all elements of H have the same fixed points. Since they have z = m as a fixed point in any case, and since they are not translations by assumption, they must have two fixed points z = 00 and z = f. Therefore, all the elements h, of H , arranged in any order with n = 1, 2, 3,. . . as a counting label, must be of the form (2.17)
(z’
-f)
= e,(z
-f)
( n = 1, 2,. . .;
I e, I
=
1)
with distinct en. The en form a multiplicative group isomorphic with H . If H is finite, the group (2.10) is both discrete and discontinuous. If H is infinite, there exist infinitely many distinct en which are points on the unit
64
DisconfinuousGroups and Triangle Tesselations
circle. Therefore, there exists a convergent subsequence em* of the en. Since the elements (2.17) form a group, the set of the E, contains also the subsequence qm =
Em*
(Em+l*)-l
which has t>helimit 1. The corresponding sequence of elements (2.17) defines an infinitesimal elliptic substitution in H , and H is not discrete. But H is also not discontinuous since the point zo f has the images q,zo f, which have zo f as a point of accumulation.
+
+
+
+
2. H contains at least one translation z’ = z T~ where ro # 0. For the subgroup T of translations of H , being discrete and being discontinuous is the same property because T will be discrete if and only if there exist two numbers w1 and w2 such that w2/wl is not real unless w2 = 0 and with the property that for any translation z’ = z r in T , we have
+
(2.18)
r
=
mlul
+ m2w2
(ml, m2 integers).
This rcsult is usually proved in the theory of elliptic functions [see, for instance, Carathdodory (1954)] (however, it is of an elementary nature). Suppose thcn that T is discrete. The images of a point zo under T form a oneor two-dimensional lattice of points zo r. Let A be the minimum distance of two points in this lattice and let A* be the distance between thc fixed points of any two elements h(e1, 61) = hl and h(e2, 62) = h2 of 11. Then (2.14) and (2.15) show that the commutator of hl and h2 is a translation z’ = z T * , whcre
+
+
(2. 19)
1 ~ * 1=
11
-el/
11
-621
A*> A
since 1 r* I 2 A. Therefore, the distance between the fixed points of hl and h? is not less than A/4, which shows that the fixed points of the elerncnts of H cannot have a point of accumulation in C if T is discrete. It remains to show that the discreteness of T also implies restrictions on the possible values of t in the elements h(c, 6) of H . By conjugating a trO is translation z’ = z r0 with an clement h( e, 6), we find z’ = z also in T. Therefore, the additive group of the c appearing in any h ( ~S,) must have a rank 5 2 . This excludes any E of infinite multiplicativc order and shows that the maximal finite order m of any E must be such that t is rational or in a quadratic number field. Therefore m = 1, 2, 3, 4, or 6, and I 1 - E I 2 1 if E # 1. This completes the proof of Theorem 2.4 since we can easily vcrify that the groups listed there are discrete.
+
+
II.3
Triangle Groups. Local and Global Relations
65
11.3 Triangle Groups. Local and Global Relations
Theorems 2.1 and 2.3 do not give us any hint of how we can construct discontinuous groups, let alone groups with a canonical fundamental region and an associated tesselation. Theorem 2.2 shows at least how number theoretical methods can be used to define a discrete and, therefore, discontinuous group. Chapter I11 will supply many more examples. On the other hand, the ancient chessboard tesselation of the euclidean plane suggests that we may be able to construct discontinuous groups b y starting with a tesselation. The next two sections contain such constructions in a particularly simple situation. We consider tesselations of the euclidean, elliptic, and hyperbolic plane and of the two-sphere with tiles of the simplest shape, that is, with triangles. We shall try to use them as tiles of a canonical tesselation by using the rcflections in the sides of the triangles as generators of a group. I n order to give a precise description of our approach, we start with the Definition of a triangle group. Let I , m, n be integers 2 2 , let (2.20)
6 =
1 l
-
+ m-1 + -1n - 1,
and let A be a triangle with angles r/l,r / m , ?r/n.If 6 > 0, A i s a spherical or elliptic triangle. If 6 = 0, A i s euclidean, and if6 < 0, A i s a noneuclidean (hyperbolic) triangle. Let Lo,Mo,No be, respectively, the sides of A opposite to the angles of size r/E,r / m , r/n, and let L, M , N , respectively, be the rejlections of the particular two-dimensional space (sphere, elliptic, euclidean, hyperbolic plane) in the straight lines (great circles in the spherical case) on which the sides Lo,M,, No lie. (We recall that, in the elliptic case, these “reflections” cannot be characterized by inversion of orientation but still have the property of leaving the line of reflection pointwise fixed and of being involutions.) The group generated by L, A l l N shall be denoted by T*(1, m, n ) in the case of spherical, euclidean, and hyperbolic triangles and by E* ( I , m, n ) in the case of elliptic triangles. The subgroup of T * ( l , m, n ) consisting of words of even length in the generators L, M , N shall be denoted by T(1, m, n ). The group T consists of the orientation-preserving (i.e., proper) motions in T* and is therefore of index 2 in T*. I n the case of E*, this argument cannot be used to construct a subgroup of index 2. We call A the basic triangle of the group T*.
66
Discontinuous Groups and Triangle Tesselations
We shall prove in the next two sections that the triangle A is, indeed, a canonical fundamental region for the groups T* in all cases and that the same is true in some but not all of the possible cases for the groups E*. We shall also give defining relations for the generators of both the groups T* and T, and, finally, we shall give matrix representations for the groups T (2, m, n ) by writing down explicitly 2 X 2 matrices for a set of generating fractional linear substitutions of T. (The matrices themselves will, in general, generate a group which is a central extension of the group T, the extension consisting of the subgroup generated by the negative unit matrix.) It is important to note that we have to read a product of motions (euclidean or noneuclidean) if represented by matrices, from right to left. Stated explicitly: If A4 and N are matrices representing elements of the group T , then M N represents the rigid motion which arises if Jirst N and then M i s carried out. I n most cases, this rule will not become apparent because it so happens that the defining relations of most of the groups which we shall encounter are symmetric expressions in the generators, meaning that they produce isomorphic (and not merely antiisomorphic) groups if read from left t o right or from right to left. The next two sections will deal with a detailed description of the possible triangle groups. W e shall frequently use the fact that the groups T*(l,m, n ) and T(1, m, n) are independent of the order in which 1, m, n are listed. The procedure outlined above for the construction of discontinuous groups from a tile which is assumed to be a canonical fundamental region of the group poses two problems. First, we have to prove that the group generated by the motions which move the basic tile A into an adjacent position has A as a fundamental region. This means that the images of the basic tile under the repeated action of the generating motions will fill the two-dimensional space without gaps and overlapping. Second, we want t o find a set of defining relations for the generators of our group. Both problems split into a “local” and a “global” part. The local tesselation problem is the question: If we consider only those images of the original tile which have a t least one point in common with it and which can be obtained by a sequence of generating motions which never move the original tile into a position completely disjoint from it, will these adjacent tiles cover a certain area completely enclosing the original one without gaps and overlapping? (The global tesselation problem would be the question formulated above involving the whole two-dimensional space.) The problem of finding defining relations has to be preceded by the following remark: To every word in the generators of the group of a canonical tesselation, there belongs a chain of images of the fundamental region. This
II.3
Triangle Groups. Local and Global Relations
67
chain has properties which will be described below and is closed if and only if the word is a relator. (In the case of the triangle groups, the generators are L, M , N . ) To describe this chain, we assign to every generator or its inverse a definite side of the canonical polygon. We call a side s of the polygon and an element g which is a generator or the inverse of a generator of the group associated if g maps the original tile (in the case of triangle groups: A ) onto an adjacent triangle gA such that A and gA have s in common. (In the case of the triangle groups, the generators coincide with their inverses; L, M , N are respectively associated with Lo, Mo, No.) If y is any element of the groupsJ we can recognize the images of the sides of the fundamental tile in its image under the action of (even if some of the sides should be equal in length). Consider now an element Y =
Qlg2 * ' * g n - l g n ,
where the g , ( B = 1 , . . ., n ) are generators or their inverses and where it never happens that a gv is followed by its inverse. Then we have: Lemma 2.3 The images I , of the canonical fundamental region I , @e., of A in the case of triangle groups T*) under the action of Yv
=
9192' * ' g "
is adjacent to the image Iv+l obtained under the action of Yv+l = g1g2
-*
g,gv+1,
and the common side of I , and I,+, is the side of I , associated with gY+l. This is also true for v = 0, if we denote the unit element by go. The proof of Lemma 2.3 consists of the following remark: If I , is mapped onto the adjacent I,+1and if y y l maps I,, onto Io,then gv+l maps loonto its adjacent neighbor which has the side sv+lof I. in common with lo, where s,+~ is associated with gVC1. Now, if y,Io = I,, then y,g,+lIo = y,+J0 has exactly the property used for the definition of IVc1. It follows that y = y n = 1 if and only if I n coincides with lo, including the original labeling of the sides. The relations in the group of a canonical tesselation correspond therefore to the closed chains of images of the fundamental region. Now we define a local relation as a relation in which all images of the fundamental region I , (including Io) have one point in common. All relations associated with chains whose members do not have one point in common are called global relations. The important fact which we wish to establish will always be that the local relations define the group. This is comparatively easy in the euclidean and the spherical or
68
Discontinuous Groups and Triangle Tesselations
elliptic cases (whenever it is true), but it is more difficult in the hyperbolic case, even if lois merely a triangle A. The condition that the angles in a triangle A be unit fractions of ?r guarantees that the local tesselation problem has a n affirmative answer if A is the basic tile. The local relations for the reflections L, M , N in the sides of A are established easily. The proof that they are defining relations in most cases will be given in the next two sections.
11.4
Euclidean, Spherical, and Elliptic Triangle Groups
A simple enumeration shows : Lemma 2.4 The only (unordered) triplets I , m, n of positive integers 2 2 for which the quantity 6 of (2.20) satisfies the condition 6 = 0 or 6 > 0 are respectively those listed in Tables 2.1 and 2.2. TABLE 2.1. 6 = 0
TABLE 2.2. 6
>0
1
m
n
1
m
n
2 2 3
3 4
6 4 3
2 2 2 2
2 3 3 3
n L 2 3 4 5
3
We shall deal with the euclidean case (6 = 0) first and prove Theorem 2.5
For 6
=
0, T*(l, m, n ) i s defined by the local relations
(2.21a)
L2 = M2
(2.21b)
(LM)n = (MN)l
=
N2 =
=
1,
(NL)"
=
1.
The subgroup T(1, m, n ) of index 2, consisting of orientation-preserving euclidean motions, i s defined by two generators u, v which are rotations with two of the vertices of A as center and the relations (2.22)
where u
Un =
=
L M and v
=
NL.
= (UV)Z =
1,
II.4 Euclidean, Spherical, and Elliptic Triangle Groups
69
Details will be given in the discussion of the individual cases.
Case 1. The group T ( 2 , 3, 6 ) . We shall use the remark that we may permute I, m, n without changing T and we shall prove: T ( 6 , 3, 2) is generated by the elements u and v, which, as euclidean motions, are represented respectively by the substitutions (2.23) z’
= ez
z’ - 1 = e 2(z - 1)
and
(E
=
e‘n‘3
=
4 + +i6).
The relations (2.24)
~ ‘ 32 ~3
=
( v u )= ~1
are defining relations for T ( 2 , 3, 6 ) . The elements vo, v2 given by
vo
(2.25)
vu4,
=
v2
=
u2vu2
generate a free abelian normal subgroup A of T which is of index 6 and has the elements u” ( Y = 0,1,. .., 5 ) as coset representatives. vo and v2 are represented, respectively, by the translations (2.26)
z’ = z
+ 1-
t2
and
z’ = z
+3-
e4.
A canonical fundamental region for T (2, 3, 6) is given by the quadrilateral Q with vertices 0,
4v3E,
+a
1,
€5.
The first three of these points are the vertices of a triangle A with angles s/6,s / 2 , a/3, which is a canonical fundamental region for T*(6, 2, 3). Figure 2c gives A (shaded) and part of the tesselation arising from A under the action of T*. The quadrilateral Q arises from A by its union with A‘, which is the reflection of A in the real axis. We start the proof of these statements with an analysis of the abstract group presented by two generators u, v with defining relations (2.24). If we add the relation V U - ~= 1, this group is mapped onto the cyclic group of order 6 generated by u. The Reidemeister-Schreier method gives us as generators for the kernel of the mapping the elements
v,
= u’Vu4-N
(p =
0, 1,. . ., 5),
and the relations v3 = 1 and ( v u )= ~ 1 and their conjugates produce the relations
vov2v4= vlv3vs = 1
and
VOv3 = v1v4 = v2v6
=
1,
which enable us to express all the v, in terms of vo and v2. In addition, they produce the relation vovz = vzvo. That T ( 2 , 3 , 6) is generated by the substitutions (2.23) is an elementary
70
Discontinuous Groups and Triangle Tesselations
geometric remark, and we see that they satisfy the relations (2.24) for u and v. However, we have to show that they do not satisfy other relations which are not derivable from (2.24). This follows from the fact that the abstract group defined by u, v, and (2.24) has the following solution for the word problem: Every element has a unique expression (2.27)
U’VO~V~’
(V
=
0,1 , . .., 5 ; k, I
=
0, f l , f 2 , . . .).
The corresponding substitution, arising from a replacement of u, vo, v2 by the rigid motions in (2.23) and (2.26), is (2.28)
x’ =
€yz
+ c ~ ~ k-( i + €2)
~ ( € 2-
€411
which is the identical substitution if and only if (2.27) is the unit element, represented by v = k = 1 = 0. It remains to be shown that the quadrilateral Q defined above produces a covering of the plane without gaps and overlappings under the action of the motions (2.28). This can be done by observing first that the action of uv ( V = 0,.. ., 5 ) in Q produces a regular hexagon (see Figure 2 c ) , and then showing that for Y = 0, the translations (2.28) move this hexagon so that its replicas form a tesselation of the plane. This task can be simplified by showing that the hexagon can be replaced by the parallelogram with the vertices 0, 1 - €2, v3 i, 1 - €4, (v3i = €2 - €4 >. Case 2. The group T (3,3,3). This is a subgroup of index two in T ( 2 , 3 , 6 ) . Its canonical fundamental region is a rhombus with vertices
0, 1, €, €2. The generating elements u and v can be represented respectively by the rigid motions (rotations) z’ = c2z
and
x’
-e
= e2(x
- c).
The elements vo = vu-l and v1 = U V U - ~ generate a free abelian subgroup A with cyclic quotient group and 1, u, u2as coset representation in T . The elements vo and v1 are represented respectively by the translations z’=z+c+l
and
z’=z+e2-1.
As a fundamental region for A , we can use the same hexagon as in the case of T(6, 3, 2). The situation is illustrated by Figure 2a. Case 3. The group T ( 2 , 4, 4 ) . The situation is illustrated by Figure 2b. It is so close to the chessboard tesselation that we need not go into the details.
11.4 Euclidean, Spherical, and Elliptic Triangle Groups
71
We now turn to the cases listed in Table 2.2 (Lemma 2.3), where 6 > 0. The details of the structure of the groups generated by reflections L, M , N in the sides of a spherical or elliptic triangle will be described separately as cases 4-7. As a summary, we state Theorem 2.6 The reflections L, M , N in the sides of a spherical triangle A generate a group T*(l, m, n ) for which A i s a canonical fundamental region. The local relations (2.21) dejine the group. I n the case of a n elliptic triangle A with the same angles, the group E*(l, m, n ) does not have A as a fundamental region in the cases where (1, m, n ) = (2, 2, 2m 1) or ( I , m, n ) = ( 2 , 3, 3 ) . In the remaining cases, E*(l, m, n ) has A as a fundamental region, but the local relations dejine merely a central extension of E* and a global relation has to be added to dejine E* completely.
+
One may say intuitively that the global rela,tion needed to define E* arises because the elliptic plane is not simply connected. We shall also discuss in some detail the nonsplitting central extensions of some of the groups T(1, m, n) which arise from the two-valued (spinor) representation of the orthogonal group in terms of unitary matrices. See Lemma 1.29. Case 4. The dihedral groups T(2, 2, n ) and E*(2, 2, n ) . Let c = eniln. We choose as the original triangle A on the sphere the triangle with one vertex at the south pole and with vertices z = 1 and z = e on the equator, which is already in the z plane. Stereographic projection maps the south pole onto z = 0. Reflection of A in the real axis produces the triangle A', which, together with A, forms a fundamental region of T. The motions u and v can be defined respectively b y the matrices
U defines a rotation with z = 0 as fixed point and with 2 r / n as angle of rotation. This will be the generator u of T . The generator v of T, defined by V or, alternatively, b y z' = i / ( i z ) , is of order 2 and represents a rotation of the sphere with z = 1 as a fixed point. We have (2.30)
U n = 112 =
(uv)2= 1,
where uv has the points fe as fixed points. From (2.30), it follows that the group generated by u, v with defining relations (2.30) is of order 2n, and that every one of its elements can be expressed uniquely in the form uyv6
(Y =
0,. . ., n - 1; 6 = 0, 1 ) .
72
Discontinuous Groups and Triangle Tesselations
The corresponding motions of the sphere move A U A f into 2 n positions which cover the sphere without gaps and overlappings. The matrices U , V themselves generate a group of order 4 n which is a nonsplitting central extension of T ( 2 , 2, n ) and can be defined by the relations
U"
(2.31)
= v 2 =
(UV)Z
where Un= V 2belongs to the center and is of order 2. [To prove this remark, observe that V 2= ( U V ) 2can be written as
U-'
=
VUV-',
which implies
U-"
=
VUnV-'
=
U".
Therefore UZn= V4 = 1 and the quotient group of the group generated by V 2is T.] This occurrence is typical for the representation of groups of spherical rotations in terms of unitary matrices; see Lemma 1.29. This group is not isomorphic with T* ( 2 , 2, n ) ,which is also of order 4 n and which has, in the case of an even n = 2m, a center element which is of order 2. But here this center clement generates a direct factor of T * ( 2 , 2, 4m). It is represented by the selfmapping of the sphere which maps every point into its diametric opposite and can bc, expressed as Num, where u = L M , v = M N . We can verify that in T * ( 2 , 2, 2 m ) , given by ( 2 . 3 2 ) L2 = M2
=
N 2 = 1, L M
= U,
MN
= V,
u~~= v2
=
( u v )= ~ 1,
the element Numbelongs to the center. It is obviously not an element of the subgroup T ( 2 , 2, 2 m ) since it docs not have an even length when expressed as a word in L , M , N . Also, Num is of order 2 since
NumNum
=
(NUN-1) m u m
= V-~U-~LVU =~ (y-luv) -mum
and V-'UVU = V - ~ ( V U=) ~1, which implies v%v = u-l_We may add that, in (2.32), L, M , N are, respectively, the reflections of C in the real axis, in the line z = et ( - CQ < t < m ), and in the unit circle. The last remarks about T * ( 2 , 2, am) allow us now to describe the groups E*(2, 2 , n ) . For n = 2m, E*(2, 2, 2 m ) is isomorphic with T ( 2 , 2, 2 m ) , arising from T * ( 2 , 2, am) by adding the relation Num = 1. This expresses the fact that the mapping of the points of the sphere onto their diametrically opposite points induces the identical mapping in the elliptic plane. The triangle A is then a canonical fundamental region for E*. Howcvcr, if n = 2 m 1 is odd, A is not a fundamental region for E * ( 2 , 2 , 2 m 1), for the following reason: There exists a mapping in the group generated by the reflections L , M , N in the sides of A which maps A (as a triangle on the sphere) onto its diametric opposite in such a manner that the vertices at z = 1 and a t z = e go, respectively, into the opposites of z = e and z = 1. If we now identify diametrically opposite points on the sphere, A is
+
+
11.4 Euclidean, Spherical, and Elliptic Triangle Groups
73
mapped onto itself by a mapping which is not the identity. Therefore, the 1), triangle with vertices 0, 1, &is a fundamental region of E*(2, 2, 2m which shows that this group is isomorphic with T ( 2 , 2, 4m 2 ) . The details of the proof for our assertions are easily derived with the help of a 1 = 3. drawing, for instance, in the case 2m
+
+
+
Case 5. The tetrahedral group T ( 3 , 3, 2 ) , and the groups T*(2, 3, 3 ) and E*(2, 3, 3 ) . A tesselation of the sphere with triangles congruent to A (which has angles a/3,~ / 3./a) , arises if we inscribe a regular tetrahedron in a sphere and mark its vertices together with the projections of the centers of the faces and the midpoints of the edges on the sphere, the center of projection being the center of the sphere. Repeated reflection of the sphere in the sides of the triangle A produces the group T* ( 3 , 3 , 2 )with generators L, M , N and defining relations
(2.33) L2 = M2
=
N2
=
(LM)3 = ( L V N )=~ ( N L ) 2= 1.
1,
This group is isomorphic with &, the symmetric group of permutation of four symbols, as may be seen from the definition of this group by Artin (1925) as a quotient group of the braid group on four strings. We merely have to identify L, M , N , respectively, with ul,u2, u3 in Artin's notation. T ( 3 , 3, 2 ) is the subgroup of T*(3, 3, 2) composed o f words of even length in L, M , N . It is generated by u = L M and v = M N with defining relations
(2.34)
u3
=
213
=
(uv)2= 1.
This is the tetrahedral group, isomorphic with A4, the alternating group of permutations of four symbols. It is represented as the group of proper motions (i.e., rotations) of the sphere which carry the original tetrahedron into itself. As its fundamental region, we may choose the union Q of A and one of its reflected image A'; the vertices of Q are now two vertices of the tetrahedron and the midpoint of a face on which both of these vertices lie. The midpoints of the edges of the tetrahedron have now disappeared from the picture; Q appears as a triangle with angles 2a/3, a/3, a/3. However, the fixed points of rot,ations of order 2 are still the midpoints of the edges, projected on the sphere. The generating rotations u, v of T(2,3,3) may be presented as unitary matrices U , V defined by
U=[ (2.35)
$(1 - i) $(-1 - i)
$ ( 1 - i) $(1
+ill
$ ( 1 - i) =
[$(l - i)
$ ( - 1 - i)
+ i)
$(1
1
74
Discontinuous Groups and Triangle Tessetatwns
U and V define both conformal selfmappings u, v of the complex plane which generate T and motions u*, v* of the elliptic plane. The group generated by the matrices U , V themselves is a central nonsplitting extension T' of the group T(3, 3, 2 ) defined by (2.34) ; the group T' can be presented b y the defining relations (2.36)
u 3
= v3 =
(UV)2.
