## Noneuclidean Tesselations and Their Groups

In 1974 Wilhelm Magnus published the fascinating book

*Noneuclidean Tesselations and Their Groups*. We reproduce below a version of Magnus's Preface to this work:In the last decades of the eighteenth century, Georg Christoph Lichtenberg (1742-1799), Professor of Physics in Göttingen and essayist, published a sequence of "Commentaries on Hogarth's Engravings" which, a few years ago, were republished in English by I and G Herden (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 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 Reidemeister-Schreier 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 Siegel (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 arc 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 Poincaré 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 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 tesselatlons in planar euclidean, hyperbolic, and elliptic geometry and also with the triangle tesselations of the sphere.

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 Mervin 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.