Outer circles. An introduction to hyperbolic 3-manifolds.

*(English)*Zbl 1149.57030
Cambridge: Cambridge University Press (ISBN 978-0-521-83974-7/hbk). xviii, 427 p. (2007).

As written in preface, the purpose of this book is to explain the geometry of hyperbolic manifolds, especially hyperbolic \(3\)-manifolds. Although there is quite a lot of literature on the subject, this book will play a special role for some of its features.

It consists of eight chapters, including 62 high quality computer-generated illustrations. Each chapter is closed with a remarkably long section of exercises and explorations. Indeed, some of them are longer than the contents of corresponding chapters.

Chapter 1 provides basic information about the geometry of two- and three-dimensional hyperbolic spaces and their isometries. Most of it is devoted to the study of Möbius transformations. The meaning of the book title “Outer circles” is explained at page 12.

Chapter 2 begins the study of groups of Möbius transformations. After introducing discrete groups, the classical classification of the elementary discrete groups is given shortly. The notion of limit sets and ordinary sets appears on stage. Finally, a brief introduction to covering spaces, orbifolds, Riemann surfaces and their uniformization follows.

Chapter 3 gathers basic properties of hyperbolic \(3\)-manifolds. The chapter includes Ahlfors’ finiteness theorem, characterizing the boundaries of hyperbolic \(3\)-manifolds, the universality of Kleinian groups, the thin-thick decomposition, fundamental polyhedra, and geometrically finite manifolds. The chapter ends with the Mostow rigidity theorem and its proof (outline).

Chapter 4 focuses on degenerate sequences of Kleinian groups. After a short review of Dehn surgery, Thurston’s hyperbolic Dehn surgery theorem is stated without proof, and the set of volumes of finite volume hyperbolic \(3\)-manifolds is explained to be well-ordered.

Chapters 5 and 6 are expository, but worth reading to non-experts in hyperbolic geometry, because the three great conjectures, that is, the tameness conjecture, the density conjecture, and the ending lamination conjecture, and the hyperbolization theorem of \(3\)-manifolds are discussed concisely.

After preparing the line geometry in Chapter 7, the last Chapter 8 discusses the formulas of hyperbolic trigonometry.

It consists of eight chapters, including 62 high quality computer-generated illustrations. Each chapter is closed with a remarkably long section of exercises and explorations. Indeed, some of them are longer than the contents of corresponding chapters.

Chapter 1 provides basic information about the geometry of two- and three-dimensional hyperbolic spaces and their isometries. Most of it is devoted to the study of Möbius transformations. The meaning of the book title “Outer circles” is explained at page 12.

Chapter 2 begins the study of groups of Möbius transformations. After introducing discrete groups, the classical classification of the elementary discrete groups is given shortly. The notion of limit sets and ordinary sets appears on stage. Finally, a brief introduction to covering spaces, orbifolds, Riemann surfaces and their uniformization follows.

Chapter 3 gathers basic properties of hyperbolic \(3\)-manifolds. The chapter includes Ahlfors’ finiteness theorem, characterizing the boundaries of hyperbolic \(3\)-manifolds, the universality of Kleinian groups, the thin-thick decomposition, fundamental polyhedra, and geometrically finite manifolds. The chapter ends with the Mostow rigidity theorem and its proof (outline).

Chapter 4 focuses on degenerate sequences of Kleinian groups. After a short review of Dehn surgery, Thurston’s hyperbolic Dehn surgery theorem is stated without proof, and the set of volumes of finite volume hyperbolic \(3\)-manifolds is explained to be well-ordered.

Chapters 5 and 6 are expository, but worth reading to non-experts in hyperbolic geometry, because the three great conjectures, that is, the tameness conjecture, the density conjecture, and the ending lamination conjecture, and the hyperbolization theorem of \(3\)-manifolds are discussed concisely.

After preparing the line geometry in Chapter 7, the last Chapter 8 discusses the formulas of hyperbolic trigonometry.

Reviewer: Masakazu Teragaito (Hiroshima)