T' has a center of order 2 generated by U3 = V3.The motions of the elliptic plane generated by U and V (or the corresponding rotations of the sphere) generate a group isomorphic with T which we shall denote by E(3, 3, 2). Similarly, the reflections L, M , N , interpreted as motions of the elliptic plane, generate a group E*(3, 3, 2) isomorphic with T*(2, 3, 3 ) . However, the description of the fundamental regions of these groups in the elliptic plane cannot simply be obtained by transferring the tesselations of the sphere to the corresponding ones in the elliptic plane. Figure 3b shows the tesselation of the sphere by triangles A* with angles r/3, r/3 , r / 2 ; the generating elements L, M , N of T*(2, 3,3) are the reflections in the sides of one of these triangles. This tesselation can be transferred to by stereographic projection; the result is shown in Figure 3a. From either one of these figures, we obtain a fundamental region for T(3, 3, 2) by combining each shaded triangle A* with one of the particular adjacent unshaded triangles which meets A* a t a vertex with angle r / 2 . The resulting polygon Q is actually a quadrange with angle r at one of its vertices; it looks, there, 2u/3. Now we may transfer the fore, like a triangle with angles ~ / 3 r/3, tesselation of the sphere by triangles A* with angles r/3, r/3, r / 2 to the elliptic (projective) plane by identifying each point of the sphere with its diametrically opposite point. By this process, pairs of shaded as well as pairs of unshaded triangles are identified, and the resulting tesselation of the projective plane is shown in Figure 3c. However, the 12 triangles in this tesselation are not the fundamental region of either the group E ( 2 ,3 ,3 ) or of E*(2, 3, 3), for the following reason: If we use the action of L, M , N to map A onto its diagonally opposite triangle A-, we will not map every point of A onto its diagonally opposite point. The two vertices VI and V2 of A a t which the angles are r/3 will be mapped respectively on the vertices Vz- and V1-, where V,- is opposite V v ( V = 1 , 2 ) . Therefore, in the elliptic plane, one half of A , i.e., a triangle A with angles r / 4 , r/3, u/2, will be the fundamental region of E*(2, 3,3), and A, together with its reflection in one of its sides, will be a fundamental region of E ( 2,3, 3). The resulting tesselation is illustrated in Figure 6, which arises from a transfer of the tesselation of the sphere by 48 triangles with angles r / 4 , r/3, r / 2 onto the elliptic plane. Fricke (1897), points out that Figure 3c represents the partition of the projective plane by the lines of a complete quadrilateral, and comments
II.4 Euclidean, Spherical, and Elliptic Triangle Groups
75
that “this shows again that the low-order groups of linear substitutions in a single variable appear everywhere in the foundations.” We shall not give the details of the proofs for our statements concerning the groups T*(3, 3, 2 ) and T ( 3 , 3, 2 ) . The group theoretical arguments, in particular the proof that (2.33) and (2.34) define, respectively, the groups Z4and A 4 and that the group with defining relations (2.36) is a nonsplitting central extension of A4, can be obtained by following the procedure described in the analysis of T ( 2 , 3, 5 ) (case 7 ) . Case 6. The octahedral group T ( 2 , 3, 4 ) and the groups T*(2, 3, 4) and E*(2, 3, 4 ) . The spherical triangle A with angles r/4, 7r/3, r / 2 produces, under repeated reflections in its sides, a tesselation of the sphere by 48 congruent replicas of A. The reflections L, M , N in the sides of A generate a group T*( 2 , 3, 4 ) with the defining relations
( 2 37a) (2.37b)
L2
=
M2
N2
=
( M N ) a= ( N L ) 2= 1.
=
1
It has a subgroup T ( 2 , 3, 4) generated by elements u = L M and v with defining relations u4 = v3
=
(UV)Z
=
=
MN
1.
u and v are rotations of the sphere which correspond, respectively, to Miibius transformations with matrices U , V defined by
(2.38) *(1 - i) v = [ $(-1 - i)
$(1 - i) + ( 1 i)
+
I.
The matrices U , V themselves generate a nonsplitting central extension of order 48 of the group T ( 4 , 3, 2 ) which can be defined by the relations (2.39)
u 4
= v 3 =
(UV)2.
T(4,3, 2 ) is isomorphic with Z4;it is called the octahedral group because the rotations u, v generate the group of all rotations which carry a regular octahedron, inscribed in the unit sphere, into itself. The group T*(4,3, 2) is the direct product of a group of order 2 generated by ( L M N ) 3 ,which represents the central inversion (i.e., which maps every point of the sphere into its diametrically opposite). This was proved by Coxeter (1963).
76
Discontinuous Groups and Triangle Tesselations
We shall not reproduce this proof here; but we shall prove the corresponding result in the case of T* (2, 3, 5). Since ( L M N ) involves an odd number of factors L, M , N , it cannot be contained in T(4, 3, 2 ) . If it also belongs to the center and is of order 2, it must therefore generatc a direct factor of T*(4, 3, 2 ) . Since the central inversion is mapped into the identity if we go from the sphere to the elliptic plane, it follows that E*(4,3, 2 ) is isomorphic with T(4, 3, 2) and can be defined by adding the relation (2.40)
(LMN)3 = 1
to those of T*(2, 3, 4). Figure 4b shows the tesselation of the sphere by _triangles with angles ~/4, sj3,s/ 2 and Figure 4a shows the tesselation of C arising from stereographic projection of the spherical tesselation. Figure 6 shows part of the tesselation of the elliptic (projective) plane which arises from the tesselation of the sphere by triangles with angles s/4 , n/3, s/2 after identification of centrally opposite points. The drawing has to be used with caution. Since the centrally opposite triangle of a shaded triangle on the sphere is unshaded and vice versa, it is impossible to keep the triangles in the elliptic plane alternatingly shaded and unshaded. This fact is hidden by the drawing since those triangles which intersect the infinite line are partly shaded and partly unshaded. The drawing is complete in the sense that it contains at least part of all lines involved in the tesselation. However, it docs not show all of their intersections. Proofs of the statements made for case 6 can be found in Fricke (1926) or may be derived in the same manner as the analogous statements in case 7.
Case 7. The icosahedral group T(2, 3, 5) and the groups T*(2, 3, 5) and E*(2, 3, 5 ) . This is the most complex and also the most important case of a spherical triangle group and we shall treat it with greater attention to detail than was given to the previous cases. However, we shall still assumc without proof the existence of a regular icosahedron which can be inscribed in a sphere. It is true that nobody has doubted the correctness of this assumption since the time of Plato, and that Euclid, in Book 13 of his “Elements” [see Heath (1956)l gave a complete construction. Figures 5a and 5b illustrate the tesselation of the sphere with congruent triangles whosc angles are s/5,r / 3 , s/2 and the corresponding tesselation of C arising froni stereographie projection. Their intricate pattern will help us to realize th a t something has to be proved here. There is no difficulty in proving that we can make a corner out of five equilateral triangles and that, at every one of the five free vertices of this corner, we can again construct a corner congruent to the first one and involving one of its triangles. But it is not obvious
II.4 Euclidean, Spherical, and Elliptic Triangle Groups
77
that we will obtain a closed polyhedron after we have used up 20 triangles and that this polyhedron can be inscribed in a sphere. Once the existence of a regular icosahedron has been established, it is not difficult to prove that the group T*(2, 3, 5 ) generated by reflections L, M , N in the sides of a spherical triangle with angles r / 5 , r/2, r / 3 contains a subgroup T ( 2 , 3 , 5 ) of index 2 consisting of (orientation-preserving) rotations of the sphere isomorphic with the group of rotations which carry the icosahedron into itself. It also follows from elementary geometric arguments that T ( 2 , 3 , 5 ) is of order 60, isomorphic with A5 (the alternating group on five symbols) , and generated by rotations u,v such that u5 =
(2,411
v3
=
(uv)2 = 1
where
(2.42)
v = MN,
u = LM,
uv
=
LN.
For the details, see Klein (1888).The group theoretical results, most of which we shall prove, may be summarized as
Theorem 2.7 Let U , V be the 2 X 2 matrices (2.43)
U=[iE3
1,
V
-1
=--[e3-E
$5
-c-2
(E
E - 1
=
€2-
e2~i/5 1.
Then, U,V define, respectively, fractional linear substitutions u,v which m a y be interpreted as selfmappings of 6 or as rotations of the unit sphere. The group generated by u,v i s afaithful representation of T ( 2 , 3 , 5 ) . It i s dejined by the relations (2.41)(that is, by the local relations). The group generated by the matrices U , V itself i s a nonsplitting central extension of T ( 5 , 3, 2 ) which will be denoted by Glzo. Its center i s of order 2 and generated by U5= V3.I t is defined by the relations (2.44)
= v 3 =
u 5
(UV)Z.
The group T*(2, 3, 5) i s dejined by the generators L, M , N and the relations (2.45a) L2 = M 2 = N2
=
( L M ) 5= ( M N ) 3 = ( N L ) 2= 1.
1,
I t i s the direct product of its subgroup T ( 5 , 3, 2) with a cyclic group of order 2 generated by (2.45b)
C
=
(LMN)5.
The quotient group of T*(5, 3, 2 ) with respect to its center (generated by C ) i s the group E* (5, 3, 2 ) N T (5, 3, 2 ) . The relation C = 1 i s the global relation needed for the dejinitimz of E*.
78
Discontinuous Groups and Triangle Tesselations
Proof: The matrices U5, V3,and ( U V ) 2must be - I , where I denotes the unit matrix. For U5, this is obvious; for V 3and ( U V ) 2 ,it follows from the fact that the traces of V and UV are, respectively, 2 cos(a/3) and 2 cos ( ~ / 2 ) It . is also obvious that U and V are unitary matrices and therefore represent rotations of the sphere. According to Fricke (1926,_p.24), z = 0 is one vertex of the icosahedron (after its projection onto C), and one of the fixed points of UV (both of which lie on the real axis) is the projection of the midpoint of the side of a triangle of the icosahedron whose opposite vertex has been projected onto z = 0. It then follows that the homographic substitutions u and v are indeed generators of T ( 2 , 3, 5) and satisfy the relations (2.41).It also follows (without geometric considerations) that U,V satisfy (2.44)and that Us= V 3generate a center element of order 2 in Glzo.Furthermore, the group generated by u and v cannot be of an order <60, since the relations (2.41) show that the group is its own commutator subgroup and therefore not solvable, and the group is not of order 1 since u,v are not the identical selfmappings of From this, it also follows that Glzo is indeed a group of order at least 120 and, in any case, of twice the order of the group generated by u and v. We can derive our statements about T ( 5 , 3, 2) and G1zOfrom
e.
Lemma 2.5 The abstract group Go on two generators U , V with defining relations (2.44)has a center of order 2. Its quotient group in Gois isomorphic with As (and Go is therefore of order 120 and, in fact, the group G1zo). Proof: Let w = (0, 1,2,3,4)and 4 = (0,3,1) be two permutations on five symbols 0,.. ., 4,written in the usual form as cycles. Then w , 4, and wt$ = (1 2)(3 4) satisfy 0 5
=
43
=
(wt$)2
=
1.
Since w , 4 E A5, and since the group generated by w , 4 cannot be solvable, 4 generate A5. Also, the mapping p : U + w , V -+ 4 defines a homomorphism of Go onto A5, and therefore Go has a subgroup H5 of index 5 consisting of those elements of Gothat are mapped by p onto a permutation that fixes the symbol 0. The group of all such permutations in A, is the subgroup A4 of order 12;it has the elements wv ( Y = 0,1, 2,3,4)as right coset representatives in A6. Therefore, H6 has right coset representatives UY in Go, and an application of the Reidemeister-Schreier method gives the generators R, So,. . ., 8 4 defined by w,
U'
=
R,
VU-3 = So, U3VTJ-'
=
S3,
UV
=
Si,
U4VU-4 =
U'VU-' = Sz, S 4
II.4 Euclidean, Spherical, and Elliptic Triangle Groups
79
for H5. Rewriting the relations
uyu5v-3u-Y
=
1)
WU5(UV)--2U--V = 1
(v =
0, 1, 2, 3, 4)
in terms of the generators of H5, we find as defining relations for H5
R = SOS& = S1SoS3= SZ3 = S3SiSo= Sd3 = S? =
= SZ83 = S S S Z
S&So = RS4So.
We can use these relations to express all generators in terms of Sz and S4, and we find defining relations for H 5 in terms of these generators to be (2.46)
SZ3 = 84'
=
(s2s4)2.
The images of Sz and S4 under p are respectively the permutations uz = (143) and 6 4 = (142) which generate Aq. I n turn, A4 contains a subgroup V4 of index three which is the four-group. The preimage of V4 in H5is a normal subgroup H I , of H5. It consists of those words in S f , S4 for which the difference of the sum of exponents of S4and of Szis divisible by 3. The elements Szx ( A = 0, 1, 2) are right coset representatives of Hi5 in H5. Using the Reidemeister-Schreier method again, we find that Hi5 is generated by the elements (2.47) 6 = Sza,
To = S~SZ, Ti = SzS4,
Tz
=
SzZS4Sz-l
and has as defining relations (2.48) % = ToB-'TzTi = TIT,B-ITz = TZTiT&' = T,2 = Tzz = %To2&'.
Using (2.48) to eliminate 6 and TI, we find that Hi, is generated by TO,Tz with defining relations (2.49)
To2 = Tzz =
It is easily seen that this is the quaternion group with To2as generator of the center and To4= 1. Since T04 = 0 2
=
Sz9 =
uzV6u--2 = 1,
it follows indeed that V6 = 1 and that V3 = U5 generates a center of order 2 in Go.The quotient group Gowith respect to its center is A5 (according t,o our construction). The center of Go = Glzocannot be of order > 2, since A5 has no center. Glz0,as central extension of A5, cannot split since otherwise the quaternion group would be a splitting extension of its center, which is not true. This proves Lemma 2.5 in all details. Incidentally, we have also
80
Discontinuous Groups and Triangle Tesselations
proved that there exists a central extension (by a center of order 2 and nonsplitting) of Ad and that it is defined by (2.46). This group appeared already in case 5, Equations (2.36). To complete the proof of Theorem 2.7, we need Lemma 2.6 The group T* with defining relations (2.45a) has a center of order 2 which is generated by the element C in (2.4513). Proof: The group T* contains the group T as subgroup of index two, and T is generated by elements u, v with defining relations (2.41). We know from the proof of Lemma 2.5 that the map p : u -+ ( 0 , 1 , 2 , 3 , 4 ), v -+(0,3, 1) is an isomorphism. Now we find, using u = L M and v = M N (L2= M 2 = N2 = I),
Since p maps the elements on the right-hand sides of (2.50a) and (2.50b) onto the identical permutation, we have shown that C commutes with both N and L and, since it commutes also with L M N , must belong to the ccnter of T*. We still have to show that C2 = 1. Now
(LMN)'O = (uv-~u-~v)~ + (0,4, 1, 2, 3)5, where the arrow indicates the action of p. Therefore C2= 1. Since T* does contain A , as a subgroup and since A5 has no center, T*(5, 3, 2) is the direct product of A , and its center. It is also the triangle group since it is of order 120. To obtain E*(5, 3, a ) , we observe that the original triangle A has a diagonally opposite one A' which is the image of A under the mapping D which carries every point of the sphere into its diagonal opposite. Transition from the sphere to the elliptic plane means a n identification of A and A'. Therefore, E* arises from T* by mapping D + 1. Since D commutes with all rotations, it belongs to the center of T* and since the center of T* is of order 2, E* N A,. This completes the proof of Theorem 2.7. Note: Seemingly, Coxeter and Moser (1965, p. 39), state our result about the structure of T* in a different form. However, our results agree with theirs if one remembers that the relation (LMN)'O = 1has to be read from right to left if L, M , N are interpreted as geometric transformations. The method used in proving Lemma 2.5 is basically the same as the method described by Coxeter and Moser (1965, pp. 12-18).
11.5 Hyperbolic Triangle Groups
81
11.5 Hyperbolic Triangle Groups We assume it to be a proved fact that a hyperbolic triangle with angles p y < 1. In the case where one of the angles is r / 2 , we shall give an explicit construction later (Theorem 2.10) which will also show that the angles determine the triangle uniquely. We shall consider triangles with angles r/l, r / m , u/n, where 1, m, n are positive integers. A triangle is considered as a closed set (we exclude, for the time being, triangles with an angle equal to zero). Two triangles will be said not to overlap if they have in common no point that is in the interior of either one. We shall use extensively the elementary geometric results for hyperbolic geometry, as presented in Section 1.4, including the theory of area. “Line” will always mean “hyperbolic straight line,” even if we use a model in which the lines are euclidean circles. We summarize our general results as a, 8, y exists provided that a
+ +
Theorem 2.8 Let L, M , N be the re$ections in the sides of a hyperbolic triangle A, with angles r/l, r/m, r/n. The images of A, under the action of the distinct elements of the group T*(l, m, n ) generated by L, M , N fill the hyperbolic plane without gaps and overlappings. T* i s dejined by the local relations
(2.51)
L2 = M 2 = N 2 = 1,
( L M ) ” = ( M N ) z= ( N L ) “
=
1.
The main difficulty in proving Theorem 2.8 consists in the proof of the statement that the images of A, fill the hyperbolic plane without gaps and overlappings. An elementary proof has been given by Carathbodory (1954b) which also includes the cases where one or several of the angles r/l, r / m , u/n are zero. At one point, Caratheodory uses a convexity argument the proof of which is supplied by Lemma 1.27 in Section 1.4. A much more general theorem has been stated by Poincard; this theorem involves all polygons which can appear in a canonical tesselation as defined in Section 11.1, characterizing these polygons by a finite number of geometric data. An elementary proof of Poincarb’s theorem has been given by Maskit (19714. His proof is based on an idea of Siege1 (1971). It uses the concept of a covering space, but none of the existence theorems in the analytic theory of Riemann surfaces and may therefore still be labeled as “elementary.” We shall not go into the proof of tkis part of Theorem 2.8 since the sources quoted are easily accessible. We shall, however, prove the group theoretical part of Theorem 2.8.
82
Discontinuous Groups and Triangle Tesselations
Let us assume first that the integers 1, m,n are different from each other. In this case, the following is true: Let 0 = W ( L ,M , N ) be a word in the generators L, M , N of T* defining a particular selfmapping (noneuclidean motion) of the hyperbolic plane. Let A = 0 (Ao)be the image of A, under the action of 0. Since the images of A, under the action of the elements of T* fill the plane without gaps and overlappings, A and A, can have an interior point in common only if A covers Ao.But in this case 0 must be the identical selfmapping since the vertices of A, must be mapped onto themselves (otherwise angles of different size would have to be congruent to each other), and a mapping which fixes three points is the identity, according to Lemma 1.17, if the points are not collinear. From here to the end of the proof of Theorem 2.9, Figure 23 provides the basis for illustrations. Figure 23 shows the tesselation of the hyperbolic plane with triangles where E = 2, m = 3, n = 7. We use Lemma 2.3 (Section 11.3) to assign a chain C of triangles A, ( v = 0,. . ., n) to 0 = W ( L , M , N ) . Since L, M , N are of order two, we may assume that W is mitten in the form
w = 9192. *gn,
(2.52)
*
where each gY ( V = 1,. . ., n ) denotes either L, M , or N and where g. and g,+l never denote the same generator. We identify I0 of Lemma 2.3 with A, and I , with its image A,, and C consists of a sequence of congruent triangles A, ( V = 0,. . ., n) each of which arises from the previous one by is reflection in one of its sides. Since gy # ghl, it never happens that reflected back into A,. In other words, A, f Av+2.Of course, AO= An. With the chain C we associate a closed polygon 11 which is constructed as follows: We mark a point Po in the interior of A, and denote by P. its image in A,. We also mark the midpoints QA(O) ( A = 1, 2, 3) of the sides of A, and we denote the corresponding midpoints of the sides of A. by Qx(,). We join P, and Pu+lwith the particular midpoint Qh(,) of the side which A, and A,+, have in common by segments of straight lines which will be, respectively, contained in A, and in A,,, because of the convexity of triangles. Then II is the oriented polygon starting in Poand going, via mid. ., Pn,back to Po.This polygon II may intersect points and points P1,. itself. If II should intersect itself, it must happen that a triangle A, coincides with a triangle A,+,, where
O
+
and either 0 < v or v p < n (or both). In this case, a subword of p symbols gP in W represents the unit element of T*, and if we choose a shortest nontrivial subword of this type we can associate with it a simple
11.6 Hyperbolic Triangle Groups
83
(i.e., nonselfintersecting) polygon II. We shall assume that the polygon II associated with our original W already has this property. Since II encloses a finite area, only finitely many triangles belonging to the tesselation of the hyperbolic plane can have points in the interior of II, since each triangle has a positive area and since, a t every vertex of a triangle inside II, only finitely many triangles of the tesselation meet since 11 itself passes only through n triangles A,. Of these, every A, has at least one and at most two vertices in the interior I* of II. This follows from Pasch’s axiom [(iv) in Section 1.41 since II intersects exactly two sides of each A, and each of these two sides must have one endpoint inside and the other one outside of II. Let P be a particular vertex inside II belonging to one of the triangles of C, and let A,, . . ., A,+kbe the consecutive triangles of the chain forming II which meet in P. (If v p 2 n,we reduce v p mod n to a number between 0 and n - 1.) We have k > 0 since at least two consecutive triangles of the chain C meet at P, namely those that have a side in common for which P and a point outside of II are the endpoints. Let A,, A,,,, . . ., A,+; & I , . . ., A,’ be the set of distinct triangles of our triangulation which meet in P, and assume that they follow each other in the order in which they are written down in such a manner that each has a side in common with the previous one and that A,‘ has a side in common with A,. (Of course, we may have s = 0.) We shall say that these triangles form the star of P. Now we replace in C the triangles Av, A,,, . . ., A+ by the sequence A,, A,‘, . . ., A,‘, A,+k and adjust II accordingly, obtaining thereby a new polygon II’. Then P is outside of II’, but II’ itself contains no point outside of II. It is not necessary anymore that II’ should be a simple polygon, but it is again the union of simple polygons connected either in points or by polygons without any interior. These then consist of combinations of line segments through which II’ runs in both directions. Since the total number of vertices of tesselating triangles in the interior of II’ is less than the corresponding number for II, our process, if continued, must terminate after finitely many steps with a “one-dimensional” polygon II* which has no interior at all. Now we shall interpret these geometric considerations in algebraic terms. The replacement of the triangles A,, A,,,, . ., A V + k by triangles A,, A,‘, .., A,’, A v + k , means that somewhere in the word W ( L , M , N ) which defines 11we have replaced k symbols of one of the cyclically written words
+
+
.
.
(LWn,
(MWZ, ( N L ) =
by the inverse of the product of the remaining symbols-which we are allowed to do group theoretically in view of the defining relations for T* given in Theorem 2.8. (Note that not the triangles but the transitions from
84
Discontinuous Groups and Triangle Tesselations
one triangle to the next are defined by the generators L, M , N . ) Doing this repeatedly will eventually change our word W into a word W* which defines a polygon II* without any interior. But then II* must be such that, a t one point, it consists of a segment immediately followed by the same segment with the opposite orientation. This means that one of the generators L, M , N in W* is followed by the same generator. In view of the defining relations L2 = M 2 = N 2 = 1, these two symbols in W* can be omitted, and a continuation of the same process must lead to the empty word. We have thus proved Theorem 2.8 in the case where I, m,n are distinct integers. Suppose now that we have m = n. Then we can bisect A0 by drawing one of its altitudes, obtaining a triangle A( with angles 77/21, r / m , r/2. These angles are distinct unless m = 21, since m > 2 if m = n. I n the case where m = 21, we repeat the same process, obtaining now a triangle A,'' with angles 77/21,
77/2m,
r/4
which are necessarily distinct because m > 2, m = 21, and the case 21 = 4 cannot arise if m = 21 since m > 2. Suppose now that A{ has three distinct angles. Then Theorem 2.8 applies, and A( is the fundamental region of a group T' generated by three reflections L, N , P with defining relations
L2
=
N2
=
p2
=
1,
=
( N P ) 2 z= ( P L ) 2= 1,
where P is the reflection in the altitude of A, which is a side of A,'. The original A, consists of A,,' and its image A,' under the action of P . We find (from direct geometric considerations) that
PNP
=
M.
Now all we have to show is that the elements L, N , PNP of T' generate a subgroup T*of index 2 with coset representatives 1, P and with the defining relations for T* where now M = PNP. The Reidemeister-Schreier method shows that T* is indeed generated by L, N , PLP, and PNP = M . Because of ( P L ) 2= 1, PLP = L is redundant. The defining relations are
Lz
=
N2 = 1 = (pLp-1)2 = (pNp-')z
=
&f2 =
1
and
(LN)ln= 1,
(PLNP-l)m = (PLP-'PNP-')"
=
( L M ) m= (LM)" = 1,
(NP)Z' = (NPNP)Z = ( N M ) ' = 1, (PN)2'= ( P N P N ) z=
=
1.
