Noneuclidean Tesselations and Their Groups

215 Pages·1974·4.116 MB·iii-xiv, 1-207\215

by  Wilhelm Magnus (Eds.)

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 ACADEMPICRE SS, INC.
ALL RIGHTS RESERVED.
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ACADEMIC PRESS, INC.
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Uniied Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NWl
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, Non-
euclidean. 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
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 Lichten-
berg (1742-1799) , Professor of Physics in Goettingen and essayist, pub-
lished 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 use-
fulness 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 mathe-
matical version of Lichtenberg’s commentaries. The pictures that moti-
vated the writing of the book are mainly those appearing in the mathemati-
cal 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 avail-
able as a reference.
In spite of its classical material, the book is not merely a salvage opera-
tion 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 stimu-
lated 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 dif-
ferential geometry, group theory, topology, or number theory. For diffe ren-
tial 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 prefer-
ence for the explicit formula wherever it appears to give information beyond
abstract formulation of a result. The first feature is inherent in the motivat-
ing 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. Non-
euclidean 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 trans-
lation of Hilbert’s axioms is given. The elliptic plane and hyperbolic three-
space 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 eu-
clidean, hyperbolic, and elliptic geometry and also with the triangle tessela-
tions 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 ma-
terial 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 fre-
quently 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 manu-
script 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.
Abbreviations and Symbols
Latin letters
An Alternating group on n symbols
CR Crossratio
En n-dimensional euclidean space
E*(Z, m, n) Triangle group with angles r/Z, r/m, r/n in the elliptic plane
G Group of fractional linear mappings (Mobius transforma-
tions) 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
Kz, & Klein’s model of noneuclidean (hyperbolic) geometry in two
or three dimensions (models in projective spaces)
0 Orthogonal group
O* Orthogonal group extended by reflections
Pz, Pa Poincard’s model of, respectively, noneuclidean (hyperbolic)
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
Mt 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
xiii
xiv Abbreviations and Symbols
Greek letters
r Restriction of G to selfmappings of upper halfplane;
PSL(2, R)
r* Restriction of G* to selfmappings of upper halfplane
rm mth principal congruence subgroup of modular group r
rl Modular group, PSL(2, 2)
zn Symmetric group on n symbols
all Fundamental group of an orientable compact 2-manifold of
genus g
Q Restriction of G to selfmappings of the unit disk
Q* Restriction of G* to selfmappings of the unit disk
Special symbols
C Field of complex numbers; the complex plane
e
Complex plane closed by a point at infinity
R Field of rational numbers
Z Ring of integers
-
x is the conjugate complex of I
I If H is a group, HI denotes its commutator subgroup.
Q Angle
--- Congruent
Elementary Concepts and Formulas
1.1 The 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: To 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 sub-
groups and, in some cases, quotient groups of subgrou-ps.W e 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 SL( 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 PSL(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.
2 Elementary Concepts and Formulas
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
(1- 2) A*2 = 1, A*gA*-’ = 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.
e
1.2 Action of G* on the Closed Complex Plane
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).T he
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 elemen-
tary 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 transforma-
tions in G) are conformal, .and those in G* but not in G reverse the oriepta-
e.
tion of The mappings t, are the only conformal transformations of C.