Over the past three decades there has been a total revolution in the classic branch of mathematics called 3-dimensional topology, namely the discovery that most solid 3-dimensional shapes are hyperbolic 3-manifolds. This book introduces and explains hyperbolic geometry and hyperbolic 3- and 2-dimensional manifolds in the first two chapters and then goes on to develop the subject. The author discusses the profound discoveries of the astonishing features of these 3-manifolds, helping the reader to understand them without going into long, detailed formal proofs. The book is heavily illustrated with pictures, mostly in color, that help explain the manifold properties described in the text. Each chapter ends with a set of exercises and explorations that both challenge the reader to prove assertions made in the text, and suggest further topics to explore that bring additional insight. There is an extensive index and bibliography.
Albert Marden is a Professor of Mathematics at the University of Minnesota.
List of illustrations; Preface; 1. Hyperbolic space and its isometries; 2. Discrete groups; 3. Properties of hyperbolic manifolds; 4. Algebraic and geometric convergence; 5. Deformation spaces and the ends of manifolds; 6. Hyperbolization; 7. Line geometry; 8. Right hexagons and hyperbolic trigonometry; Bibliography; Index.