These relations either agree with those for T* or are derivable from them. This settles the case where A' has three distinct angles. In the case where
11.6 Hyperbolic Triangle Groups
85
only A" has three distinct angles, we have to build up A, from four replicas of Ad' and we will obtain T* as a subgroup of index four of the group of reflections in the sides of A;'. This then completes the proof of Theorem 2.8. Comments on the Proof: To illustrate the complexity of the situation, we remark that there exist tesselations of the hyperbolic plane by very simple polygons which are not the canonical fundamental region of any group of noneuclidean motions. For example, the tesselation with triangles whose angles are a / 2 , r / 3 , a/7 (see Figure 23) is a subtesselation of a larger tesselation with quadrilaterals which have four equal sides. These quadrilaterals arise by joining together the four triangles which meet at a point where the angles are a / 2 . It can be shown that such a quadrilateral cannot be made into the fundamental region of any group that is generated by motions shifting the quadrilateral into adjacent ones which have exactly a full side in common with the original quadrangle. It is plausible that the local relations will also be global relations only if the tesselated region is simply connected, and our proof of Theorem 2.8 (which uses the Jordan curve theorem for polygons) actually depends on the simple connectivity of the noneuclidean plane. A simple counterexample (due to Macbeath) is the following one: Take a curvilinear quadrangle bounded by the arcs of the curves I z I = 1, 1 z I = > 1, which are located between the real axis and the ray z = ei+t, 0 5 t (4 fixed, 0 < 4 < 7r/2). Then the mappings z' = Xz and z' = ei+z map the quadrangle onto adjacent ones in the manner of a canonical tesselation. But if 4 is not of the form 27r/n, where n is a positive integer, the quadrangle is not the fundamental region of the group generated by these mappings, and even if (b = 27r/n, the defining relations of this group are not the local relation (which consists of the relation stating that the two generators commute) but are supplemented by the global relation stating that z' = ei+z is a mapping of order n. We can use the geometric considerations in the proof of Theorem 2.8 to solve the word problem for the group T* in a simple algebraic manner. We shall prove: Theorem 2.9
Let 1, m, n be positive integers such that 1 1 1 -+-+-
Let W be a word in the generators L, M , N of the group T*dejined in Theorem
Discontinuous Groups and Triangle Tesselations
86
2.8 and assume that W i s normalized in the form (2.52). Then, if W = 1, there appears somewhere in W a subword W' of one of the cyclically written relators (2.53) (LW", (MNI2, (NL)m whose length exceeds half of the length of the relator. Theorem 2.9 allows a solution of the word problem for T* since it shows that we can shorten every word W = 1 by replacing the larger part of a relator (which appears as a subword in W ) with the inverse of the shorter part of the same relator. The rcsulting word W' can then be put into the normalized form (2.52) without increasing its length, by using the relations L2 = M 2 = N2 = 1. If W = 1, a repetition of this process must change W into the empty word. To prove Theorem 2.9, we return to the closed II associated with W and defined in the proof of Theorem 2.8. We inscribe within II a n interior polygon III the sides of which are also sides of the triangles A, and which is defined as follows. Let Q1 be the vertex of A, inside II which is also a vertex of A,. Assume that Q1 is also a vertex of As,.. ., Ak, but not a vertex of A,,+,. Then Ak,+lmust have a second vertex Q2insideII and QZ must also be a vertex of Ak,+2.We continue in this manner, obtaining an oriented closed poygon III with vertices Q1, Q 2 , . . ., Q h which is entirely inside II. If 111 has only one vertex, then W must be a cyclic permutation of one of the relators (2.53), and our theorem is proved. Otherwise, In must enclose some area since the normalization of W excludes the case that an interior angle of In can be zero. (Of course, III need not be a simple polygon and may enclose several finite parts of the plane, all of which are fully tesselated with replicas of Ao.) Assume now that the oriented part In* of III with vertices Q 8 , . ., is a simple closed polygon which encloses some area. Then at least one of the interior angles of this ( t 1)-gon must be less than T since, generally speaking, the sum of the interior angles of a hyperbolic polygon with ( t 1) vertices is less than ( t - 1)T . Let Q be a point where the interior angle a is less than T . Then a will be a multiple of r / N , where N denotes one of the numbers 1, m, n, and there will be 2[1 - ( a / r ) ] N > N consecutive triangles A, which belong to the chain C , and which meet in Q . The corresponding subword W' of W satisfies the conditions of Theorem 2.9. An important result about the structure of the triangle group is
Q8+
+
+
Theorem 2.10 Let T(1, m, n ) be the subgroup of index 2 in the triangle group T* (1, m, n ) which con.sists of orientaticm-preserving selfmappings of the
11.5 Hyperbolic Triangle Groups
noneuclidean plane; T is generated by A the relations
An
(2.54)
=
B1
=
=
LM and B
=
87
M N and defined by
( A B ) m= 1.
T h e n the elements # 1 of finite order in T are conjugates of powers of A , B, or AB. Proof: The elements of T map onto itself the tesselation of the noneuclidean plane defined by the fundamental region FR of T . Suppose W # 1is of finite order. It is represented by an elliptic substitution and has exactly one fixed point # in the noneuclidean plane. Let FR* be a n image of FR such that # lies inside or on the boundary of FR*, and let 0 be the element of T which maps FR onto FR*. Then W* = OW@-' has a fixed point #* inside or on the boundary of FR. If cp* were a n inner point of FR, then there would exist distinct points (close to +*) inside FR such that W* would map one of them onto another one. This contradicts the definition of FR. Therefore, #* is on the boundary of FI1 and W* must map FR onto one of its images that have a point in common with FR. But then +* must be one of the vertices of FR. These, in turn, are the fixed points of the powers respectively of A , B, A B , and BA. This proves Theorem 2.10.
Corollary 2.10 If N i s a normal subgroup of T such that, under the homomorphism T + T I N , the elements A , B, A B of T a r e respectively mapped onto elements of the same order n, 1, m in T / N , then N i s torsion free (i.e., the unit element is the only element of finite order). Proof: The conjugates of powers # 1 of A , B, A B do not belong to N but to other cosets of N in T since their images under the mapping T + T / N are not the unit element of T I N . Although it can be shown in general that a noneuclidean triangle with angles a!, p, y exists and is uniquely determined if a, 0, y are nonnegative and a! /3 y < r , we shall give an explicit construction of a right triangle and a n explicit representation of the subgroup of proper noneuclidean motions in the group generated by reflections in the sides of the triangle. We shall use the unit disk as a model for noneuclidean geometry, with the circles orthogonal to the unit circle serving as straight noneuclidean lines. The angles will then be the euclidean angles. We summarize our results as
+ +
+
< r/2. Let Theorem 2.11 Let a > 0 and /3 2 0 be angles such that a 0, Q, P be three points in the u n i t disk I z I < 1, defined respectively by their coordinates ZP = XP '$/P, 20 = 0, ZQ = XQ,
+
88
Discontinuous Groups and Triangle Tesselathns
[sin a cos (a
YP =
+ 8)I / P
and (cosz8 - sin2 a)lt2.
p =
Then 0, Q, P are the vertices of a noneuclidean triangle with angles a , n/2, 8, respectively, at 0, Q, P. The sides OQ and OP are respectively parts of the real axis and of the straight euclidean line joining 0 and P. The side Q P i s part of the circle with center at z = xc, where xc = (cos 8 ) / p , and with radius r = (sin a )/ p . Let L, M , N denote, respectively, the rejlections in the sides OQ, QP, PO of the triangle. They generate a group T* which has a subgroup T of index 2 consisting of orientation-preserving noneuclidean motions and i s generated by a
(2.55)
=
LM,
b
=
LN.
T o a, b there correspond respectively matrices A , B of Mobius transformations which are given by (2.56)
A
=
-[i
s1na
C O S ~
-cosp
-p
1,
B=[e0 ia
'1
e-ia
and which map I z I = 1 onto itself. If a = n/m and 8 = n / l , where 1, m are integers, then A , B define a group T of Mobius transformations with the defining relations (2.57)
A2
=
Bm
=
( A B ) ' = 1.
Theorem 2.11 is verifiable by easy calculations. We shall not go into the details, but we shall exhibit some special cases which will be used in Section 11.6. Corollary 2.11 relations
The following groups on two generators A , B with defining
(i)
A2
B6
=
(AB)4 = 1
(ii)
A2 = B6
=
( A B ) 6= 1
(iii)
A2
=
( L ~ B=) I ~
=
=
B8
have faithful representations in terms of Mobius transformations which m a p
11.6 Hyperbolic Triangle Groups
Iz I 5
89
1 onto itself. .They are represented respectively by the following matrices:
+
1 (2 3- f i ) 1 / 2 i(2 2 0
B=-[
-
0
(2
+
tlz)1/2
- i(2 - f i y 2
I.
Note that in all cases the entries are algebraic integers, but that in case (iii) they do not all belong to a cyclotomic field since [2(fi
+1)]y(2 +
.\/z)1/2
=
21’4,
and the splitting field of x4 - 2 = 0 has a nonabelian Galois group over the rationals. Using (1.47), we can replace A, B by real matrices, but then we lose their property of having algebraic integers as entries. The entries in A and B will be algebraic integers for rational values of ( Y / T and p / x if and only if g = (cos p ) / (sin a) is an algebraic integer. I f (Y = ?r/mand p = r / l , and if M is the least common multiple of I, m, we have
where p m = A1 = M . According to a lemma in the theory of cyclotomic fields, the following is true: Let { = ezri/*. Then i - p will be a unit if v and N are coprime and N is divisible by a t least two distinct prime numbers. If, however, N is a power of a prime number, then ( p - l)/({- 1) will be a unit and, therefore, an algebraic integer. For a proof, see Hilbert (1932). This remark allows us to show in many cases that the matrices in (2.56) have entries which are algebraic integers. Six of our figures may serve to illustrate hyperbolic triangle tesselations. I n Table 2.3, we list the triangles by giving their three angles and the twodimensional space which is triangulated. I n all of these drawings, any two triangles which can be carried into each other by an even number of
90
Discontinuous GTOUPS a d Triangle Tesselations TABLE 2.3
Figure
Angles of tesselating triangle
Tesselated space
15 16 17 18 23 27
7/21 7/3,0 */2, r / 3 , 0 r/2, 7/3,0 0, 0, 0 */2, 7/37 7/7 */2, 7/3, 7/8
Upper halfplane Conic (Klein’s model) Unit disk Unit disk Unit disk Unit disk
reflections (i.e., by an orientation-preserving motion) are either both shaded or unshaded. Figure 20 shows the image of a strip in the upper z halfplane in the w plane where
w
= &ir
The lines are those arising from the tesselation of the x halfplane by triangles with angles n / 2 , r / 3 , 0 (Figure 15).
11.6
Some Subgroups of Hyperbolic Triangle Groups
In this section, we shall give a few typical examples of subgroups of triangle groups. There exists a highly developed theory of discontinuous groups of motions of the noneuclidean plane. However, the proofs of these theorems require sophisticated methods and tools from group theory as well as from analysis and they would not fit into the framework of an introductory text. We shall give a survey of some of the group theoretical results in Section 11.7, where we also shall indicate some of their relationships with problems in the theory of Riemann surfaces. We shall begin with a rather obvious general remark: Lemma2.7 Let I’ be a discontinuous group of planar noneuclidean motions with a canonical fundamental region FR, and let S be a subgroup of r. Then S has a connected fundamental region which consists of the union of congruent replicas of FR. If FR has a finite area a,and if j is the index of S in r, then the fundamental region of S has the area ja. Proof: We take as generators of I’ those which go with the canonical fundamental region. We decompose r into right cosets of S and use as right
II.6 Some Subgroups of Hyperbolic Triangle Groups
91
coset representatives words in the generators which form a Schreier system (i.e., every initial segment of every word presenting a coset representative is itself a coset representative). Using Lemma 2.3, we see that the union U of the images of the fundamental region of H under the action of the coset representatives is connected (in the sense that any two interior points can be embedded in a connected open set within U ) . The action (from the left) of the elements of S on U then carries U onto a set of disjoint (except for boundary points) replicas of U which cover the plane. The theory of Riemann surfaces [see Siege1 (1971) ] provides us with the following information: Any closed Riemann surface of genus g 2 2 can be mapped conformally onto a noneuclidean polygon I&, with 49 sides and vertices, where the sides are identified in pairs. (Since IIro is also part of the complex plane, e.g., of the upper halfplane or the unit disk, there is a natural metric in ILUwhich allows us to define the term "conformal mapping.") lT4, will then be a canonical fundamental region for the fundamental group @, of the Riemann surface which is defined in terms of 29 generators A , and B, ( v = 1, . . ., g j and the single defining relation AlBlA1-'B1-'AzBzA1'B2-'.
(2.58)
*
~AuBuAu-'Bu-'= 1.
The noneuclidean motions A,, B, produce a tesselation with congruent replicas of lLUin such a manner that at every vertex, 4g of these replicas meet and that every interior angle of 114, appears at any given vertex in one of the replicas meeting there. It follows from this remark that we have Lemma 2.8
The area of the fundamental region of 9,is 4 r ( g - 1).
Once we have a representation of aU as a discontinuous group of noneuclidean motion for g = 2, we also have a representation for any g > 2. To show this, we use the Reidemeister-Schreier method to compute generators and defining relations for the normal subgroup @* of %. which consists of all words in for which the sum of exponents in Al is a multiple of n > 1. We find that AlV ( v = 0 , . . ., n - 1) are coset representatives of @* in @2 and that @* is generated by the elements
E,
=
AlvBIAl-Y,
C,
=
A1vA2A1-v,
D, = AIYBzA1-Y,
X
=
Al".
The defining relations of @* then turn out to be (2.59)
E,+lEv-lC,D,Cv-lD~--l= 1
(V =
0, 1 , . . ., n - 2)
XEOX-1En-1-~Cn-1Dn-1C"-1-lDn-l-l = 1. By using (2.59) to eliminate El,. . ., En+ we find that @* is a single relator
92
Discontinuous Groups and Triangle Tesselations
group, the remaining defining relation being
which shows that @* is isomorphic with
cDntl.
Example 1 4 as a subgroup of T ( 2 , 8, 8 ) . The fundamental region of T ( 2 , 8, 8 ) consists of the union of two right triangles with angles 7r/2, r/8, r / 8 . The resulting quadrilateral has again the shape of an isosceles triangle A with angles r/4, r / 8 , r/S, the fourth vertex being the midpoint of the base. The area of this fundamental region is r / 2 . We need eight replicas to obtain a fundamental region with area 27r (as required for a fundamental region of %) , and we choose (tentatively) the octagon which may be described as the star of eight triangles A meeting at a common vertex where each triangle has the angle r / 4 . As coset representatives of a2 in T , we will have to choose the elements By ( v = 1, 1,. . , 7). This still leaves us several choices for the subgroup S of index 8 in T which has these coset representatives. We can choose as generators for S any set
.
(2.60)
C,
=
BYAB-f(,)
(v
=
0, 1 , . . ., 7),
where the integers f(v) form a permutation of order 2 of the integers 0,. . ., 7. [The permutation must be of order 2 since all of the words
BYA2B-
(2.61)
must be expressible in terms of the generators (2.60) by the ReidemeisterSchreier process. This means that the word (2.61) must be equal to
BYAB-f(Y,B,(Y,AB-f','Y,, and therefore we must have f ( f (v) ) 3 v mod 8.1 If v = f (v) for any v, then S will have elements of order 2 and cannot be isomorphic to @Z. Also, % cannot be normal in T,because this would mean that f(v
+ 1) = f(v) + 1mod 8
and then the permutation v +f(v) could not be a product of four twocycles. We now choosef( v) according to the following table: Y:
f(u):
0 1 2 3 4 5 6 7 2 3 0
1 6 7 4 5
11.6 Some Subgroups of Hyperbolic Triangle Groups
93
The relations for the resulting subgroup S are (2.62)
cycf(u)= 1,
~
O
~
~
~
=
1.
Z
~
1
~
4
~
The first eight relations in (2.62) merely state that C , and Cj(v)are inverses of each other. They become redundant if we replace respectively Cz, C3, Ca, C7 by Co-ll Cl-I, C4-ll CS-’,and we are left with four generators and the single defining relation (2.63)
~
~
~
~
-
1
~
~
-
1
~
~ =~
1,4
~
~
-
1
~
4
-
1
~
~
which shows that our subgroup S is isomorphic with @z. The generators A , B and, consequently, the generators CO,C1, C4, c5 of are defined as Mobius transformations. Kowever, these Mobius transformations are given [according to (iii) in Corollary 2.111 by 2 X 2 matrices. We now use (2.63) to prove a result the significance of which will be explained in Section 2.7: Lemma 2.9 If the generators Co, C1, C4, c5 are replaced by the matrices of the Mobius transformations which they represent, then the right-hand side in (2.63) will be the matrix +I (the unit matrix) and not -I. Proof: If we replace the generators C, in (2.63) by their expressions (2.60), the left-hand side in (2.63) is freely equal (in A , B ) to
(2.64)
( ABA-lBA-lBA B )zB-8.
If we replace A , B by the matrices in (iii), Corollary 2.11, and use the fact that A-’ = -IAand ( A B ) s = Bs = -I, wefind that (2.64) goesinto + I . Example 2 aZas a normal subgroup of T (2, 6, 4 ) . The area of the fundamental region of T(2, 6, 4) is r/6. We shall construct (Pz as a normal subgroup of index 24. We start with generators A , B of T satisfying (2.65)
A2
=
=
( A B ) 4= 1
and construct first the normal subgroup of index 2 generated by B and ABA-l = B1.The relations are
=
Bo
Bo6 = Bl6 = (BoB1)’ = 1. Next we construct in this subgroup a normal subgroup of index six generated by C , = BOvB1BO1-” ( V = 0, 1,. . ., 5),
~
~
94
Discontinuous Groups and Triangle Tesselations
which is defined by the relations
c,2 = 1,
c,c,c4c,c,c, = 1.
Finally, we introduce U,+, = C,+lC, (v = 1, 2, 3, 4) and find that the Uv+l generate a subgroup of index 2 in the group generated by the C, which is defined by the single relation (2.66)
Uz-' LTC'
U4-I
U5-I U4 Uz U5U3 = 1.
By a change of generators, this relator can be put into the standard form of the defining relator for a2. We may describe a2as the normal closure of ( B z A ) 2in T. The quotient group T/a2has a center of order 2 generated by ( A B ) 2 .Its quotient group in T / a 2is a dihedral group of order 12.
Example 3 4 as a subgroup of T (2, 3 , 8 ) . The area of the fundamental region of T is t / 1 2 , and 'pz appears as a subgroup of index 48. This case has been investigated thoroughly by Dyck (1880). I n his paper, he shows a drawing (our Figure 27) in which the fundamental region of 4 appears as an equilateral 16-gon with angles which alternatingly have the sizes t / 4 and t / 2. This 16-gon is composed of 48 replicas of the fundamental region of T , which in turn consists of the union of a shaded and a n unshaded triangle. The triangles themselves are fundamental regions for T*(2, 3, 8). If we label the sides of the 16-gon clockwise from 1 to 16 1 and 2v 6 (both taken mod 16), and identify sides with numbers 2v we obtain a closed, orientable two-dimensional manifold of genus 2. For details, see Dyck (1880).
+
+
Example 4 Three subgroups of T(2, 3, 7). Since T is its own commutator subgroup, it cannot have any solvable proper quotient groups. The normal subgroup of lowest index in T is isomorphic with as.Its index is 168, and T / 4 is the simple group PSL(2, Z,), which is the group of fractional linear transformations
where a,/3, y, 6 are elements of the Galois field of order 7. Figure 24 shows the fundamental region of +3 tesselated by 168 replicas of the fundamental region of T , which is the union of a shaded and a n unshaded triangle. The fundamental region of is a regular 14-gon with angles 2?r/7. If we label the 1 sides counterclockwise from 1 to 14 and identify the sides 2v
+
II.7
General Theorems. A Survey and References
95
+
( v = 0,. . ., 6) with the side labeled 2v 6 (mod 14), we obtain a closed orientable surface of genus 3. The points with a n odd label and the points with an even label are respectively represented by a single point on the closed surface. For the details see Klein and Fricke (1890, pp. 369ff). Some subgroups of T ( 2 , 3, 7) which are neither normal nor fundamental groups have been studied by Fricke and Klein as tools for the construction of other subgroups. We mention them here mainly because of the drawings which go with them. Figure 25 shows the fundamental region of a subgroup of index 56 in T(2, 3, 7) and its tesselation by replicas of the fundamental region of T . For details, see Klein and Fricke (1890, pp. 462ff). Figure 26 shows the tesselation of the fundamental region of a subgroup of index 63 . in T. The fundamental region is a regular heptagon with angles ~ / 2 The tesselation is asymmetric since we can reflect the heptagon in a diagonal going from a vertex to the center of the opposite side. This reflection carries the heptagon into itself but not the tesselation. For details, see Fricke and Klein (1897, p. 621). Subgroups of the modular group T (2, 3, c4 ) will be dealt with in Section 3.2.
11.7
General Theorems. A Survey and References
The discontinuous groups described in this chapter are very special cases of large classes of important groups. We shall try to give a brief description of a somewhat more general framework into which the special cases fit, and we shall also mention some other instances in which several of them occur, seemingly without any direct connection with their properties as discontinuous groups. Finally, we shall mention some of the advances of pure group theory by means of which it is now possible to prove certain algebraic results without using the geometric methods which provided the original proofs. However, our account will be confined to results which can be stated in a simple form, using a rather elementary terminology.
Euclidean Space Groups
The discontinuous groups of two-dimensional euclidean motions studied in Section 11.4 are either finite or they have the property of leaving an infinite lattice invariant. By this, we mean that they map the set of points nlxl %x2 onto itself, where xl,x2 are two linearly independent vectors
+
96
Discontinuous Groups and Triangle Tesselations
(the coordinate vectors of two points) and nl, nz are arbitrary integers. These groups are called two-dimensional space groups if they contain two linearly independent translations. In an analogous manner, we define an m-dimensional euclidean space group S, as a group of m-dimensional euclidean motions which map the set of lattice points m
C n,x,
(cr = 1 , . . ., m; n,
E 2)
CI-1
onto itself, where S, contains translations by m linearly independent vectors. The elements of S, can be represented in the form of linear mappings (2.67)
Y-+JY+t,
where J is an m x m matrix with integral entries and determinant f l and t is a vector with m integral components. The vector y is the coordinate vector in euclidean m-space with xl,. . ., x, as basis vectors. The matrices J form a group P, under multiplication which is called the point group of S,. We may consider the group of matrices J as a matrix representation over the integers of the abstract group P,. If this representation is irreducible over the ring of integers, we call S , irreducible. Now we have the Theorem (D. McCarthy) T h e irreducible space groups S, coincide with the class of injinite groups that contain a normal abelian subgroup A , of rank m and have the property that all of their proper quotient groups are Jtnite.
For a proof and for references to the theory of the groups S m , see McCarthy (1968). It has been shown in general by Bieberbach (191 1) that a discontinuous group of m-dimensional euclidean motions must be a space group if it has a fundamental region of finite volume. For a n example of an unusual tesselation of the plane with spiral-shaped tiles, see Voderberg (1937).
The Group GlZOt Dehn (1910) showed that Glzo is the fundamental group of a so-called Poincard space. This is a closed three-dimensional space in which the first homology group is trivial but not the first homotopy group. (Equivalently,
t See Theorem 2.7, Section 11.4.
II.7 General Theorems. A Survey and References
97
we may say that the fundamental group of a Poincarb space is a nontrivial group which coincides with its commutator subgroup. This is obviously true for GI*,,.) Dehn also constructed an infinite set of Poincar6 spaces whose fundamental groups are infinite. Glzo still is the only known finite group which is the fundamental group of a Poincar6 space. It is also the only known (nontrivial) finite group which coincides with its commutator subgroup, and can be presented in terms of two generators and two defining relators.
