The Geometry of Discrete Groups

This text is intended to serve as an introduction to the geometry of the action of discrete groups of Mobius transformations. The subject matter has now been studied with changing points of emphasis for over a hundred years, the most recent developments being connected with the theory of 3-manifolds: see, for example, the papers of Poincare [77] and Thurston [101]. About 1940, the now well-known (but virtually unobtainable) Fenchel-Nielsen manuscript appeared. Sadly, the manuscript never appeared in print, and this more modest text attempts to display at least some of the beautiful geo­ metrical ideas to be found in that manuscript, as well as some more recent material. The text has been written with the conviction that geometrical explana­ tions are essential for a full understanding of the material and that however simple a matrix proof might seem, a geometric proof is almost certainly more profitable. Further, wherever possible, results should be stated in a form that is invariant under conjugation, thus making the intrinsic nature of the result more apparent. Despite the fact that the subject matter is concerned with groups of isometries of hyperbolic geometry, many publications rely on Euclidean estimates and geometry. However, the recent developments have again emphasized the need for hyperbolic geometry, and I have included a comprehensive chapter on analytical (not axiomatic) hyperbolic geometry. It is hoped that this chapter will serve as a “dictionary” offormulae in plane hyperbolic geometry and as such will be of interest and use in its own right.

Table of Contents

CHAPTER 1 Preliminary Material

CHAPTER 2 Matrices

CHAPTER 3 Mobius Transformations on R

CHAPTER 4 Complex Mobius Transformations

CHAPTER 5 Discontinuous Groups

CHAPTER 6 Riemann Surfaces

CHAPTER 7 Hyperbolic Geometry

CHAPTER 8 Fuchsian Groups

CHAPTER 9 Fundamental Domains

CHAPTER 10 Finitely Generated Groups

CHAPTER 11 Universal Constraints on Fuchsian Groups

About the Author

Alan Frank Beardon (April 16, 1940) is a British mathematician. Beardon obtained his doctorate at Imperial College London in 1964, supervised by Walter Hayman. In 1970 he was appointed as a lecturer in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge, with promotions to readership and professorship until his retirement in 2007. He is an emeritus fellow of St. Catherine’s College, Cambridge.