Test for Discontinuity of Subgroups I't I', the group of noneuclidean orientation-preserving motions, will be presented here as PSL(2, R), that is, by Mobius transformations with real coefficients. Nielsen (1940) showed that a subgroup H of r will be discontinuous if all of its transformations which are not the identity are hyperbolic. His proof uses geometric arguments. A generalization, which has been proved algebraically, is the following.
Theorem (Siegel, 1950) A subgroup of r which has more than two limit points is discontinuous in the upper halfplane i f and only if it contains no injinitesimal elliptic substitutions.
Groups Generated by Reflections
The triangle groups T* of Section 11.3 are very special cases of a wellinvestigated class of groups which arise geometrically if we assume that the reflections in the faces of an n-dimensional polytope generate a discontinuous group with the polytope as fundamental region and the local relations as defining relations. (The space may be euclidean or noneuclidean.) As abstract groups, they can be defined b y a finite set of generators r, ( v = 1,. . ., n ) and defining relations (2.68)
(evp= 1, evr
=
e,, 2 2
(possibly
m)
for
Y
# p).
A survey of the groups (2.68) is given by Coxeter and Moser (1965). For a detailed discussion (with proofs) in the finite case, see Benson and Grove ( 1971). The solution of the word problem for the groups T*( I , m, n)
t See Theorem 2.3, Section 11.2.
98
Discontinuous Groups and Triangle Tesselations
which we stated as Theorem 2.9 (Section 11.5) has been generalized b y Tits (1969) to the case of the groups (2.68). As examples of current research, we mention Vinberg (1967), who studied groups generated by reflcctions in noneuclidean space, and &arcman (1970) , who discussed the representation theory of groups generated b y reflections.
Fuchsian Groups, General Theory
All of the orientation-preserving discontinuous groups of two-dimensional noneuclidean motions discussed in Sections 11.5 and 11.6 are very special cases of fuchsian groups, which can be defined both geometrically and algebraically as follows. Geometrically, a fuchsian group F is defined as a finitely generated discontinuous group of orientation-preserving noneuclidean motions or, equivalently, as a discontinuous group of Mobius transformations which map a circular disk (e.g., the upper halfplane or the disk I z I 5 1) onto itself. If all points on the unit circle are limit points, it is called of the first kind, otherwise of the second kind. (There also exist investigations of infinitely generated "fuchsian" groups.) It is an important and difficult theorem first established by Fricke [see Fricke and Klein (1897, pp. 182190) ] and proved with a different method by Heins (1964), that the groups of the first kind can be defined algebraically as follows. Theorem (Fricke) The fuchsian groups can be characterized by a sequence of exponents e, (v = 1 , . . ., n if n > 0 ) which are integers 1 2 and, possibly, co, and a n integer g 2 0 (the genus). W e call the combination of the e, (whose order does not matter) and of g the signature of F and denote a specific group F by F(e,; g) or, explicitly, ( i f the e, are given numerically) by
F(e1,. . ., en; 9 ) .
(2.69)
+
Then the group F in (2.69) has n 2g generators c,, ax, bx (A less g = 0 ) and the dejning relations (2.70)
cyey= 1,
-
B
clcz- -cn
(a~bxax-'bx-*)
=
X-1
The choice of the e, and of g i s arbitrary provided that n
(2.71)
F ( F ) = 29 - 2
+ C (1 - ev-I) > 0. ".=I
1.
=
1,. . ., g un-
II.7
General Theorems. A Survey and References 99
The number p ( F ) i s called the “measure” of F. The area of the fundamental region of F i s r p ( F ). I n a representation of F as a group of Mobius transformations, the c, are represented by elliptic transformations if e, < 0 0 . For e, = co, the c, correspond to parabolic transformations. The ax, bx are always represented by hyperbolic transformations. The groups T(1, m, n ) discussed in Section 11.5 are the groups F(1, m, n ;0 ) . The fundamental groups agappearing in Section 11.6 are the groups F (- ; g ) . Like the free groups, the fuchsian groups (as abstract groups) form a class of groups in which simple subgroup theorems hold. For instance, every subgroup of finite index in a fuchsian group is again a fuchsian group (the same is true for the subset consisting of the group aq).If F is a fuchsian group and S one of its subgroups of finite index y, then (2.72) P(S) = YP(F). A finitely generated normal subgroup of a fuchsian group is also of finite index. [For this and related results, see Greenberg (1960) .] Until recently, all proofs were based either on theorems of noneuclidean geometry [e.g., (2.72), which simply states that the fundamental region of S has y times the area of that of F j or on matrix theory or on simple topological arguments, in particular on the remark that the fundamental groups of unramified covering spaces of a given (two-dimensional) space are the subgroups of the fundamental group of the given space. The more sophisticated theorems in the theory of fuchsian groups, in particular the question of when two groups given by systems of generators and relators of type (2.70) are isomorphic, can still be approached only by using nonalgebraic tools [see, for instance, Wilkie (1966) or Macbeath (1967)l. However, Hoare et al. (1971) proved (2.72) for fuchsian groups and their subgroups purely group theoretically, defining a fuchsian group by a presentation of type (2.70). Also Curran (1972) showed that p ( F ) 2 0 is a necessary and sufficient condition for F to be infinite, using the methods of homological algebra. A remarkable new result on fuchsian groups [which are now defined abstractly by a presentation (2.70)] was proved algebraically. We have the
Theorem (Hoare et al., 1972) A subgroup of infinite index in a group with a presentation (2.70) i s a free product of cyclic groups.
If one of the exponents ev in (2.70) equals 0 0 , the group F itself is a free product of cyclic groups and the theorem follows from the Kurosh sub-
100
Discontinuous Groups and Triangle Tesselations
group theorem. However, if co > e, 2 2 for all v , F is not a free product of cyclic groups unless it is cyclic itself, in which case g = 0, n = 1.
Torsion-Free Subgroups of Finite Index in Fuchsian Groupst
Fenchel conjectured that every fuchsian group has a subgroup (and, therefore, a normal subgroup) of finite index which is torsion free. Such a subgroup will then either be free or a fundamental group 9,( 9 > 1). [That the 9, are torsion free is a special case of a result found by Karrass et al. (1960).] The first complete proof of this conjecture was given by Fox (1952). Fox constructed permutation groups in which the generators have prescribed orders and satisfy the relations of a given F. Using then a generalization of Theorem 2.10 which states that in all fuchsian groups the elements of finite order are conjugates of powers of the cy in (2.70), he proves Fenchel’s conjecture. Mennicke (1967,1968a) showed that Fenchel’s conjecture can be reduced to the case of the triangle groups T(1,m, n) and then proves it for them by finding a 3 X 3 matrix representation for these with entries which are algebraic integers. Using then a suitably large prime ideal J in the ring of these algebraic integers and taking the representation mod J, he obtains the same result. For the T(1, m, n ) , Feuer (1971) found a particularly simple proof by using 2 x 2 matrices with entries from a suitable Galois field to obtain a representation of T ( l , m, n ) in which the generators and their product have the prescribed order. All of these proofs rely on the generalization of Theorem 2.10 for fuchsian groups, for which, so far, no complete proof has been obtained without the use of the theory of fundamental regions for fuchsian groups. However, if one merely assumes that the abstract fuchsian groups defined by a presentation (2.70) have a faithful matrix representation in terms of finite matrices with complex entries, then Fenchel’s conjecture is answered by the following.
.
Theorem (Selberg, 1960) Let M , ( p = 1,. . , r ) be finitely many nonsingular matrices with entries from the field C of complex numbers. Then the group generated by the M , has a torsion-free subgroup of finite index. Unfortunately, no simple proof is known which shows the existence of a finite faithful matrix representation of abstract fuchsian groups over C. It is therefore of interest that there exists a purely algebraic proof of
t See Theorem 2.1.0 and Corollary 2.10.
II.7 General Theorems. A Survey and References
101
Corollary 2.10 in the cases where all of the exponents e, of a fuchsian group (2.70) are sufficiently large [see McCool (1968) and Schupp (1970)l. Schupp’s proof is based on a new method which also sharpens earlier results found by Greendlinger (1960). These in turn offered the first purely algebraic proof for the simple solution of the word and conjugacy problems for the groups 9, which Dehn (1912) had given, using arguments from noneuclidean geometry.
Homogeneous Representation of Fuchsian Groups?
The generators of fuchsian groups-defined now as discontinuous groups of noneuclidean motions-are given as fractional linear transformations of a complex variable z. Each transformation is represented not by one but b y infinitely many 2 X 2 matrices M since M and AM define the same Mobius transformation. Fricke and Klein (1897, pp. 200-209) discussed the following question. Suppose, for a fuchsian group F with abstract presentation (2.70) , we can find a representation in terms of Mobius transformations such that the generators c,, ax, bx are, respectively, represented by transformations with the 2 X 2 matrices C,, Ax, Bx. Will it then always be possible to find complex numbers y y , ax,PXsuch that (2.73a) (2.7313)
(YYcY)ev =
-
1,
0
II (
yiCiyL’z* *ynC%
d x )
(PxBx) (WAA)-’(PABA)-’ = 1,
A-1
where I is the 2 X 2 unit matrix? It is obvious that the ah, can be chosen in any manner without influencing the result, and that the answer to Fricke’s question is “yes” if n > 0 and if a t least one e, = 00. I n the remaining cases (either n = 0 or all e, finite), Fricke states that we cannot only satisfy (2.73a) (which is obvious) but that we always can satisfy (2.7313) if n is even, in particular if n = 0. Example 1, Section 11.6 verifies this result for n = 0 and g = 2 in a particular case. Since we know (see proof of Lemma 2.8) that a2contains a subgroup isomorphic with 9, which is of index g - 1 in %,we can show that Fricke’s result is correct a t least for one particular 9, for all g 2 2. Fricke’s proof of his result rests on what he calls “considerations of continuity.” They are of a geometric nature and involve continuous deformations of the normalized fundamental regions of the groups F with the same
t See Example 1, Section 11.6.
102
Discontinuous Groups and Triangle Tesselations
signature. His arguments are not sharply focused and look plausible rather than rigorous. A modern proof, at least for the case n = 0, has been given by Bers (1958, 1965). This proof also uses a continuity argument. However, it is based on a sophisticated and rigorously founded theory which was not available to Fricke. We mention briefly that the problem raised by Fricke was studied in great generality by Schur (1904),who investigated the connection between the homogeneous and fractional linear representations of a group G . Schur showed that this connection is given by an abelian group associated with G which was later called the Schur multiplier and now is called the second cohomology group of G (over C). For details see MacLane (1963).
Signi$cance of the Groups T(2, 4, 6 ) and T(2, 3, 7)t
Let Q(t,, tz, t3) be a ternary quadratic form with integral coefficients. Assume that Q is indefinite but that Q is a nonzero form, i.e., Q # 0 for integral values of tl, tz, tB which are not all zero. Let S be the (symmetric) matrix of Q and'let U be the multiplicative group of all matrices M with integral coefficients and determinant f1 such that
MtXM
=
S.
We call U the group of units of Q, and we denote its quotient group with respect to the subgroup consisting of f I (I = unit matrix) by Uo. According to Section 1.4 (definition of Kz), Uo is a group of projectivities of projective two-space which leaves a quadratic form (1.52) fixed. According to Lemma 1.14 (Section 1.4), this group is induced by a group of noneuclidean motions, represented by conformal selfmappings of the upper halfplane. We denote by Uo+the subgroup (of index 1 or 2) in Uowhich is induced by proper (orientation-preserving) selfmappings of the upper z halfplane. We call Uo+the associated group of units of Q. Now we have the (Mennicke, 196813) Let Q be an inde$nite, nonzero ternary quadratic form with integral coeficients. Then the associated group of units of Q is in all but ajinite number of cases a subgroup of T(2, 4, 6 ) . In the case of the quadratic form
Theorem
t?
- 3122 - 3t32,
the associated group of units i s exactly T(2, 4, 6).
t See Examples 2 and 4, Section 11.6.
II.7 General Theorems. A Survey and References
103
We have here an example of a number theoretical definition of a fuchsian group. Chapter I11 will deal with other examples of this nature. The significance of the triangle group T ( 2 , 3, 7) is based on the following
Theorem (Hurwitz, 1893) A Riemann surface R, of genus g 2 2 with a fundamental group a, admits at most 84(g - 1) conformal selfmappings. The selfmappings form a .finite group H which i s a quotient group of a fuchsian group F with respect to a normal subgroup isomorphic with a,. The order of H assumes its maxinzal value 84(g - 1) if and only if F i s isomorphic with T ( 2 , 3, 7).
If H is of maximal order, it is called a Hurwitz group. The Hurwitz groups are the nontrivial finite quotient groups of T ( 2 , 3 , 7 ) . The group T ( 2 , 3, 7) plays such a distinguished role because the area of its fundamental region is smaller than that of any other fuchsian group (Hurwitz, 1893; Greenberg, 1963). To indicate the proof, we add the following remarks : The canonically dissected Riemann surface R, can be mapped conformally onto the fundamental region of a,, such that the elements of a, act as a discontinuous group of conformal selfmappings of the unit disk 1 z 1 < 1. The conformal selfmappings of R, also induce conformal selfmappings of z 1 < 1, and together with those of ag1 they must produce a discontinuous group F since H is finite. The fundamental region of F must therefore produce a subtesselation of that of and the order 1 H j of H is therefore the quotient of the areas of these two fundamental regions. For the case F = T ( 2 , 3, 7), we find from Lemma 2.8 (Section 11.6) and Fricke’s theorem (in this section) that I H 1 = S4(g - 1). [For proofs of the underlying results from the analytic theory of Riemann surfaces, see Siege1 (1971).J It is not automatically true for all fuchsian groups F (and not even for all triangle groups) that a normal subgroup of finite index j > 1 is a group ag. However, this is true if F = T ( 2 , 3, 7 ) ,because of Corollary 2.10 (Section 11.5) since the normal closure of any one of the cyclic subgroups of orders 2, 3, or 7 in T ( 2 , 3, 7) is the whole group. We can use Theorem 2.11 (Section 11.5) to show that there exist infinitely many Hurwitz groups. From the remarks after Corollary 2.11, it follows that the entries in the matrix representation (2.56) for T ( 2 , m, 1) are not always algebraic integers if m = 3 and 1 = 7. However, the denominators in the entries are powers of either 2 sin(s/3) or of 2 sin(?r/7), according to our arrangements, which permit us to choose either ?r/3 or a/7 for a. Since 2 sin(?r/3) = 16and since u = [2 sin(a/7) 1” satisfies the equation ua - 70‘ f 1 4 ~ 7 = 0,
*,,
104
Discontinuous GTOUPSand Tria?$e
Tesselations
it follows that the algebraic integers 2 sin(r/3) and 2 sin(r/7) divide, respectively, the rational prime numbers 3 and 7. Consider now the algebraic number field K which is generated by the entries of a representation (2.56) of T(2, 3, 7 ) , and let r o b e a prime ideal of K . Since ?ro cannot divide both 3 and 7, we may assume that the denominators in the entries are coprime with ?ro. Taking now the elements of these matrices mod ?rol we obtain a representation of T(2, 3, 7) as a subgroup of the group PSL(2, q) of Mobius transformations with coefficients in a Galois field of q = p' elements where p is the uniquely determined rational prime number that is divisible by ro. It is easy to see that this representation is never the trivial one, and thus defines a Hurwitz group. These remarks could be used to show that, in fact, infinitely many groups PSL(2, q ) are Hurwitz groups. However, we shall not pursue this line of investigation, since the question has been settled already with a different method. We ha,ve the Theorem
(Macbeath, 1968)
The group PSL(2, q ) is a Hurwitz group if
(i) q = 7, (ii) q = p = f l mod7, (iii) q = p3, where p = f 2 or f 3 mod 7,
and for no other values of q. Macbeath also lists the lowest values of g which can occur. They are 8 = 3 for q = 7; g = 7 for q = 8 ; a n d g = 14for q = 13. The caseg = 3, q = 7 is illustrated by Figure 24. That there are other Hurwitz groups, in particular all alternating groups
A , for a sufficiently large n, is a much quoted but unpublished result of G. Higman. In Section 111.2, the problem of finding the finite quotient groups of T(2, 3, co ) (the modular group) will be discussed in detail. Every Hurwitz group is, obviously, also a quotient group of the modular group.
Exercises 1. Case 6 of Section 11.4 deals with the group T(4, 3, 2 ) , which contains the dihedral group T (2, 2,3) as a subgroup of index 4. Use Figure 4a or 4b to find a fundamental region for T(2, 2, 3). [Hint: This fundamental region will be the union of replicas of the fundamental region of T(4, 3, 2) corresponding to the coset representatives of T(2, 2, 3) in T(4, 3, 2).1
Exercises
2.
105
Enumerate the different ways in which a square can be considered as a tile of a canonical tesselation of the euclidean plane. Enumerate the possible groups of motions arising from the various constructions and show that only one of them is abelian and without elements of finite order.
3. Find a fundamental region for the group generated by the two proper euclidean motions a, b defined by a:
Z'
= EZ,
b:
Z'
=z
+ 1,
where B is a primitive sixth root of unity. 4.
Using Figure 23, show that a quadrilateral with four equal sides and angles 2 r / 3 and 2 r / 7 , respectively, a t the two pairs of opposite vertices can be used as a tile for a tesselation of the hyperbolic plane but that this quadrilateral is not the canonical fundamental region of any discontinuous group of hyperbolic motions. [Note that either one of the diagonals of the quadrilateral divides it into two congruent parts each of which is a fundamental region for T (2, 3, 7) .]
5. Let
I, m, n be positive integers such that 1 1 1 -+-+-
Let S*, T* be respectively the Mobius transformations with matrices S, T. Show that p > 1 and that S*, T* generate a group in which the relations *T*1n s*l = T*m = (8 -1 hold. Show also that I , m,n are, respectively, the exact orders of S*, T*, S*T*. 6 . Show that S*, T* in Exercise 5 map the interior of the unit disk onto itself, and, by computing the fixed points of S*, T*, S*T* within the unit disk 1 z I < 1, show that these are the vertices of a noneuclidean triangle with angles r/l, r/m, ?r/nand that S*, T* actually generate the triangle group T(1, m, n).
106
Discontinuous Groups and Triangle Tesselations
7. Show that the triangle group with generators S, T and defining relations s7 = T2 = (ST)a = 1 has a (nonnormal) subgroup H of index 7 with coset representatives 1, S, P, .. ., SB and find generators and defining relations for H . 8. Let I, m, n be odd positive integers 2 3 and let p be a prime number such that p - 1 is divisible by I, m, n. Show: There exists a group of 2 X 2 matrices with entries from the Galois field of p elements, generated by two matrices A , B such that A , B, and A B have respectively the exact orders I , m, and n. [This leads to the construction of a torsion-free subgroup of T(E, m, n) . See Feuer (1971) for generalizations.]
Chapter
I11 Number Theoretical Methods
111.1 The Modular Group
As usual, we denote the ring of integers by 2 and the group of fractional linear substitutions 2’ =
+ cz + d
az b -
(ad - bc
=
1; a, b,
C,
d E 2)
by PSL(2, 2) and call it the modular group or (an older notation) the elliptic modular group, since it appears in the theory of elliptic functions. PSL(2, Z) is obviously discrete and therefore (Theorem 2.3) discontinuous in the upper halfplane, which is mapped onto itself by the elements of the group. We shall also need the homogeneous modular group SL ( 2 , 2), which is the multiplicative group of the matrices
[:
3
(ad
- bc = 1;
a, b, c, d E 2).
The center of SL(2, Z) is of order 2 and consists of the matrices * I , where I is the 2 X 2 unit matrix. PSL(2, 2) is the quotient group of SL(2, 2) with respect to its center. We need first
108
Number Theoretical Methods
Lemma 3.1 SL(2, 2) is generated by (3.3)
A=[;
-3
0 B=[l
-1
Proof: We define
(3.4) and observe that A-'B = T I and A-'B2 = Tz, and T1-'TZ = B and T,T2-lT1 = A-l. Therefore, S L ( 2 , 2) is generated by A , B if and only if it is generated by TI, Tz. Multiplication of the matrix in (3.2) on the left by TIXreplaces a by a Xc and leaves c unchanged. Similarly, multiplication by T P replaces c by c pa and leaves a unchanged. If I a I 2 I c 1, we can choose Xso that 1 a Xc 1 < 1 a 1, andif I a 1 I I c 1, wecanchoosepsothat 1 c p a 1 < 1 c I. Therefore, we can, by repeated multiplication of the matrix in (3.2) by a word in T I , Tz, obtain a matrix which has one of the two forms
+
+ + +
(3.5) Since the determinant of these matrices must still be +1, we have a' = d' = f l or b" = -c" = f l . Therefore, the first matrix in (3.5) is of one of the forms TI@or A2Tl@and the second one is of one of the form ATP or A3T16,where p = f b ' , 6 = f d " . Therefore, we can express (3.2) in terms of A , B or of T l , T2. We shall denote the Mobius transformations (3.1) corresponding to the matrices A , B, T1,Tz respectively by a,8, 71, r2 and we shall prove
Theorem 3.1 SL(2, 2) is defined by the relations (3.6)
A2 = B3,
A4 = 1,
and P S L ( 2 , 2) is defined by the relations (3.7)
a2
=
p3
=
1
for its generators a,p. Therefore, PSL(2, 2) is isomorphic with the triangle group T ( 2 , 3, 00 ) .
Since it is clear that A2 = -I is the only element #I in S L ( 2, 2) that induces the identical transformation (3.1), and since (3.6) shows that A2
111.1 The Modular Group
109
is of order 2 and in the center of SL(2, 2) [regardless of the possibility of other relations not derivable from (3.6) which may hold for SL(2, Z)], it follows that (3.7) will be defining relations for PSL(2, 2) if the relations (3.6) define SL(2, 2).It also follows from (3.6) that every word in A , B can be written in one of the following forms:
P(T1, Tz)Ael
(3.8)
AP(T1, Tz)Ael
where P ( T I ,T 2 )is a word in Tl, Tzwith nonnegative exponents only and e may assume any one of the values 0, 1, 2, 3. We have proved Theorem 3.1 if we can show that (3.8) cannot be the unit matrix unless P is the empty Tz (i.e., unless all exponents of Tl and Tzin P are zero). I n that word in T1, case, the expression (3.8) becomes a power of A , and it is indeed true that A is of order 4, as stated in (3.6)) and not of a lower order. To prove that P ( T1,T z ) equals I only if P is the empty word, we define the action of a matrix (3.2) on a quadratic form
Q(x, Y)
=
Lx2
+ MXY+ Ny2
by the mapping
Q(x, Y)
+
&(ax
+ by, cx + dy)
=
Q'(x, Y) = L'x2
+ M'XY+ N'y2,
where L', M', N' are homogeneous linear functions of L, 211,N. I n particular, we have, respectively, for T I and T2
TI:
L'
=
L,
Tz:
L'
=
L
Therefore, if L
M'=2L+M,
+ M + N,
M'
=
M
+ 2N,
N'=L+M+N,
N' = N.
> 0, N > 0, and M 2 0, we have in both cases L' + N' > L + N .
+
Applying now P(T,, T 2 ) to the special form x2 y2 = QO, we see that, unless P is the empty word, the resulting form QO' cannot be Qo since the sum S' of the coefficients of x2 and y2 in QO' will be greater than 2. Therefore P ( T I ,T z )cannot be the unit matrix. But the same is true for the expressions in (3.8) ,because the action of A does not change the sum of the coefficients of I and y2. This completes the proof of Theorem 3.1.
Theorem 3.2 T h e discontinuous group PSL(2, Z), whose action on the upper halfplane i s defined by (3.1), has the following fundamental region FR: Let z = x iy, and let FR be dejined by
+
(3.9)
-g<x<3,
x P + y 2 2 1.
Number Theoretical Methods
110
+
T h e points z = i and z = ( - 1 i 6 > / 2 are, respectively, theJixed points of a and 8. If we view FR a s a noneuclidean quadrangle with vertices at the points p , ( V = 1 , . . ., 4), where (3.10)
p1
=
pz
00,
=
(-1
+i6)/2,
p3
=
i,
and denote, respectively, the sides with vertices p,, p,l maps sz onto s3 and a@ m a p s s1 onto s4.
(V
p4
=
(1
+ i6)/2,
mod 4) by s, then a
+
Proof: We show first that every point zo = xo iyo, yo > 0, can be mapped into FR by an element of P S L (2, 2). Let z,, = xn iy,, be a sequence of images of zo under the action of elements of P S L (2, 2) which is defined as follows: zzm+l
=
+ iy2m+1 = xzzm - d2m + iy2m
+
( m = 0, 1, 2 , . . .)
~zm+l
where dz, E 2, z2,,,
=
xz,,,
+ iyzm
=
I
~2:2m+i
I F B, ( m = 1, 2, 3,. . .).
-l/~z,,,-~
Obviously, zzm+l is the image of z2,,, under the action of rl-d*m and zzm+2 is the image of z~,,,+~under the action of a. If it happens, for a finite value of n, that 2, is in FR, Theorem 3.2 is true. Othenvise, the nuykers zZ,,,+lare all in the strip -f 5 x 5 3, and I zZwl 1 < 1. We claim that in this case Y2*3
> Yzm+1
since yZm+3 = yZm+Z
=
yZm+l/j
&m+l
1'
and 1 Z Z ~ I I< 1. It follows that the are distinct and that there exists a convergent subsequence of the zzm+1whose limit is in the upper halfplane, and this, by definition, contradicts the fact that PSL(2, 2) is discontinuous (Section 11.2). Now we shall show that two interior distinct points of FR cannot be equivalent under P S L (2, 2). Suppose that zo = xo i y and ~ zo' = XO' iyo' are both in FR, that yo' 2 yo, and that
+
+
+
b zo' = azo ___ CZO
+
+d .
+
Then yo' = y o / ]czo d l2 and I czo d l2 2 czyoz. It follows from yo' 2 yo that c2y025 1 and since yo2 2 3 (because yo E FR), we find c = 0 or c = f l . The case c = 0 is trivial since then a = d = f l , and either b = 0 (in which case zo' = zo) or I x i I 2 3. For c = fl,we find
I czo + d 1' >
(*XO
+
2 ( 1 d I - BI Z
111.1
The Modular Group
and therefore by the same argument as before, I d I = 0 or Without loss of generality, we may assume c = fl. Then z; = a
- 1/z0
or
z;
=a
Id 1
111 =
1.
- l/(zo f 1).
If zo E FR, both zo and zo f 1 are outside of I z 1 < 1. Therefore - l/zo and -l/(zo f 1) are inside the disk I z I 5 1, and parallel translation of this disk in the direction of the real axis by multiples of 1 produces only regions which have no interior point in common with FR. (The situation is illustrated by Figure 15, which will be discussed after Corollary 3.2 in some detail.) The remaining statements of Theorem 3.2 involve only simple calculations, which will be omitted. We also have: Corollary 3.2 The noneuclidean rejlections dejined by the selfmappings pll PZ, p3 of the upper z halfplane (3.11)
pi:
Z’
= -2;
pz:
Z’ =
1/E;
p3:
Z’ =
-1 - 1;
generate the group T*(2, 3, GO) (Theorem 2.8, Section 11.5), which has T ( 2 , 3, w ) = P S L ( 2 , 2) as a subgroup of index 2. The relations are (3.7) and (3.12)
PI&?=
p2p3 =
8,
p3p1
=
71-’,
p? =
1.
The fundamental region of T*(2, 3, w ) i s the left half of the fundamental region of P S L ( 2 , 2) as defined in Theorem 3.2, after bisection by the y axis. The proof of Corollary 3.2 consists in elementary calculations. Figure 15 shows part of the tesselation of the upper halfplane with replicas of the fundamental region FR* of T* ( 2 , 3 , w ) .The regions are shaded or unshaded according to the parity of the number of reflections (3.11) which are needed to carry the original FR* onto the particular replica. The cusps of the replicas of FR are the images of z = co under the action of P S L ( 2 , 2).They coincide with the rational points of the real axis [the points whose coordinates are a/c, a, c € 2.Since for every coprime pair a, c, there exists a matrix (3.2) , these are all of the rational points]. This shows that P S L ( 2 , 2) is not discontinuous on the real axis. It is the simplest ilIustration of the fact that discontinuity of a group depends on the space on which the group acts. Figure 17 arises from Figure 15 by mapping the upper halfplane and its tesselation conformally onto the unit circle.
112
Number Theoretical Methods
111.2
Subgroups and Quotient Groups of the Modular Group
The subgroups of the modular group and the quotient groups of the normal subgroups of finite index have been studied extensively. The origin of the interest in this topic is to be found in the theory of the elliptic modular functions and its connection with the theory of algebraic equations with a given Galois group. The most comprehensive presentation of the algebraic aspects of the theory seems to be that of Fricke (1926). Klein and Fricke (1890, Vol. 2) should also be mentioned, although there the function theoretic problems dominate the scene. For later developments of the theory and for references, see Newman (1972). We shall discuss first three simple examples which are connected with the drawings reproduced in the text and then prove a few results referring to the concept of “principal congruence subgroups.” Finally, we shall report briefly on nonparabolic subgroups. They appeared first implicitly in a paper by Neumann (1933). Since the modular group is the free product of two cyclic groups (of order 2 and 3, respectively), the abstract nature of all of its subgroups is governed by the Kurosh subgroup theorem, which states that they are free products of cyclic groups of orders 2, 3, or C Q , and that the free factors of finite order are generated by conjugates of a and p. At the same time, the number theoretical definition of the modular group allows us to define a n infinite number of its subgroups by imposing arithmetic conditions on the entries of the matrices (3.2). We define the mth principal congruence subgroup of PSL(2, 2) as the subgroup of elements with matrices (3.2) where (3.13) a = d = f l m o d m ,
b=c=Omodm
( m E 2 ; m > 1)
and denote it by r,. By extension of this definition, we shall also write for PSL(2, 2) itself.
rl
Example 1 Free subgroups of index 6 . r2is a free group freely generated by r12and r 2 (Section 3.1). Its fundamental region in the unit disk consists of
the union of a shaded and an adjacent white triangle, as shown in Figure 18. The central shaded triangle is, in turn, the union of the six triangles meeting at the center of the unit disk as shown in Figure 17. r2is the normal closure of r12in rl;the quotient group rl/rz is therefore defined by its generators a,/3 and the defining relations (3.14)
a2
=
p3 =
(ap)2 =
1.
It is isomorphic with the dihedral group of order six, which is also the group
111.1 Subgroups and Quotient Groups
113
PSL (2,2)of fractional linear substitutions in one variable with coefficients in a Galois field of two elements. To prove these statements, we start with Figure 15, which shows the tesselation of the upper halfplane with triangles of angles z/2, ~ / 0;3 each ~ of these is a fundamental region of T* (2,3, 03 ) . We choose the six triangles id3/2 as coset representatives of a submeeting a t the point zo = -4 group S* of T*. According to Corollary 3.2 (Section I I I . l ) , these triangles may be considered as representatives of the elements
+
(3.15)
pZ,
p31
p2p3i
pZp3PZ7
pZp3pZp3,
all of which have xo as a fixed point. The elements (3.15)are indeed coset representatives of a normal subgroup S* of T*(2, 3, a),namely of the normal closure of pl. Consequently, S* can be generated by forming the conjugates of pl by the elements (3.15). The Reidemeister-Schreier method then shows that the relations of T* allow us to present S* as the free product of the three cyclic groups of order 2 generated by (3.16)
u1
= pl,
UZ
= P3plp3,
u3
= pZP3plP3pZ.
The subgroup S of index 2 in S* that consists of words with a n even number of symbols ul, uz, u3 (and, therefore, of orientation-preserving selfmappings of the upper halfplane) is then freely generated by dluZ = (Plp3)' =
712j
(TI03
=
(&')'
=
72'.
This subgroup is of index 12 in T* (2,3, 00 ) (since S* is of index 6 ) , and therefore S is of index 6 in PSL ( 2 , Z ) . The relations (3.14) must hold in the quotient group of S and since they determine uniquely the only noncommutative group of order 6 [which is known to be both dihedral and isomorphic with PSL(2, a)], our algebraic statements are proved. As for the geometric statements, we may map the upper halfplane conformally onto the unit disk. The six triangles meeting a t zo then go into the six triangles meeting a t the center of the unit disk as shown in Figure 17. They fill a triangle with angles 0, which is reproduced as the shaded central triangle in Figure 18 and which is a fundamental region for S*. One of the sides of this triangle is the image of the part of the imaginary axis in the upper halfplane, going from z = i to z = 03. Since p1 is the reflection in the imaginary axis. (Corollary 3.2), the rest of our geometric statement is immediate. It remains t o show that 712 and 72' actually generate r2. (They are obviously contained in rz.) Now S has six coset representatives in PSL(2, 2) which, according to (3.14), can be chosen as 1, a, P, P2,
4 4 .P2.
114
Number Theoretical Methods
None of these elements except 1 is represented by a matrix defining a n element of r2.Therefore, S = r2. Since F2 is free of rank 2, it follows that the modular group contains free subgroups of every finite and of countably infinite rank. r2is a maximal free subgroup, but, it is not the only one, since the commutator subgroup I’ll of I’l is also free of rank two. It is freely generated by the commutators of the elements of the free factors, that is, by ffpff-1p-1
= ffpffp,
apa-1p-2
=
ffp%Yp,
and also of index six but with a cyclic quotient group. Neither of its generators is in r2. There exists a relationship between rl’ and the solutions of the diophantine equation (3.17)
22
+ y2 +
22
=
xyz.
Fricke’s theory of the characters of fuchsian groups (which is outlined briefly in Chapter IV) shows the following: If 71, y2 are any two generators of rl‘,and if x, y, 2 are, respectively, the traces of the matrices of the substitutions 71, y2, y1y2,then x, y, z satisfy (3.17). It has been proved by Rosenberger (1972) that conversely all solutions of (3.17) b y positive integers x, y, z can be obtained in this manner. Relations between “Markov triples,” that is, triples of integers a, b, c satisfying a2
+ b2 + c2 = Sabc,
and rl’,and the connection between Markov triples and the automorphisms of a free group of rank 2 were discovered by H. Cohn (1968). Example 2 Subgroups of index 12 in T*(2, 3, 00). (Figures 19a, 19b; Dyck, 1882). The shaded large quadrilateral Q in Figure 19b appears also in Figure 17, where it is subdivided into 12 shaded and unshaded triangles with angles r/2, r/3, 0. Its vertices on the unit circle I z 1 = 1 have the coordinates z = -i, e+/6, eir/‘j, i. The reflections in the sides of Q generate a group with four generators of order 2 and no other relations; it contains a free subgroup of rank 3 which is of index 2. The central shaded quadrilateral Q1 in Figure 19a is again the fundamental region of a group with four generat)ors of order 2 (the reflections in the sides of Q1) and no other defining relations. The resulting group G12 is not itself a subgroup of T*(2, 3, a),but it is conjugate with such a subgroup of index 12 in T* within the group of conformal selfmappings of the
fII.2 Subgroups and Quotient Groups
115
unit disk. To show this, we merely have to map the quadrilateral Q* in Figure 17 onto Q1 by a conformal selfmapping of the unit disk such that the vertices -i, e--is/6, i, -eirI6 of Q* go, respectively, into the vertices -i, 1, i, -1 of Ql.We sce that Q* is again tesselated by 12 triangles with angles T j 2 , */3, 0. Example 3 Fundamental region of r,. (Figure 2 2 ; Klein and Fricke, 1890, Vol. 1, p. 373). The congruence subgroup r7 (see Theorem 3.3) is of indcx 168 in rl. Its fundamental region will consist of 168 replicas of the fundamental region of the modular group and is made up of 336 shaded and unshaded triangles. Figure 22 shows a strip S in the upper halfplane of width 3, part of which is tesselated by 24 shaded and unshaded triangles of angles ~ / 2 ,~ 1 30., Let S' be the reflection of S in one of its vertical sides. We tesselate S' again by shading the images of the unshaded triangles in S and vice versa. The union S* of S and S' is then (in part) tesselated by 48 triangles. These form a fundamental region for a nonnormal subgroup of index seven in the modular group. By combining S* with 13 of its copies 1 (I = 1,.. ., 13), we obtain a arising by the parallel translations z' = x fundamental region for r7.The details are complicated, but the proof is clearly outlined by Klein and Fricke (1890). Let ml, mz, ma,. . . be an infinite, monotonically increasing sequence of C positive integers such that m, divides m,,l for v = 1, 2, 3,. . .. Then rmVtl rnp,and the intersection of all the r,, is the unit element. Therefore, the rn, define a subgroup topology for rl = P S L ( 2 , 2) in the sense of Hall (1950). This implies, in particular, that if y1 # y2are two distinct elements of rl,then thcre exists an integer m such that y1 and y2 have distinct images The example of rz (dealt with in Example 1) in the quotient group rl/rm. might induce one to assume that the rmare actually a basic set of normal subgroups of finite index in r in the sense that every such subgroup contains a r,. For it is indeed true that every normal subgroup of finite index in rl contains the elements rlm and 7 . p for a sufficiently large m. This follows from the observation that the fundamental region of every such subgroup must have a cusp and that therefore the subgroup has a parabolic substitution. (Observe that the fundamental region of a subgroup consists of finitely many replicas of that of rl,which has a cusp.) Now every parabolic substituion is, in rl, conjugate with a power rlm of rl, and r1 is, in turn, a conjugate of r2-l. However, for m 2 6, it is not true that rmis the normal closure of rim (Reiner, 1961). Part of this result (namely Theorem 3.3) seems to have been known for a long time, according to a remark by Klein (1880) (however, I have not been able to find a contemporary proof for it).
+
116
Number Theoretical Methods
We shall prove:
Theorem 3.3 There exists a normal subgroup N of the modular group which i s of$nite index and contains none of the principal congruence subgroups r,. The most remarkable feature of Theorem 3.3 is the fact that it shows that the two-dimensional unimodular group is an exceptional case. Bass et al. (1967) and, independently, Mennicke (1965) showed that the group XL(n, Z) of n x n matrices with integral entries and determinant +1 has, for n > 2, the following properties: (i) All normal subgroups of finite index contain a principal congruence subgroup. (ii) The only normal subgroups of infinite index are finitc and belong to the center.
It is clear that PSL(2, Z) , and therefore XL(2, Z), does not have property (ii) , since its second commutator subgroup would provide a counterexample because the first commutator subgroup is free and of rank 2. It is more difficult t o prove that the modular group does not have property (i) , and we shall prove this fact (i.e., Theorem 3.3) by an application of the Jordan-Holder theorem. For this purpose, we shall establish three lemmas. Lemma 3.2 Let Z, be the ring of integers mod nr, and let PSL(2, Z,) be the group of Mobius transformations (3.1) with a, b, C, d Zm. Then the quotient group rl/rmof the mth principal congruence subgroup in the modular group is isomorphic with PXL ( 2 , Z,) . Lemma 3.3 The only nonabelian quotient groups that can appear in a composition series of PSL(2, Z,) are the groups P S L ( 2 , Z,), where p is a prime number. [This implies p 2 5 , in which case PSL(2, Z,) is simple.) Lemma 3.4 The alternating group All on 11 symbols is a quotient group of the modular group, and All is not isomorphic with any group P S L ( ~z,,) = rl/r,. The kernel of the homomorphic mapping rl-+ All provides the subgroup described in Theorem 3.3 because the simple group A11 cannot appear as a quotient in a composition series of any group rl/rm.
1II.g
Subgroups and Quotient Groups
117
Proof of Lemma 3.2: We shall deal with the homogeneous modular group SL(2, 2) of matrices ( 3 . 2 ) rather than with I’l and we shall prove Lemma 3.2 by showing: Let a, b, c, d be integers representing residue classes mod m such that ad - bc = 1 mod m. Then there exist integers a‘, b’, G‘, d’ such that
a
E
b = b‘,
a’,
c
= c‘,
d
= d‘mod m,
a’d’ - b‘c’ = 1.
From this it follows that the natural mapping of S L ( 2 , 2) i n t o S L ( 2 , 2,) [which arises by replacing the integers in a matrix of SL(2, 2) by their residue classes mod m ] is, in fact an onto mapping. Let g be the greatest common divisor of a and b, and let a* = a/g and b* = b/g. Then there exist (according to Euclid’s algorithm) integers c*, d* such that a*d* - b*c* = 1. We have
b]r*:I][“”
Now let a”d”
[:
=
d
1
+ km and (c”
c*
- k)d” = t.
”]”
k
d“
Then
] - [1 0]
1 0 t
1
modm,
0 1
and the matrix (3.18)
has the desired property. Proof of Lemma 3.3: We show that SL(2,Zm)is the direct product of the groups SL(2, 2,))where q denotes any one of the distinct prime powers qy = p , ” ~the product of which equals m. Let again a, b, c, d be integers representing residue classes mod m, such that ad - bc = 1 mod m, and let a,, b,, c,, d, be integers such that,
= a,,
b
d , = 1,
b,
a
b,,
c
= 0,
c,
E
= c,, E
0,
d
= d,
d, = 1
mod qy modq,, for
1.1
z v.
According t o the “Chinese remainder theorem,” the a,, b,, c,, d, exist and
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Number Theoretical Methods
are uniquely determined mod m. Let
Then the product M of the M , has the property to be congruent M , mod qy for all V, regardless of the arrangement of the factors. Therefore,
M E
[z
:]modm.
Since ad - bc E a,d, - b,c, = 1 mod q., we have a unique decomposition of the elements of SL(2,2,) into a product of pairwise commuting matrices M , which define elements of SL(2, Z q V )Therefore . SL(2, 2), is indeed the direct product of the SL(2, Z q r ) , and a quotient group that appears in the ), must also appear as a quotient group in a composition series of PSL ( 2 , 2 composition series of an SL(2, 2,). Next we show that the quotient groups of a composition series of SL(2, 2,) are all abelian, with the possible exception of PSL(2, Z,), which is the quotient group of SL(2, 2,) with respect to its center. The proof is based on the remark that SL (2, Z,v+i) contains a normal abelian subgroup N , whose quotient group is SL ( 2 , Z,.), where N , consists of the matrices P, defined by
up7
”’
1 - xp’
] mod
pU+l,
in which A, p , u are representatives of residue classes mod p . From direct calculations we see that the P, form an abelian group of exponent p and rank 3 which is normal in SL(2, Zpy+l), and that M-lM* is a matrix of the type of P, if M , M* t SL(2, Z P y + 1 ) and M = M* mod p’. This completes the proof of Lemma 3.3. Proof of Lemma 3.4: We observe first that the simple group A11 cannot be isomorphic with any group PSL(2, Z,), because the latter group is known to be of order p ( p 2 - 1 ) / 2 , and p is the greatest prime number dividing its order. Since 11 is the greatest prime number dividing the order of All, we would have p = 11. But All has order 1 1 ! / 2 ,which is larger than 11 ( 1l 2- 1 )/ 2 . Now we show that A,, is a quotient group of the modular group PSL(2, 2). To prove this, we merely have to show that A11 can be generated by a.n element a* of order 2 and an element p* of order 3. Using
ZII.2 Subgroups and Quotient Groups
119
the standard notation for a permutation as a product of cycles, we choose (3.19a)
a* = (1, 5) (6,
7) (8, 10) (9, 11)
(3.19b)
8* = (1, 2, 3) (4, 5, 6) (7, 8, 9).
Then 8*a* = (1, 2, 3, 5, 7, 10, 8, 11, 9, 6, 4).
Since p*a* is an ll-cycle, the group generated by a*,8* is certainly transitive. Since 11is a prime number, the group is also primitive. If we can show that it contains a three-cycle, it must be All or Xll (the symmetric group), according to a theorem about primitive permutation groups [see Wielandt (1964, Theorem 13.3, p. 34)]. Since a*, P* E All, we merely have to compose a three-cycle out of these permutations to complete the proof of Lemma 3.4. Now p*-la*fl*
p*(Y*p*-1a*
=
(2,s)(478) (7,111 (9, l o > ,
= (175, 11, 10, 6 , 7 , 879) (3,4),
(,*a*j3*-1a*)4= (1,6) (5,7) (8, 11) (9, lo), y* =
p*-'(~*p*(p*a*p*-~a*)~ = (1, 6, 2) (4, 11, 5, 7, 8).
Therefore y*5 = (1, 2, 6) is a three-cycle. This proof of Lemma 3.4 is based on unpublished work by Constance Davis. A much more general result has been proved by Miller (1901), who showed that all alternating groups A , for n 2 12 are quotient groups of PSL(2, 2). Miller uses Bertrand's postulate, according to which, for any positive integer n, the sequence n 1, n 2,. . ., 2n contains at least one prime number. Dey and Wiegold (1971) have shown that all symmetric and alternating groups on n 2 9 symbols can be generated by two elements of order 2 and 3, respectively, and are therefore quotient groups of PXL(2, 2). They do not use Bertrand's postulate, giving explicit expressions for the generators. However, they suppress practically all calculations. We already mentioned that every subgroup of finite index in the modular group rl contains parabolic substitutions (remark preceding Theorem 3.3). There exist, however, nonparabolic subgroups of infinite index. We have :
+
+
Theorem 3.4 A subgroup A of rl i s a maxinzal nonparabolic subgroup if either me of the following two conditions i s satisjied;
(i) The elements rlX(A = 0, f1, f 2 , . right coset representations of A.
. .) forma complete spstem of distinct
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Number Theoretical Methods
(ii) Consider the homogeneous group A* generated by the 2 X 2 matrices dejining th,e elements of A. Then every ordered pair of coprime integers a, c appears exactly once as the pair of elements in the jirst column of one of these matrices.
Condition (ii) implies (i), and (i) implies (ii) i f a (Theorem 3.1) i s in A. Neumann ( 1933) investigated subgroups of the homogeneous modular group SL(2, 2) which are defined by condition (ii). As a motivation, he refers to the work of Schmidt on the foundations of geometry, in which a construction of a group A* containing the matrix A in (3.3) is needed. Neumann showed by a n explicit Construction that the cardinality of the distinct groups A* containing A is that of the continuum. His results will be summarized here. To prove Theorem 3.4, we first state the near-trivial Lemma 3.5 Let T be a parabolic element of rl. Then T has a fixed point with a rational coordinate a/c, where a, c are coprime integers. If p rl has the first column entries a, c, then p%p is a power of T ~ .
Proof: p maps 00 onto a/c. Therefore p%rp is parabolic with fixed point (We read from right to left, since we are using matrices.) Now we show: (ii) implies that A is maximal nonparabolic. First, we observe that A cannot contain a parabolic substitution T. Otherwise, it would contain a power of T~ (Lemma 3.5). But then any elements p and j u l k for a n appropriate k would be elements of A which have matrices with the same first column. It follows that the cosets A7LX must all be distinct since otherwise a power #1 of r1 would be in A. Next, it follows that the cosets A n X contain all of rllbecause, given an element y E rllthere exists a n element p in A which has the same first column in the corresponding matrix as y,,and p-ly would be a power of rl. This also shows that A cannot be contained in a larger nonparabolic subgroup and that (ii) implies (i) . By using the same argument, it follows that a group A satisfying (i) is maximal nonparabolic. First, if (i) is satisfied, A cannot contain a parabolic element T.Otherwise, there would exist an element y = p 7 I X ( p E A ) in one of the cosets of A such that 00.
y-'Ty
=
71-h/L-1Tp71h
would be a power # 1 of T~ (Lemma 3.5). But then p&rp E A would be a power # 1 of 71, which contradicts (i). Next, adjoining any element y E rl to A would mean adjoining an element p~~ (A # 0 ) , and the enlarged group A would cease to be nonparabolic.
I I I . 2 Subgroups and Quotient Groups
121
To show that (i) implies (ii), m-e have to assume that A contains an element of order 2, e.g., a. Because then the square of the matrix defining the element of order 2 is the negative unit matric, and if a, c are the entries in the first column of an element of A, the entries -a, -c also appear in the first column of an element of A. Our proof will be complete if we can show: Given a rational number r, there exists a matrix in any A with property (i) whose first column entries a, c are such that a/c = r. Such a matrix p certainly exists in rl. If p E A n x , then p ~ E~ A -and~ has the same first column as p. This completes the proof of Theorem 3.4. We cannot claim that all of the possible maximal nonparabolic subgroups are described by Theorem 3.4. It could happen that such a subgroup has more right coset representatives than the r I hbut cannot be enlarged to a subgroup which has only the rlx as coset representatives. The construction of the groups A satisfying (i) of Theorem 3.4 is based on the following. Lemma 3.6 Let X .+ f (X) ( X C 2) be a permutation of the integers which satisfies the conditions
(3.20) f ( f ( x )
=
X,
m - 1)
=
1
+ f ( f(x) + 11,
f(o) = 0.
Then f ( X ) defines uniquely a group A satisfying (i) of Theorem 3.4 and containing a. The generators and defining relations of A are, respectively, (3.21) yx
= ~ i ~ a ~ i - f ( ' ) ,~ x ~ f ( x=)
1,
~ x ~ f c x ) + i ~ x - i - '= 1.
Proof: Since a and T~ generate rl,the Reidemeister-Schreier method shows that a group A must be generated by elements yx if (i) of Theorem 3.4 is satisfied. The conditionf (0) = 0 is equivalent to a C A. The condition f ( f ( X ) ) = X in (3.20) arises from the fact that (3.22)
71xa27,-x
= r l A a T l - f ( x ) ~ l j ( x ) a ~ ~ - ~ ( ~ ( x ) )= 1
which also proves that yxyfcx) = 1. The last condition in (3.20) and the second set of defining relations in (3.21) arise from the fact that 1,
T ~ ~ ( ( Y T ~ = ) ~ T ~ - ~
together with an application of the relations yxy/(x) = 1. That different permutations j ( X ) define different subgroups A follows from the remark that yx is defined by the matrix (3.23)
122
Number Theoretical Methods
Since the first column of this matrix determines yx uniquely as an element of A, it follows that the matrix (3.23) cannot appear in a subgroup A’ belonging to a function f’( A) if f ( A ) # f’(A). The main difficulty consists now in the construction of permutations f (A) which satisfy (3.20). Since the original paper by Neumann (1933) is not easily accessible, we summarize its results as Neumann’s Theorem Let hl ( 1 = 1, 2, 3,. . .) be a n infinite sequence in which each hl i s either 0 or 1. Let kl = 1 - hz and define gt by
Then gl+1 - gl = 1 if hz = 1 and gl+l - 91 = 6 if hz = 0. Every integer 1 2 can be represented in exactly one manner as a n expression of one of the forms gz,
(82
+cr)h
(u = 1, 2, 3,4, 5 ) .
The permutations of the integers defined in Table 3.1 satisfy the conditions (3.20). Diferent sequences hl define diflerent permutations and different subgroups A with the properties stated in Lemma 3.6. TABLE 3.1
The proof of Neumann’s theorem now merely requires verification. The 1 is denoted by /3 in Neumann’s paper. function f (A )
+
IIZ.3 Groups of Units 111.3
123
Groups of Units of Ternary Quadratic and Binary Hermitian Forms
We shall report in this section on some groups of noneuclidean motions which are presented as fractional linear substitutions of z, leaving a circle in the z plane fixed, and which are discontinuous within the circle. The coefficients of the substitutions in these groups will be elements of quadratic or multiply quadratic number fields, and they will define homogeneous linear substitutions which leave certain hermitian or quadratic forms invariant. The restricting conditions on the Coefficients will be number theoretical. The results will be presented without proofs. Most of them have been derived by Fricke and Klein (1897, Part 3, pp. 502-634) with geometric methods. Some of the results have been derived in a more detailed manner in the papers by Fricke (1890, 1891). Our report is confined to a discussion of the simpler parts of the theory and to the presentation of some explicit formulas covering special cases. It does not nearly cover all of the material by Fricke and Klein (1897).
Unit Groups of Ternary Quadratic Forms
We consider a ternary quadratic form 3
(3.24)
f ( t Y )
= f(tl, tz, t 3 )
c
= Y
bYJYtLl
(b, =
bpv)
,P-1
where the b, are integers, and we call a linear substitution 3
(3.25)
t, =
c uv,rtr’
(v = 1, 2, 3)
Ll-1
with integral coefficients a,,,, a unit off if 3
3
(3.26)
f (C UY,PtLl’)
c bY,,t”’t,’
=
p-1
Y
44-1
identically in the indeterminants t”’. Using matrices, this means that (3.27)
AtBA
=
B,
where A , B denote, respectively, the matrix (u,,,) and the symmetric matrix (by,). If det B # 0, we have det A = f1,and the matrices A with det A = +1 form a group G ( f ) which is called the group of units off and which is a
124
Number Theoretical Methods
subgroup of the group SL(3, 2) of 3 X 3 matrices with integral coefficients and determinant 1. If f is positive definite, the group G( f ) is finite. We shall consider only the special indefinite forms
+
f = [ p , q, r ]
(3.28)
= ptI2 - qh2 - rt32,
where p , q1 r are positive integers which are coprime in pairs and none of which is divisible by the square of a prime number. In Chapter I, Section 4, it is shown that a Mobius transformation induces a linear transformation of three variables AI, A,, A3 which carries the quadratic form A1A3 - AZ2into itself [formula (1.5.4)]. By transforming [ p , q, r ] into this form in the variables A,, we can show: Lemma 3.7 Let B [ p , q, r] be the diagonal matrix with entries p , -ql -r in the main diagonal and let a,0, y, 6 be real numbers with a6 - , 8 = ~ 1. Then the 3 X 3 matrix A [ p l q, r ] defined by
(3.29)
!
+ B2 + + 62) (P/q)’’’ x (a7 + Pa)
+(a2
~ [ pq,, r]
=
+
(q/P)1’2(aP 76) + ( r l P > ” 2 ( a Z- B2
Y2
+(p/r)1’2(a2
+
82
a6
+ Br
+ Y2 - 6 9
(r/Q)1f2
x (w- B6) ( q / r ) l I 2 ( a @- 76)
- y2 - 62)
+(a2
- Y2 + 6 9
satisfies (3.27), and every A with this property and det A form (3.29). It follows that the Mobius transformation (3.30)
Z’ =
+ YZ
+6
(a,8, y , 6
- 6%
=
-
+1 has the
real; a8 - BY = 1)
defines a unit of [ p , q, r] whenever a,P , Y, 6 are chosen in such a manner that the entries of A [ p l q, r ] are rational integers, and that all units of A can be obtained in this manner. Fricke and Klein (1897, p. 537) state the exact conditions which a,/3, y, 6 have to satisfy if (3.30) is to define a unit of [ p , q, r ] . We shall give these conditions only in the special case where p = r = 1, without giving a proof, as Theorem F1 Let q be square free and let q = qlq2 be a n y decomposition of q into the product of two positive integers ql, qz. Then the Mobius transformation
III.3 GTOUPS of Units
(3.30) defines a unit of [l, q,
125
11if and only if
(i) q is even and (3.31)
a!
=
a(ql)1/2, p
=
b(qz)1'2,
y =
c(qZ)ll2,
6 = d(q1)'l2,
c(qz)'/zl
6 = d(q1)'/2,
where a, b, c, d are rational integers and qlad - q2bc = 1.
(3.32) (ii) q is odd and either a(ql)1/2, @
a! =
( 3I331
a
=
+b +c +d
b(q2)lI2,
3
y =
0 mod 2,
qlad - q2bc
=
1,
or (3.34)
a!
=
~ ( q ~ / 2 ) l /0~=, b ( q ~ / 2 ) ' / ~y,
=
c ( q ~ / 2 ) ' / ~6, = d ( q 1 / 2 ) ' / ~ ,
and qlad - qzbc = 2. I n both cases, a, b, c, d are rational integers. The basis for the construction of a fundamental region of the unit group G ( [ l , q, 11) is the following.
Theorem FZ
The Mobius transformation
(3.35)
+ + +
(where a b c d 3 0 mod 2 i f q i s odd) define a normal subgroup H of finite index in G ( [ l , q, 11). The group H i s called the principal subgroup of G ( [ 1 , q, 11). I t i s conjugate with the subgroup H , of the modular group defined by the matrices (3.36)
[iC3
(ad
- qbc
= 1).
Since H , is of finite index in the modular group (in other words, since the modular group and G([1, q, 11) are commensurable) , the construction of a fundamental region and of generators for G([1, q, 13) can be derived from our knowledge of the modular group. For every q, a fundamental region for H , consists of finitely many replicas of the fundamental region of the modular group. By conjugation with the matrix
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Number Theoretical Methods
H , goes into H . Since D is a dilatation of the plane with z = 0 as its center which increases the values of I z 1 in the ratio 1 : 1 q Ill2, we can choose as a fundamental region for H a contraction of the fundamental region of H,. Since G ( [l, q, 13) is a finite extension of H , we have then to subdivide the fundamental region FR* of H into finitely many parts in order to obtain the fundamental region FR of G. Since the matrix
v1=
[-1
O
'1
0
belongs to G(C1, q, l]), we will start our construction by considering the intersection of FR* with its image FR** under the action of V1. This construction is illustrated in part for q = 6 in Figure 21, which shows two superimposed tesselations arising from the one of the modular group. The unit circle, which will appear as part of the boundary of FR, is not drawn in Figure 21. Nor would it give FR, since H is of index eight in G([l, 2, 11). Details will be found in Fricke and Klein (1897, p. 549). Whereas there exists a systematic approach for the construction of the fundamental regions of the groups G ( [l, q, 13) , the situation gets much more involved in the general case G ( [ p , q, r]) since most of these groups are not any more commensurable with the modular group. We shall reproduce here the algebraic and number theoretical results stated by Fricke and Klein (1897, pp. 546-565) for some special cases without proofs. The important general result found by Mennicke (1968b) has already been mentioned in Section 11.7. A glance at the formulas to be presented here confirms Mennicke's theorem insofar as it shows that in the abstract groups G ( [ p , q, r ] ) , which are given explicitly, the elements of finite order have indeed orders which can appear in a subgroup of T(2, 4, 6). Note: We shall list the generators of various groups G([p, q, 7-3) by giving the 2 X 2 matrices which define them. However, the defining relations are those satisfied by the corresponding fractional linear substitutions.
(i)
G([1, 6, 11). Generators:
v l = [ -1O
07 ,
v2=
l a 11'
[o
III.3 Groupsof Units
127
DeJining relations : v 1 2
(ii)
=
G(C1, 5, I]).
v1=[
v 3 2
=
v 4 2
=
v,v,v,v,
1,
=
1.
Generators:
1,
1 2 6
0 1 -1 0
v2 =
co
117
(iii) G(C3, 1, 11). Generators:
v,=q
]
1 1
f i - 1 1 '
v - - [1 "fi
',"I,
0 I-*
I. DeJining relations : VI4 =
Vz2 = V3'j
=
1,
VIVzV3
=
1.
This group is isomorphic with T(2,4, 6 ) . It is conjugate with the group (i) in Corollary 2.11 (Section 11.5). As a triangle group, it does not contain any parabolic substitutions. Therefore, it cannot be commensurable with the modular group since every subgroup of finite index in the modular group contains parabolic substitutions. This follows from the fact that the fundamental region of a subgroup of finite index consists of finitely many replicas of the fundamental region of the whole group. (iv)
G(C3, 5, 11). Generators:
1,
0 1 0
v1=[ -1
128
Number Theoretical Methods
Defining relations : v12 = vz2 =
v3fJ=
$742
=
1,
v,v,v3v4
= 1.
(v) G(C3, 13, 11). Generators:
(vi) G(C7, 1, 11). Generators:
v1=
" 7,
v5
-1
1
,
1
0
V3=/i"-3
1
v4=-[
2/2
2/5+31 0
2-2/5
-1
(vii) G(C11, 1, 13). Generators:
v , v5 = q -1
1 l],
Dejining relations : v14 = vz2 = v32= v 4 3 =
1,
v1vzv3v4 =
1.
I I I . 3 Groups of Units
(v%)
G([15, 1, 11). Generators:
v , = q 2/2
.=-[1
l],
-1
0
4 3 4 3 - 6
1
0
.\/3+.\/511
Defining relations : v14= vz2=
(ix) G ( [1, 3, 7)).
v1=[
v3z=
v42
Generators:
=
1,
v1v2v13v4
--[1
0 2 - ffil - 3
-1O 0 l],
=
1.
47+31, 0
(x) G(C7, 5, 11). Generators:
v1= [- 1O
7,
0
[
vz = -
d1 f l - 03
@ +0 3 i 1
129
130
Number Theoretical Methods
(xi) G(C23, 1, 11). Generators: 1 1
2/2
-1
1
v - “
4-z
0
- + 50i 7
fi-5
v 5 = 1- [ 4 2
3 9
9 - 2 a
-3+
2
~
Defining relations: v14= v23= v32= v42 = v52=
1,
v1v2v3v4v5
=
1.
Davis (1964) has constructed a finite extension of G ( [ p , 1, 13) in the case where p is a prime number and p = 1 mod 4,which acts transitively on the triplets of coprime integers t,, h, t3 satisfying pt12
- t22
-
t32
=
0.
Unit Groups of Some Indefinite Hermitian Forms
We shall consider the linear substitutions (3.37)
21
= a21’
+ pzz’,
22
= yz1’
+
822’
(a8 - p y = l ) ,
where a, 0, y, 8 are ganssian integers (i.e., numbers of the form n
+ im,
I I I . 3 Groups of Units
131
where n, m are rational integers), which have the property that, for a positive integer D, (3.38)
~iZi- DzzZz = ZI’Z~’- Dxz‘Zz’,
identically in the complex variables zl, x2. The group of Mobius transformations
+ +
z’ = ___P YZ 6
with gaussian integers a, P, y, 6 for which (3.38) holds will be called the group of units of the hermitian form zlZl- Dx2Z2 and will be denoted by HD. The groups HD are subgroups of Picard’s group (Theorem 2.2, Section 11.2), but in contradistinction to Picard’s group, H D is discontinuous in the disk 1 x I 5 which it maps onto itself (Fricke and Klein, 1897, p. 471). According to Fricke and Rlein (1897, p. 474) , the group of units of an indefinite binary hermitian form with gaussian integers as coefficients and with determinant D is commensurable with the modular group if and only if it contains parabolic substitutions. It is shown (Fricke and Klein, 1897, p. 475) that this will happen if and only if the congruence x2 13 0 mod D has a solution, that is, if all prime factors of D xhich are =3 mod 4 appear an even number of times in the factorization of D. In a few special cases, explicit results are available. We have:
+
Theorem (Fricke-Mennicke) of the Mobius transformations
For D
=
3, 5, 7, the group H D consists
where a , P are gaussian integers, and where a bar denotes the conjugate complex number. Fricke and Klein (1897, pp. 477-490) give generators for H s and H7 (in the case of D = 7, only for an extension H?* of H7 in which H , is of index two) and describe t,he presentation of Hs and H7 as abstract groups in terms of generators and defining relations. For D = 3, Mennicke (1961) shows that Hli is generated by six elements V , ( v = 1 , . ., 6 ) given by the matrices
.
132
Number Theoretical Methods
which satisfy the defining relations v12 == v22= v32= v42= v52=
V62 == 11
v1vZv3v4V5v6 =
1.
It follows that H 3 has a subgroup of index 2 which is isomorphic with the fundamental group +2 (Section 11.6). Fine (1973) has shown that Picard's group contains a subgroup isomorphic with @2 which has two loxodromic generators. This subgroup cannot map any circular disk onto itself, and it is not known whether it is discontinuous anywhere in the complex plane and, if so, what set of limit points it has. Generators and defining relations for this subgroup are
.=I-'
2
- 3+4i 2-2i
C = [ 3 + 2i
1
4i
-2
B
1
1,
=
[;
-12i]>
D=[-l-2i
-1-2i
2+2i
ABCDA-'B-'C-'D-'
=
2-4i -1+6i
1.
Exercises
1. Consider the group H3 defined in the Fricke-Mennicke theorem (end of Section 111.3) and its generators V1,. . ., v6. Show th a t the elements of H3 map one particular circle onto'itself and that H3 contains a subgroup of index 2 isomorphic with a2.Show that H3 is conjugate (within G ) with a group with real coefficients and show that these can be chosen so that they all lie in a quadratic number field. 2. Assume that H is a subgroup of finite indexn in the modular group rl, such that H has a fundamental region which has exactly one boundary point on the real axis. Show that H is conjugate with a subgroup whose coset representatives are z' = z
Show that for n
=
+
3 and n
0, 1 , . . ., n - 1). 4,such a subgroup H actually exists.
(T =
T
=
Exercises
133
3. Show that the permutations
A
= (1, 3, 5 ) (2, 4, 7) (1, 2) ( 3 , 4) ( 5 , 6>1 generate the symmetric group on seven symbols, which therefore is the quotient group of the modular group with respect to a normal subgroup N . Show that N does not contain any principal congruence subgroup of the modular group (Drillick, 1971).
4.
=
It follows from the Kurosh subgroup theorem that all elements of order 2 in the modular group are conjugate. Show that this has the following consequence: The images of the point z = i under the action of the modular group are exactly the points (a+i)/c, where a, c are integers, c > 0, and c is a divisor of a2 1. Hint: Consider the fixed points of elements of order 2. [The exercise is implicit in a paper by Taussky (1949) .] z =
+
Chapter
IV Misce I lanv J
IV. 1 Examples of Discontinuous Nonfuchsian Groups
A discrete subgroup of P S L ( 2 , C) which is conjugate with a subgroup of P S L ( 2 , R) has the property that all of its limit points are on a circle, e.g., the real axis. (A limit point is a point of accumulation of the images of any given point under the action of the group elements.) Such a group will then be discontinuous both inside and outside of the circle carrying the limit points. The set of limit points is a perfect set which may or may not be dense on a circle. However, we are not concerned here with these more subtle questions. We want to present a few examples of subgroups of PSL(2, C) which, although discontinuous in a part of 6, will have sets of limit points which do not lie on any fixed circle but form complicated point sets like a nowhere differentiable Jordan curve. We shall present mainly material developed by Fricke and Klein (1897, Part 11, Chapter 3, pp. 399-445). For some modern developments see Maskit (1965a, b, 1968, 1971b) and the survey article by Bers (1973). The construction of the groups to be considered is based on a result found by F. Klein: Let U , and V,, ( V, p = 1, 2,. . .) be, respectively, generators of discontinuous groups U and V such that the boundary of a fundamental region of either group i s contained in the fundamental region of the other group. T h e n the U , and V,, together generate a discontinuous group which i s isomorphic with the free
I V.1 Examples of Discontinuous Nonfuchsian Groups
135
product of U and V and has the intersection of their fundamental regions as fundamental region. Klein called this construction the composition of groups. We shall not give a general proof, but merely a few examples. Figure 28 shows two pairs of circles. All four circles are disjoint. The part of the plane outside of the first pair of circles is the fundamental region of an infinite cyclic group generated by a n element V1 which maps the exterior of the first circle of the pair onto the interior of the second circle of the pair. Similarly, the exterior of the second pair of circles is the fundamental region of an infinite cyclic group generated by the element V z which maps the exterior of one circle onto the interior of the other. The intersection of these fundamental regions has been shaded. It is the fundamental region of a free group V freely generated by V1 and Vz.I n this case, it is easy to see that the action of any element of V , represented by a freely reduced word W in V1, V,, sends the common (closed) exterior E of the four circles onto part of the interior of one of the circles. Therefore, not any two interior points of E can be mapped onto each other by a n element of V . This means that E is a t least part of a fundamental region of V . That it is all of it can be shown by proving that the images of E under the action of V leave only a set of measure zero in C uncovered. An outline of such a proof is given by Frickc and Klein (1897, pp. 126-131). The groups considered here are the simplest cxarnples of so-called Schottky groups. For details, scc also Lehner ( 1964, pp. 118-1 19) . The free group generated by V1, V 2need not be fuchsian, even if the four circles in Figure 28 have a common orthogonal circle (in which case the group may be fuchsian). We can choose V1 or V 2 or both as loxodromic substitutions, and, since the trace of a loxodromic element is not real, it cannot be conjugate with an element of P S L ( 2 , R). By construction, VI and V zwill never be elliptic or parabolic if the closed disks bounded by the four circles are completely disjoint in pairs. A more complicated example of “composition” of groups is illustrated in Figure 29. The interior of the large crescent is the fundamental region of a cyclic group of order n, generated by an element Vz which maps the largcr arc of the crescent onto the smaller one; the angle between the arcs is 2 7 / n . Similarly, the common exterior of the small circles inside the crescent is the fundamental region of a group of order m, where 2 7 / m is the anglc between these circles. The generator Vl of this group maps one exterior arc of one circle onto the exterior arc of the other. Vl and V2together generate a free product of cyclic groups with defining relations Vlm
=
vz72”= 1.
The shaded region is a fundamental region of this group.
136
Miscellany
We shall discuss in detail a single type of groups which are defined in a particularly simple manner but have a set of limit points which consists of a continuous, simple curve on which both the points where the curve is not differentiable and the points where the curve has a tangent are everywhere dense. Such curves had been shown to exist before, but they had not appeared in any “practical” problem of analysis. A passage from Poincar6 (1899) [a longer passage is quoted by Nin e (1972) ] illustrates this remark: “In former times when one invented a new function it was for a practical purpose; today, one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that.” We shall follow closely the presentation given by Fricke and Klein (1897, pp. 415-428), which is very lucid and of amazing simplicity. It seems that it has not been reproduced anywhere else. We have made it more elementary (but a little longer) by proving directly some statements which Fricke derives from general but rather difficult theorems. The publication of PoincarB’s remark after the appearance of Fricke and Klein’s work is somewhat puzzling since Poincar6 is one of the founders of a systematic theory of discontinuous subgroups of P S L (2, C) , and it is hard to explain why he should not have known Fricke’s presentation or, if he did know it, why he should have thought it to be irrelevant. We shall prove: ( Y = 1, 2, 3, 4) be four circles in the finite part of the complex plane such that C, and C,+1 (with Cs = C,) have exactly one point in common and the open disks bounded by the C, are pointwise disjoint. A s s u m e that there does not exist a circle which i s orthogonal to all of the C,, and let R, denote the reflection in C,. T h e n the group ll generated by the R, has the following properties: I t i s discontinuous everywhere in the plane outside of a set P of limit points which lies entirely within the closed disks bounded by the C,. The common exterior of the C, together with the C, but without their points of contact forms a fundamental region FR for D. The set P has the following properties:
Theorem 4.1 Let C,
(i) P has two-dimensional measure zero, (ii) P i s a one-one continuous image of the unit circle (and m a y be considered as a simple closed curve). (iii) T h e following types of points are everywhere dense in P : Points f,, in which P has a tangent, points f h , in which vectors f r o m the point to points in an arbitrarily small neighborhood o n P enclose an angle of a size bounded away f r o m 0 and T, and points f l such that P winds itself infinitely m a n y times
IV.1
Examples of Discontinuous Nonfuchsian Groups
137
around them. T h e points f,, fh, f i are, respectively, $xed points of parabolic, hyperbolic, and loxodromic elements of D. P r o o f : Every element # 1 of D can be written as a word (4.1)
Wn = R,,Rt&,..-Rt,
(iy= I1, 2, 3, 411,
where i, # iyelfor v = 1 , . . ., n - 1. (By that, we mean that we carry out the reflections R,, in the order in which they appear, R,, first, etc.). We call n the length of W , and we shall write simply
. ., in)
(4.2)
(iy),
or
(il,.
for W , if no confusion can arise. We have: Lemma 4.1 Distinct words W , define distinct elements of D, which therefore has the defining relations R,2 = 1 (i = 1, 2, 3, 4 ) . The subgroup D+ of D that consists of elements represented by words of even length is freely generated by the parabolic elements
Xi
=
R1R2,
Xz
=
RzR3,
X B = R3R4
of PSL(2, C).
P r o o f : If U , V are two words of type (4.1),possibly of different length but in any case not identical, then U-lV can be changed (by using R,2 = 1) into a word Wnwith n > 0. But W nwill never represent the identity because it defines a mapping of C onto itself under which FR (the common exterior of the C,) is mapped into the interior of one of the C,. Therefore, the R? = 1 are defining relations. That X,, X z , X3 and also (4.3)
Xq
=
XiX2X3
=
RiR4
are parabolic follows from a simple calculation where we may assume that onecircleisgivenbyIz( = l a n d t h e o t h e r b y x = l + i t ( - w < t < m ) . The fixed point A , is the common point of C, and C,+I. There are 4 . 3n-i distinct elements W,. We consider now their action on FR. The circles C, form what may be called a simple chain of circles, which we shall denote by Lo. They are shown in Figure 38. The closed disk bounded by C, is denoted by i in this drawing. Now we reflect the circles Cj for j # i in C,. The resulting circles form again a chain, which we call L,. The closed disk arising from Cj by reflection in 6, is denoted by the symbol j , i. The chain L2 is formed by the closed disks with symbols starting, say, a t 1, 2 and going via 4, 2 and 3, 2 to 1, 4, etc., and ending with 2, 1. Each C, contains now three disks j , i ( j # i) in its interior (see Figure 3 8 ) . Now we reflect all disks with symbols j , i in Ck(k # i ). The resulting image
138
Miscellany
will be denoted by j , i, k. Then the disks j , i, k for j # i # k will be contained in the closed disk i, k. By induction, we see that the image of the closed disk bounded by Gj under the action of W , in (4.1) and denoted by (4.4)
. .
.?,21,. . ., in
(j #
il
#
- - - # in)
is contained in the closed disk denoted by 21,*
* *
, zn,
each of which contains the three disks corresponding to the three values of il. The disks (4.4) form the chain L,, which consists of 4.3" closed disks which are arranged in cyclic order such that each disk has exactly one point in common with the preceding disk and this point is distinct from the point it has in common with the succeeding disk. The fundamental region FR has a boundary consisting of two quadrilaterals Qo and Q, (the boundary lines being circular arcs) which have their vertices Ai in common. Qo is finite and Q , is the boundary of the part of the z plane which contains z = CQ . The images of Qo under the action of the Ri are indicated in Figure 38. We have
.j #
Lemma 4.2 Any two distinct points in the interior of Qo or of Q- are nonequivalent under the action of the elements 6 # 1 of D. The boundary points of Qo and Q , other than the vertices Ai are equivalent to themselves under the action of exactly one 6 # 1, which is then one of the Ri. Proof: Any word W , of D with n > 1 represents an element which maps Qo and Q, inside the chain Ln.-l,which, for n 2 2, has only the points A i with Qo and Q , in common.
Lemma 4.3
Let a(L,) be the area covered by the disks whose union is
L,. Then (4.5)
lim a(L,)
=
0.
n-00
Proof: We use the fact that reflection in a circle is a conformal mapping (apart from the inversion of the orientation). Therefore, it is locally a similarity transformation consisting of the shrinking or expanding of the distances between a point and those in a sufficiently small disk around it by a distortion factor 6 . For points having a distance r d from the center of the reflecting circle with radius r, we find for any d 2 0
+
(4.6)
6=
+ d)2 5 1.
?-2/(r
I V.1 Examples of Discontinuous Nonjuchsian Groups
139
Now consider the four circles Ci whose radii are respectively ri. Let C* be a disk or the interior of a polygon bounded by circular arcs outside of C i and such that the distance of any point of C* from C ; is a t least d > 0. Let a ( C * ) be the area of C* and let 6o = max[ri2/(ri
+ d ) 2 ] < 1.
Then the image of C* after reflection in Ci has a n area at most equal to 602a(C*).Now we apply these remarks to cy(L,). We obtain L,+1 from L, by reflecting (for i = 1, 2, 3, 4) the part of L, not inside C ; in the circle Ci. Therefore, a(L,+l) 5 cr(L,). Let L,* be that part of L, that remains if we remove four disks with radius p and centers Ai from the plane. Then the points of L,* outside Ci have a minimal distance d = d(p) from Ci. The area of that part of Lnil that originates from reflecting points of L,* is a t most 602a(L,*).The area of those parts of L,+l that originate from the reflection of points inside the disks of radius p with centers a t Ai is a t most 4rp2 since the area of a region outside of a circle is never increased by reflection in the circle. Since, for any fixed p , 60 = 60(p) < 1, and
(4.7)
lim6 P(p )
=
0,
n+w
we can enclose Ln+lin an arbitrarily small area if n is sufficiently large. This proves Lemma 4.3.
Lemma 4.4 The set P of limit points of D consists of the points common to all chains L,. Let s be the arc length on the unit circle, where 0 5 s 5 27r. Then there exists a continuous function f (s) with values in C such that, for 0 5 s1 < s2 < 27r, f(sd f f ( 4 , f(O> = f(2T)l and the points of P are in one-one correspondence to the values f(s) (0 5 s < 2*). Proof: The L, consist of unions of closed disks. Any point in FR or L, but not in L,+1 cannot be a limit point since it is the center of a n open disk all points of which are images of a t most finitely many points in FR (namely their images under the action of words of length i n ) and whose images, with finitely many exceptions, lie in L,+l. On the other hand, the points of contact between any two circles in L, are limit points since they are fixed points of parabolic elements of D (see Lemma 4.5). Since the radii of the circles in L, tend to zero as n -+ 00 (Lemma 4.3), the limit points are certainly dense in P. It follows easily [see, e.g., Ford (1951)l that a limit point, of limit points is a limit point. Therefore, P consists entirely of limit points.
140
Miscellang
We now map the unit circle onto the points of P . For this purpose, we consider the group rz (the principal congruence subgroup mod 2 in the modular group), which has been discussed in Section 111.2 (Example 1 ) . As shown in Figure 19b (together with Figure 17, of which Figure 19b is a simplified part), rzis isomorphic with a subgroup of index 2 in a group D* generated by the reflections in four circles, which now have the unit circle as a common orthogonal circle. Figure 19b shows only the parts of these four circles lying within the unit circle. One of them is now a straight line. The biggest shaded region in Figure 19b corresponds to the region Qo in Figure 38. The limit points of D* all lie on the unit circle since the limit points of rzall lie on the real axis. (Note that r2has no elements of finite order and consists of parabolic and hyperbolic elements with fixed points on the real axis.) T hat the parabolic fixed points of rzare dense on the real axis and, therefore, the parabolic fixed points of D* are dense on the unit circle can now be shown directly. This is not necessary, however, since Lemma 4.3 and Lemma 4.4 are valid even if the four circles C, have a common orthogonal circle. Now we establish a one-onr and onto mapping of the unit circle onto the points of P. To each of the four circles C, we assign one of the four circles C,* which form the boundary of the fundamental region of D* (as described above with the help of Figure 19b) in such a manner that again C,* and C,+l* have exactly one point in common. We define Ll* as the chain fornied by the closed disks bounded by the C,*. Next we define the chain L,* exactly the same way in which we defined L,, only now we start from the disks bounded by the circles C,* instead of using the disks bounded by the C,. Since reflection in a circle does not change the propcrty of the reflected circles to have exactly onc point in common or to be outside of each other, we have a uniquely defined one-one correspondence between the 4.3, disks of L,* and those of L,. However, the disks of I,,* are all bounded by circles orthogonal to the unit circle, and their common intersection consists exactly of the unit circle. As before, the radii of the disks in L,*tend to zero as n 3 a. Consider now a sequence of closed disks An* such that An* belongs t o L,* and that An+l* is a subset of An*. Then there exists exactly one point on the unit circle which belongs to all of the An*. It can be described by a fixed value so of the arc length s on the unit circle. The disks An in L, corresponding t o A,* also contract to a point as n + a,and we denote the value of the complex coordinate zo of this point byf(so). This is the mapping of the unit circle onto P. Now we have to show that it is continuous. Let s; be the coordinate of another point on the unit circle. Let 1 s; - so 1 = e and let pn* be the smallest value of the radius of a disk in L,*. We assume that n is so large that none of the disks of L,* contains more than one-quarter of
I V.1 Examples of Discontinuous Nonfuchsian Groups
141
the perimeter of the unit circle. Then the arc has a length not less than This follows because the boundaries of the disks are orthogonal to the unit circle. We thus find: If e 2 1 6 p n * , the points on the unit circle with coordinates so and so' are either in the same or in two adjacent disks of L,*. Now let rn be the maximum value of the radii of the disks in L,. Then f ( s 0 ) and f(so'), which are the counterparts in P of, respectively, the points with coordinates so and so', lie in the same or in adjacent disks of L,, and we find fipn*.
(4.8)
I f(so') - f(so) 1 5 4r,
if
I so' - SO 1 5 l 6 p n * -
This proves the continuity of f(s) (and completes the proof of Lemma 4.4) since r, -+ 0 as n -+ 0 0 . So far, we have proved statements (i) and (ii) of Theorem 4.1. In order to prove (iii) , we need Lemma 4.5 There exist chains Lo of four circles C,, each touching Cz+l (with C5 = C,) , each in the exterior of the others, such that there exists no circle orthogonal to all C, and such that C, and Ct+2 have no points in common. The group D generated by the reflections R , in C, contains parabolic and hyperbolic elements and also loxodromic elements whose trace is not purely imaginary (Le., whose squares are not hyperbolic).
Proof: The existence of chains Locan be established by considering two parallel lines a r d two circles between them which touch each other and each of which touches one of the parallel lines (Figure 39a). Mapping a point zo between the parallels and outside of the circles onto the point x = 00 produces the desired chain Loprovided that the straight line joining the three points of contact between circles and parallels is not orthogonal to the parallels. (Note: This argument shows that the points of contact in a chain Lo always lie on a circle.) Now a circle orthogonal to all of the C, would have to go through their four points of contact, and by construction, this uniquely determined circle is not orthogonal to the C, in these points. The elements X , = R,R,+, of D+ are parabolic. This is seen by mapping A , (thc point of contact of C, and C,+,) onto co. Then C, and C,+l are mapped onto parallel lines, and the product of reflections in two parallel lines is a translation. Therefore, X , is conjugate with a translation and is parabolic. Now take three consecutive circles of Lo, for instance, CI, CZ,Ca. There exists a t least one circle K orthogonal to all three of them (Lemma 1.26). K must pass through Al and Az and must be orthogonal to the tangents of C, in A , and Az. If these tangents intersect in a finite point, this is the center
142
Miscellany
of K , which then is uniquely determined. If the tangents are parallel, the centers of C1, C2, Ca are collinear and the straight line joining them is the (only) common orthogonal circle. The parabolic element,s XI = RlR2 and X2 = R2R3 map K onto itself. By mapping K onto the real axis and A , onto 00 , we see that XI is conjugate with a translation 2’ = z t ( t real), and X 2 will then be parabolic but not a translation since A1 and Az are different points. Therefore, X2is then conjugate with a real matrix
+
(4.9)
[:
.1
2:
(2a - a* - bc
=
1 ; c # 0)
and the product X1X2 is therefore conjugate with (4.10)
+
which has the trace 2 tc # 2. Therefore, XIXz is nonparabolic unless 2 tc = - 2 . I n this case, Xl-lXz has trace 2 - tc = 6 and is hyperbolic. It cannot be that X,X2 is elliptic since X1, X 2 generate a free group according to Lemma 4.1, so X1X2cannot be of finite order. Also, it cannot be elliptic of infinite order since then D+ could not be discontinuous anywhere. Therefore, the group generated by X1, Xz contains hyperbolic elements under all circumstances. Since it leaves K fixed, it will not have loxodromic elements. To show the existence of these, we must use the fact that the circle K is not orthogonal to Cq and does not pass through either A3 or Aq. (If it would pass through A1,Az, AB, it would be uniquely determined by these data and would, therefore, be the circle through all four Ai that we had shown earlier to exist. In addition, it would be orthogonal to all of the Ci, which is against our assumption.) By conjugation of D+ with an appropriate element of P S L ( 2 , C), we can map K onto the real axis. We may then assume that XI and X2 are represented by real matrices, that Xl is again a real translation, and that X2 is of the form (4.9). The parabolic element Xa will be of the form
+
(4.11) y
2--a
and its fixed point ( a - l ) / y will not be real. We find for the trace of XlnX3the value 2 n y t , where t is real and t # 0. If y is not real, we can choose n so that XlnX2is loxodromic with a trace which is not purely imaginary. If y is real, a cannot be real. In this case we compute, for
+
IV.1 e =
Examples of Discontinuous Nonjuchsian Groups
143
f l , the elements
(4.12)
x z t x 3 x 2 t=
[;
Be
2 - a.
1.
If one of the y e is not real, we can obtain the desired loxodromic element as before. Assuming now that both yt are real, we find after some calculations that both y-'[c( 1 - a) - y (2
- a)l2 =
y+
and
y-'[c(
1 - a)
+ yaI2 = y-
must be real. Since c, a , y are real, c # 0, and QI is not real, this is possible only if both c ( 1 - a ) - 27
+ ay
and
c(1 - a )
+ya
are purely imaginary. But then the difference 27 of these terms would be purely imaginary, which is against our assumptions. Therefore, D+ contains loxodromic elements whose square is not hyperbolic. Lemma 4.6 The fixed points of parabolic as well as those of hyperbolic and of loxodromic elements of D+ are dense everywhere on P. Proof: The points of contact of any two circles in the chain I,, are fixed points of parabolic elements since they are images of the fixed points of the elements X i . By construction, they are dense on P. But then there are images of fixed points of both hyperbolic and loxodromic elements in every neighborhood of the fixed points of parabolic elements, since an appropriate power of a parabolic element carries any given point into any prescribed neighborhood of the fixed point. Lemma 4.7 At the fixed points of parabolic elements of D,,the curve P has a tangent.
It suffices to prove Lemma 4.7 for the point of contact A , of C2 and C3. To show this, we map A4 onto z = ot). Then C1 and C4 are mapped onto parallel lines and Cz and Ca are mapped on circles touching each other such that C1 and C4 are, respectively, tangents of C2 and C3.Figure 39a shows the resulting configuration. We denote by K j the (uniquely determined) circle orthogonal to the three circles Ci with i # j . In our arrangement, Kz and KX will then be straight lines orthogonal to the paral!els C1and C4, and they will pass respectively through the centers of C3 and Cz (see Figure 39a).
144
Miscellany
Similarly, K1 and K4 will now be circles orthogonal, respectively, to Cd and C1 and having A 2 as a point of contact. Consider the closed region B (shaded in Figure 39a) which contains Az and is bounded by arcs of the K, (i = 1 , 2 , 3 , 4 ) .We shall show that the part of P in the neighborhood of A’L is entirely confined to B. If this is true, P has a tangent a t Az in our particular setup. However, a Mobius transformation will map P onto a curve with the same property for the image of A2 since Mobius transformations are both analytic and conformal. Our argument would apply to the A # A Z as well. Since all the points of contact of circles in 11, are images of the A , under Mobius transformations of D (which map P onto itself since the image of a limit point under the action of D is a limit point), our lemma will be true if we can prove our statement that P i s confined to B in the neighborhood of A,. We observe that the reflections in the C, map the strip S bounded by the parallel lines K2 and K 3 into itself. This is obvious for the reflections in C1 and C4 (which are now straight lines), but it is also true for the reflections in Cz and C3 since the centers of these circles lie, respectively, on K3 and Kz. Thercfore, R2 will map KBonto itself and will map K 2onto a circle inside X, touching K3 a t the center of C2. Thc corresponding statemcnt holds for Ra. Thus. it follows that a limit point in S must stay in S under the action of all elements of D. Since the A , are in thc closed region S (including z = w ) , and since their images are dense on P , it follows that P lies in S. But then it follows that P must also lie in the common exterior E of K1 and Kq (including the boundaries of these circles). This is seen by mapping A z onto z = 00 (instead of A4, which we did in order to obtain Figure 39a). T h e common part of S and E consists now of B and of certain parts of thc plane which lie in S above the line C4 and below the line C1. Thereforc, all points on P in the neighborhood of A z lie in B, which proves Lemma 4.7, since the boundary of B a t A 2 consists of circular arcs which touch a t A,. Lemma 4.8 X,X, = R& is a hyperbolic clement of D+. Let M be onc of its fixed points. Then vectors from M to the points of P in a n arbitrarily small neighborhood of M enclose an angle bounded away from 0 and A.
It suffices to prove Lemma 4.8 by using a suitable coordinate system arising from x by its replacement with a properly selected 2’ =
(a*z
+ b * ) / ( c * z + d*) .
We choose the system so that the circlcs C1 and Cs become concentric circles. This can always be achieved as follows: Draw the straight line I
IV.1
Examples of Discontinuous Nonfuchsian Groups
145
through the centers of C1 and C3 and compute the crossratio CR of the four points of intersection of 1 with C1 and Cz. Then there exist concentric circles with centers on 1 for which the corresponding crossratio has the same value CR. Now four points on a line can be mapped onto any four points with the same crossratio by a Mobius transformation, and this is the one we need. The configuration of the C; is shown in Figure 39b. C1 and Cs are concentric, and C2 and C4 are congruent. The circles K j orthogonal to the C ; with i # j are now in two cases straight lines, K 2 being the line joining M with the center of Cqand Kq being the line joining M and the center of C2. These lines must enclose an angle a # T since otherwise the four Ci would have a common orthogonal circle. Reflections in C1 and C3 keep M fixed, which, therefore, is a fixed point of R1R3 = XIX2, the other fixed point being a t 00. According to the proof of Lemma 4.5, X,XZ is parabolic or hyperbolic (see 4.10). It appears now that XlXz must be hyperbolic since it has two distinct fixed points. The lines Kz and K4 are orbits. They go through points of contact between some of the circles C;, and these points are fixed points of parabolic elements of D+ and therefore belong to P. So do all of their images under the action of the elements (X~XZ)", all of which lie either on K z or on K 4and which have M as a point of accumulation. This proves Lemma 4.8.
Lemma 4.9 Let F be a fixed point of a loxodromic element Y of D+, and assume that Y 2is not hyperbolic. Then P winds itself infinitely many times around F. Proof: Y has two distinct fixed points F , F'. There must exist a closed disk A in a chain L, for sufficiently large n such that F lies in A but F' lies outside of A. F cannot lie on the boundary of A because the points of P on the boundaries of the disks of L, are fixed points of parabolic elements of D+. Since D+ is discrete, it is impossible that a fixed point of a parabolic and a loxodromic element should coincide. (This can be seen by putting the common fixed point a t z = 00 and conjugating the parabolic element by the powers of the loxodromic one. We obtain infinitesimal translations.) Now we conjugate D+ by an appropriate element of PXL( 2, C) such that F is located a t z = 0 and F' is located a t z = ca . Then Y is represented by a matrix
(4.13)
[
(p, a
real; p
> 0;
-T
< < T), Q
146
Miscellany
where we may assume that 0 < p < 1. (Otherwise, we shall replace Y by Y-1.) Next, we mark on the boundary of A the two points T and T* which belong to P (see Eigure 39c). The construction of L, shows th a t T and T* are the two points of contact between A and the two adjacent disks of L,. Let T be such that the distance TF is not larger than the distance T*F. Let A' be the image of A under the action of Y . We claim that A' is in the interior of A and that, with the possible exception of T , no boundary point of A belongs to A'. First of all, part of A' is in A, and not all of A is covered by A', since p < 1. Next, A', too, is a disk of a chain L,. Since A' does not contain all of A it is therefore inside of A. Since a common point of two circles bounding disks in the chains L, and L, must belong to P , A and A' can touch only in T (since p < l ) , and T must be the image of T* under the action of Y . Should this happen, we shall replace Y by Y z ,which is still loxodromic, and we now have the situation, shown in Figure 39c, where no boundary point of A belongs to A'. The image T' of T lies on the boundary of A' and belongs to P. The vectors from F to T and T' enclose a n angle a which is not a n integral multiple of 27r. Also, the points T and T' can be connected by a chain of disks belonging to a chain LMwith a suitably large M such that the first disk A1 touches the rim of A from the inside a t T and the last disk, AZ say, touches the rim of A' from the outside a t T'. I n between, A i touches AX+^ and Ax-1 for X = 2,. . ., 1 - 1, and all of the disks AI,. . ., A, are inside A and outside A'. The limit point curve P then goes from T to T' inside the closed disks Al, . . ., Ac. Applying our loxodromic mapping to the pair of disks A, A', we obtain now the pair of disks A', A", where A'' has no point in common with the boundary of A' and lies entirely inside it. The image of T' is 1"' on the rim of A". The vectors FT' and FT" form again the angle a, and P goes from T' to T" inside a chain All,. . ., Ac' of disks which arise from AI, . . ., A Zby the application of the loxodromic element. The Ah' are now inside A' and (except for T") outside of A", Also, the angle between the vectors FT and FT" is 2a (and not zero). Continuing in this manner, we see that Lemma 4.9 is true. Figure 30 shows an approximation to the curve P of limit points of D. Only parts of the original circles C , are shown in this drawing, and one of the C , is represented by a straight line. It is, of course, impossible to draw the curve P itself. Fricke and Klein (1897, pp. 418-420) connect the various fixed points of elements of D+.with infinite sequences of reflections R;, which are then represented by infinite sequences of type (4.2). Fricke shows that the fixed points correspond to sequences which, from a certain point on, are periodic. I n his arguments, he also uses the theory of characters of finitely generated kleinian groups. The basic algebraic ideas of this theory will be presented in Section 4.2.
I V.1 Examples of Discontinuous Nonfuchsian Groups
147
The following descriptions are taken from Fricke and Klein (1897, pp. 428-445). Figure 31 shows a limiting case of the situation investigated in Theorem 4.1 and represented by Figure 30. I n Figure 31, the circles C1 and C3 also touch. The quadrilateral Qo in Figure 38 is now decomposed into two zeroangle triangles, whereas Q, remains a quadrangle. The limit points of the group D still fill a curve P with the properties described in Theorem 4.1, but in addition we have now, inside P , an infinitude of circles which are filled with limit points of D. These circles are drawn heavily in Figure 31. An even more extreme limiting case is obtained b y choosing the circles Ci so that each one touches all of the others. In this case, the fundamental region of D degenerates into four zero-angle triangles which form the remainder of a disk A, from which we have removed three interior disks which touch each other as well as the rim of Al. After subdividing each of these triangles into six triangles (meeting at one point and alternatingly shaded and unshaded) and then carrying out the repeated reflections in the circles bounding the four disks, we obtain the tesselation of the plane shown in Figure 32. Here there exist infinitely many circles filled with limit points. However, these limit circles do not contain all limit points of D. There exist sequences of limit circles (with radii converging toward zero) such that points on these limit circles have points of accumulation not belonging to any one of the limit circles themselves. Figure 33 shows the tesselation of the plane arising from a group D of reff ections in four pairwise disjoint circles. One of the circles contains three others in its interior. The fundamental region consists of the common part of the exterior of three smaller circles and the interior of the big circle containing the smaller ones. The cardinality of the set of limit points is that of the continuum, but the open set obtained by removing the limit points from the plane is connected (and infinitely connected). Figure 34 indicates the tesselation of the plane with replicas of the fundamental region of a group which is generated by the reflections in five circles, four of which form a chain of the type considered in Theorem 4.1 and three of which touch in pairs. The limit points fill infinitely many limit circles and also infinitely many continuous nondifferentiable curves of the type described in Theorem 4.1. Figures 35 and 36 indicate (very sketchily) the tesselation of the plane arising from the composition (in the sense of F. Klein, as defined a t the beginning of Section IV.1) of rather elementary groups. In Figure 35, we have t,he result of the composition of two (euclidean) triangle groups T ( 2 , 3, 6) (Section 11.4). Figure 36 indicates the result of the composition of two tetrahedral groups (Section 11.4). For explanations, see Fricke and Klein (1897, pp. 433-436).
148
Miscellany
IV.2 Fricke Characters In this section, we shall introduce very briefly a method which was developed by Fricke (Fricke, 1894; Fricke and Klein, 1897, pp. 285-410; 1912, pp. 286-439) for the classification of discontinuous subgroups of PSL(2, C). Although Fricke's results are frequently based on geometric arguments which today appear to be intuitive rather than rigorous, some of them have been confirmed and even expanded in recent times (Keen, 1971; Purzitsky and Rosenberger, 1973) or utilized for related purposes (McKean, 1972). The algebraic aspects of Fricke's theory of the characters of elements in a finitely generated subgroup of the special linear group SL(2, K) over a commutative ring K of characteristic zero and with an identity element have been developed by Horowitz (1972). We shall not go into the details but mention only a few basic facts. Let A , (i = 1 , . . ., n ) be 2 x 2 matrices of determinant 1 with elements in C.Let G, be the group generated by the A , and, for any M E G,, let t r M denote the trace of the matrix M . In particular, lct (4.14)
tzlz2...tv
( l l v l n ;
=
t r A,,,,. * - A q p
l l i l < i 2 <
-..< i , I n ) .
Then t r M is a polynomial with integral coefficients in the 2" - 1 traces (4.14). For example, (4.15)
tr Ad2-'
(4.16)
t r A1A2A1-lA2-l
=
tltz - t12,
=
tlz
+ t22 + t122- t1t2tl2 - 2.
We always have tr M = t r M-l. Also, matrices which are conjugate in SL(2, C) have the same trace. In particular, this is true for elements conjugate in G,. Horowitz has investigated the question of to what extent the conjugacy classes of G, can be separated by the values of the traces of their elements. Fricke calls the values (4.14) the invariants of G,. His aim is the following: Let H , be an abstract group which has at least one representation as a fuchsian (or, more generally, as a kleinian) group. Find, for a suitable N , all N-tuples of invariants for representations of H , as discontinuous subgroups of PSL(2, R) or PSL(2, C) such that every system of conjugate representations is given by one N-tuple. The conditions to be imposed on the values of the invariants may be algebraic equations or inequalities or both.
Exercises
149
Instead of trying to make this description more precise, we shall give an example which appears already, in part, in Fricke and Klein (1912) and which has been fully established by Purzitsky and Rosenberger (1973). Let H2 be the abstract group (4.17)
(a, b ; (abu-'b-')k
=
1)
(k 2 2).
Let A , B be the matrices of two real Mobius transformations a,P such that (4.18)
(aPa-'P-')
is the identity mapping. Let t r A = u,
(4.19)
tr B
=
v,
tr AB
=
w
and assume that (4.20)
U '
> 4,
V'
> 4,
U '
+ v2 + w 2 - uvw = 2 - 2
COS(~/~).
Then a + a, b + P is a faithful representation G2 of H2 as a fuchsian group, and any fuchsian group isomorphic with H2 is conjugate with a group G2. For any choice of the real numbers ,u, TJ, w satisfying (4.20) there exist real matrices A , B satisfying (4.19). The automorphisms of HZinduce birational transformations of the real variables u, v, w, mapping them onto polynomials u', v', w' in u, v, w such that U'Z
+ + w'2 - u'v'w' 0'2
=
2 - 2 cos(r/k) ,
identically in u, v, w. This group of birational transformations is discontinuous on the part of the cubic surface defined by (4.20). A fundamental region can be given explicitly. For k = 2, we can choose
A = r 0
O ] , r-l
B = [ l/t(rz - 1) 2/t(rz - 1 ) z t ( l
tr2
+ T4)/(r2- 1 )
1
where r, t are real and r2 > 1 and t # 0 (Magnus, 1973).
Exercises
1. We use the notations of Section IV.2. Let Gz be generated by two matrices Al? Az with determinant one and assume that ti, &, tlz all have
150
Miscellany
values different from f 2 and that
+ +
tI2 h2 tlZ2 - tltztlz # 4 and # 0 Show: I n this case, the matrices Al, Az are uniquely determined up to conjugation by one and the same matrix T. I n particular, if tl = 2 cos (r/Z), t2 = 2 cos (rim), t12 = 2 cos ( r / n ), where I, n, m are positive integers, and (1/Z) (l/m) (l/n) < 1, then G2 is conjugate with the triangle group T(1,m, n ). 2. Show: tlZ3 satisfies a quadratic equation with coefficients which are polynomials in tl, h, t 3 , t 1 2 , t 1 3 , t23 [for the definition of terms, see (4.14)]. For a solution see Fricke and Klein (1897).
+
+
Groups That Are Discontinuous in Hyperbolic Three-Space
I n this chapter, we shall describe some subgroups of the extension G* (Section 1.1) of PSL(2, C) which are discrete but not discontinuous anywhere in C. According t o Theorem 2.1 (Section 11.2), such groups will be discontinuous in hyperbolic three-space, which we may represent either by the model P , (Lemma 1.12) or by the model K 3 (Lemma 1.8) of Section 1.3. The easiest way to define groups which are discontinuous in three-space but nut in C is based on number theoretical considerations. Thcse will be presented in Section V.l. In Section V.2, we shall report briefly on geometric constructions of such groups.
V.l
Linear Groups over Imaginary Quadratic Number Fields
Let m be a positive square-free integer and denote b y Rm the ring of algebraic integers in the imaginary quadratic number field Q The group PSL (2, R,) , that is, the group of Mobius transformations with determinant +1 and coefficients in R,, will be denoted briefly by k m . The group was mentioned in Section 11.2 as Picard’s group. All of the groups k, are discrete, and none of them is discontinuous anywhere in c. (If R, is a ring with a euclidean algorithm, this can be shown in the same manner as for P,in Section 11.2.)
(4-m).
152
Discontinuous Groups in Hyperbolic Three-Space
The following problems have been investigated : ( I ) To find a fundamental region for q, in Pa. (11)
To obtain generators and defining relations for q,.
(111) To obtain an insight into the subgroup structure of the abstract group qm. The most comprehensive results for (I) and (11) were obtained by Swan (1971). He showed how to solve (I) and (11) in general, obtained theorems for all sufficiently large m, and detailed information for the values 1, 2, 3, 5, 7, 11, 15, and 19 of m. His investigation of problem (11) is based on the solution of the geometric problem (I). For the values 1, 2, 3, 7, 11 of m, R , has a euclidean algorithm. I n this case, the results found by P. Cohn (1968) have enabled Fine (1973) to solve problem (11)by purely algebraic methods and to solve the word problem in the abstract group q, by using the theory of free products with anialgamated subgroups. For m = 1, this had been done earlier by Waldinger (1965) and, in a different manner, by Drillick (1971). The best investigated group is the Picard group \kl. Fricke and Klein (1897, pp. 79-93) give a survey about the older literature and a detailed construction of the fundamental rcgion of in Pa.The starting point for which is generated by four orientathis construction is a larger group ql*, tion inverting mappings A , B, C, D defined by
A:
Z+Z,
B: ~-+--i-l,
c:
Z-+l/Z,
D:
Z-+
-iZ,
which satisfy the defining relations
A2 (AC)'
=
=
(AD)'
B2
=
= C2 =
0 2
=
1
( B C ) 2= ( B D ) 4= ( C D ) 2= 1.
The fundamental region of \kl* in Pa is the strongly drawn noneuclidean tetrahedron T* in Figure 37. Its faces are given (in the notation of Lemma 1.12, Section 1.3) by the four equations y=o,
22+1=0,
x2+y2+u2=1,
s+y=o.
Figure 37 also shows the reflection of T* in three of its faces and several other replicas of T* with cusps in the plane u = 0. The fundamental
V.2 Some Geometric Constructions
region of
9 1
153
is then given by the inequalities
- x < + , - -2‘ <
o
22+y2+u321.
ikl is of index 4 in PI*. It is generated by four Mobius transformations S , T ,
U , V with corresponding matrices
‘1,
S _ t [ i0 -i
-:I,
T-[i
and defining relations 8 2
(us-1)2 =
=
u2 =
(vT-1)2 = (sT-1)2
vz = 1 = (uT-1)3 = (Vu-1)3 =
1.
,4n equivalent presentation has been derived by Sansone (1923). The analogue of Theorem 3.3 (Section 111.2) has been proved for \kl by Drillick (1971). It states that there exist subgroups of finite index in \kl which do not contain a principal congruence subgroup of These are defined here by using the ideals in the ring of gaussian integers (which happen to be principal ideals). Drillick’s proof is explicit and follows the pattern of the proof of Theorem 3.3. Specifically, he shows that the alternating group A , on n symbols is a quotient group of ikl if n is of one of the forms 16t
+ 1,
16t
+ 6,
16t
+8
.
( t = 1, 2, 3,. .).
He also shows that infinitely many symmetric groups S, are quotient groups of \kI,in particular S7.
V.2 Some Geometric Constructions Subgroups of G* [the extended group PSL(2, C)] which are discontinuous in hyperbolic three-space can be constructed by using the analogue of the geometric methods which led, for instance, to the construction of the triangle groups (see Section 11.5). Numerous constructions of this type have been carried out by Best (1971). His paper also contains a brief account of earlier work on this problem and may be used as a standard reference. Best uses the model P3 (Section 1.3) for hyperbolic three-space. We shall report here in some detail on an example found by Gieseking (1912) (which uses the model K3, Section 1.3) because it has been published only
154
Discontinuous Groups in Hyperbolic Three-Space TABLE 5.1
as a privately printed Ph.D. thesis (instigated by M. Dehn) which is nearly inaccessible. Using the notations of Section 1.3 and formulas (1.31)-(1.34), we define four points A , B, C, D on the euclidean sphere which forms the boundary of the model K3 of hyperbolic three-space. Homogeneous coordinates t, ( v = 1, 2, 3, 4) for these four points are listed in Table 5.1. A , B, C, D are the vertices of an (improper) tetrahedron Toin three-space. Its faces are represented by the faces of the euclidean tetrahedron with thcse vertices. We can define two orientation-reversing noneuclidean motions U* and V* each of which maps Toonto an adjacent tetrahedron in such a manner that U* maps A-A,
C-B,
D+C
B+A,
C-B,
D-D.
and V* maps
U* and V* are induced, respectively, by the orientation-reversing mappings of C onto itself given by the formulas:
v*:
z--
a’ =
CW’J
y’J
+ p’ + 6’
3(1 + iG), 8’ =
6‘ = + ( 1 - i f i ) .
-a(% + i f i ) ,
y’ =
0,
Exercises
155
U* and V* generate a group D* with a single defining relation U*ZV*2 =
v*u*.
The tetrahedron Tois a fundamental region for D*. A subgroup D of index 2 in D* and consisting of orientation-preserving noneuclidean motions is generated by the elements
u*2 = x,
y*2
=
y
with the single defining relation
XY-'X-'YXYX-'Y-'XY
=
1.
Here X , Y are Mobius transformations defined, respectively, by the matrices X o , Y owhere 1
6 - i a
xo = 1'2[ 4d6
-dG-ifi] 18 + i d E
7
The parabolic substitutions Xo, Yo are equal, respectively, to OX&' and BYl&', where O is of the form
and
Xl
=
[A ;I,
Yl
=
Yo.
Finite quotient groups of D* have been constructed by Arrhenius-Wold (1971). For D,the corresponding problem has infinitely many obvious Y , modulo any ideal coprime solutions since we can take the elements in X1, G ). with 6 in the algebraic number field Q ( i
+a +
Exercises
1. Let A , B, C , D be, respectively, the matrices
156
Discontinuous Groups in Hyperbolic Three-Spaee
Show : The Mobius transformations a, b, c, d belonging respectively to A , B, C , D generate Picard’s group. Defining relations for these generators are a3 = b2 = c3 = d2 = (ac)2 = (ad)2 = (bd)2 = ( b ~ ) 2 = 1. 2. Using the notations of Exercise 1, show that the homomorphism
a
+
(1, 2, 3) (4, 5,
w,
c-+ (1, 2 , 7 ) ( 4 , 5 , 8 ) ,
b
-+
(1, 2 ) (4, 5) (3, 6)
d-+ (1, 2 ) ( 4 , 5 ) ( 7 , 9 )
maps the Picard group onto the symmetric group on nine symbols. Show that the kernel of this mapping cannot contain any principal congruence subgroup of the Picard group. (The congruence subgroups are defined by taking the matrices which are congruent to the unit matrix or its negative modulo any ideal in the ring of Gaussian integers.) (Drillick, 1971)
3. Let U*, V* be generators of the group defined in Section V.2. Dctermine the finite quotient groups of this group which are defined by the mappings
u*+ and
u*
(2, 4, 5, 3),
v*-+ (1, 2, 4, 3) v* -+ (1, 2, 3)
( 2 , 3, 41, into permutation groups on five and four symbols, respectively. [Arrhenius-Wold (1971), where many more examples of this type are given.] -+
Illustrations
The line in square brackets indicates the volume and page where the original figure appeared and the section and page where the figure is discussed in the present work. D, Dyck (1880); F, Fricke (1926); F-K, Fricke and Klein (1897, 1912) ; K-F, Klein and Fricke (1890). Fricke, 1916 refers to: Fricke, R., “Die elliptischen Funktionen und ihre Anwendungen,” Vol. 1. B. G. Teubner, Leipzig.
Figure 1 (a, b)
Tesselation with proof of the Pythagorean theorem [II.l, 531
159
Figure 2 (a, b, c )
Triangle tesselations of the euclidean plane [K-F, 107 11.4, 69,701
160
Figure 3 (a, b)
Tesselation of the tetrahedral group on plane and sphere [K-F, 104 11.4, 741
Figure 3 (c)
Tesselation of the tetrahedral group and a dihedral group in the elliptic plane [F-K, 70 11.4, 74)
161
Figure 4 (a)
Tesselation of the octahedral group on plane [K-F, 75 11.4,761
162
Figure 4 (b)
Teaselation of the octahedral group on sphere [K-F, 76 11.4, 76)
163
Figure 5 (a)
Tesselation of the group of the icosahedron on sphere [F, 46 11.4, 761
164
Figure 5 (b)
Tesselation of the group of the icosahedron on plane [I?, 46 11.4, 761
165
Figure 6
Octahedral tesselation in the elliptic plane [F-K, 71 11.4, 741
166
Figure 7
Icosahedral tesselation in the elliptic plane [F-K, 72 11.4, 763
167
Figure 8 (a)
Orbits of hyperbolic substitution m-F, 206 1.2, 101
168
Figure 8 (b)
Orbits of elliptic substitution m-F, 206 1.2, lo]
169
,
Figure 9
Orbits and their orthogonal trajectories of a parabolic substitution m-F, 207 1.2, lo]
170
Figure 10
Figure 11
Orbits and their orthogonal trajectories of loxodromic substitutions with fixed point at infinity w-F, 168 1.2, lo]
Orbits and their orthogonal trajectories of loxodromic substitutions with finite fixed points w-F,172 1.2, 107
Figure 12
Mapping of halfplane into an ellipse [F-K, 23 1.6,42]
171
Figure 13 (a, b)
Orbits of hyperbolic and of elliptic substitutions in Klein’s projective model [F-K, 34 I.6,42]
172
Figure 14
Orbits of parabolic substitutions in Klein’s projective model [F-K, 35 1.6, 421
173
Figure 15
Tesselation of upper halfplane induced by the extended modular group 11.5, 90; 111.1, 1111 [F, 30
174
Figure 16
Tesselation of ellipse induced by extended modular group [K-F, 239 11.5, 901
175
Figure 17
Tesselation of unit disk induced by extended modular group 111.1, 111, 112; IV.l, 1401 m-F, 112
176
Figure 18
Tesselation of unit disk induced by repeated reflection of zero-angle triangle [K-F, 111 111.2, 1121
177
Figure 19 (a)
Tesselation of the unit disk induced by subgroup of the extended modular group 111.2, 114; IV.l, 1401 [D, 20
178
Figure 19 (b)
Tnsselation of the unit disk induced by subgroup of the extended modular group 111.2, 114; IV.l, 1401 [D, 20
179
Figure 20
Image of part of the tesselation induced in the upper halfplane by the extended modular group under an exponential mapping 11.5, 901 [Fricke (1916, 300)
180
Figure 21
Superposition of two tesselations induced by the modular group [F-K, 549 111.3, 1261
181
Figure 22
Part of tesselation induced by the extended modular group [K-F, 373 111.2, 1151
182
Figure 23
Tesselation of the unit disk belonging to the triangle group T*(2, 3, 7) [K-F, 109 11.5,901
183
Figure 24
Fundamental region of a subgroup of index 168 in T(2, 3, 7) [IK-F, 370 11.6, 941
184
Figure 25
A heptagon and its seven neighbors in the tesselation of T(2, 3, 7) [K-F, 464 11.6, 951
185
Figure 26
Unsymmetric teaselation of a regular heptagon by 63 triangles with angles r/2, r/3, r / 7 [F-K, 621 11.6, 951
186
Figure 27
Tesselation by the triangle group T*(2, 3, 8) [D, 17 11.6,941
187
Figure 28
Composition of two cyclic groups of infinite order
[F-K, 191
IV.l, 1351
Figure 29
Composition of two cyclic groups of finite order [F-K, 191 IV.1, 1351
188
Figure 30
Tesselation and nondifferentiable curve of limit points for nonfuchsian group [F-K, 418 IV.1, 1461
Figure 31
Tesselation of nonfuchsian group with infinitely many limit circles [F-K, 429 IV.1, 1471
190
Figure 32
Degenerate form of tesselation in Figure 30 [F-K, 432 IV.1, 1471
191
Figure 33
Tesselation arising from reflections in four disjoint circles [F-K, 437 IV.1, 1471
192
Figure 34
Tesselation with nondifferentiable curve of limit points and with limit circles [F-K, 440 IV.l, 1471
193
Figure 35
Tesselation arising from composition of two euclidean triangle groups [F-K, 433 IV.1, 1471
194
Figure 36
Tesselation arising from composition of two tetrahedral groups [F-K, 435 IV.l, 1471
195
Figure 37
Fundamental region of Picard’s group [F-K, 82 V.l, 1521
196
Figure 38
First step in the construction of the tesselation induced by a specid nonfuchsian group CIV.1, 1371
197
Figure 39
Behavior of nondifferentiable limit point curve near: (a)parabolic fixed point [IV.l,143]; (b) hyperbolic fixed point CIV.1, 1451; (c) loxodromic fixed point CIV.1, 1461.
198
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Index
.I
Angle, 47 Archimedian axiom, 31, 48 Area, 31 Associated, 67 Axioms of geometry, 4 3 4 9 Axis of projectivity, 42
Elliptic substitution, 9 Equivalent, 57 Euclidean space groups, 96 Euclidean triangle groups, 68 Extended orthogonal group, 40 Exterior of a circle, 5
F B
Basic triangle, 65 Between, 26, 45
C Canonical tesselation, 55 Circle, 3, 35 Circle group, 19 Closed complex plane, 2 Commensurable, 125 Complex conjugate, 2 Composition of groups, 135 Congruence, 28, 46 Congruence subgroup, 112 Conic, 22 Conjugacy clam, 6 Crossratio, 3 Crossratio on a line, 21
D Deficiency of a triangle, 32 Discontinuous group, 57 Discrete group, 57 Distance (noneuclidean), 29 Dehn (Lemma 1.16), 28 Dehn-Gieseking group, 154 Dihedral group, 71
E Elliptic geometry, 37-41 Elliptic modular group, 107
Fixed point, 6 Fricke characters, 148 Fricke-Mennicke theorem, 131 Fricke’s theorem, 98 Fuchsian groups, 62 first and second kinds, 98 Fundamental group, 91 Fundamental region, 57 G
Gaussian integers, 60 Global relations, 67 Global tesselation problem, 66 Groups generated by reflections, 97 Groups of units, 123
H H a j b theorem, 54 Hauptkreisgruppe, 19 Higman theorem, 104 Hilbert’s axioms, 43-49 Hoare-Karrass-Solitar theorem, 99 Homographic substitution, 3 Homogeneous modular group, 107 Hurwitz groups, 103 Hurwits theorem, 103 Hyperbolic circle, 34 Hyperbolic substitution, 9 Hyperbolic three-space K3, Pa, 13, 17 Hyperbolic twwjpace, 20, 23 205
206
Index I
Icosahedral group, 76 Incidence, 26,44 Infinitesimal substitution, 57 Interior of circle, 5 of conic, 23 of triangle, 27 Invariants (Fricke), 148 Involution, 4 Irreducible, 96
L Leg of an angle, 47 Length of an interval, 29 Limit point, 56 Line element, 13,32, 39, 41 Local relation, 67 Local tesselation problem, 66 Lorentz group, 17 Loxodromic substitution, 9
M Macbeath theorem, 104 Markoff triple, 114 McCarthy theorem, 96 Mennicke theorem, 102 Mobius transformation, 2 (Formula 1.4a) Modular group, 107
N Neumann theorem, 122 Noneuclidean area, 32 distance, 29 geometry, 13 line element, 13, 18, 32, 34 three-space, 13, 17 twc-space, 20 Nonparabolic subgroup, 119
0 Octahedral group, 75 One-parameter groups, 8 Orbit, 9, 56
Order, 45 Orientation, 5 of circle, 5
P Parabolic substitution, 9 Parallel postulate, 31, 48 Pasch’s axiom, 27, 45 Picard’s group, 60, 152 Poincar6 space, 96 Point group, 96 Pole (of a projectivity), 42 Polygon, 27 Positive (orientation of a circle), 6 Positive side (of a line), 5 Principal congruence subgroup, 112 Principal subgroup, 125 Projective coordinate, 21 Projective special linear group, 1 Projective three space, 11 Projective two space, 20 Projectivity, 11 Proper angle, 27 Pythagorean theorem, 53
R Ray, 27 Reflection in a circle, 4 in a noneuclidean line, 28
S Schreier system, 91 Schur multiplier, 102 Selberg theorem, 100 Side of circle, 5 of line, 5 Siege1 theorem, 97 Simple polygon, 27 Space group, 96 Special linear group, 1 Spherical geometry, 37-41 Star, 83 Stereographic projection, 11 Substitution, 2 Symplectic, 17, 43, 52
Index T Tesselation, 52 Tetrahedral group, 73 Tile, 52 Torsion free, 87 Trace, 6 Transfer principle, 26 Transformation, 1
A 4 B 5 C 6
D 7 E 8 F 9 G O
H I 1 2 J 3
Triangle groups, 65 elliptic, 71 euclidean, 68 hyperbolic, 81 spherical, 71
U
Unit groups, 123
207
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