Outer Circles. An Introduction to Hyperbolic 3-Manifolds
Three-dimensional geometry is a fascinating topic, which has gone through an enormous evolution over the last 40 years. There are already several books describing this topic (e.g. by W. Thurston, R. Benedetti and C. Petronio and J. G. Ratcliffe). There are also books covering the topic of Kleinian groups. This book offers an overview of both fields (coming from the side of Kleinian groups). The book has two parts. The first part covers the fundamental facts of the theory. The first four chapters cover basic facts on hyperbolic space, Riemann surfaces, discrete groups, basic properties of hyperbolic manifolds and 3-manifold topology. The last two chapters are devoted to specific topics (line geometry and hyperbolic trigonometry). This part of the book is mostly written in the style of a textbook. The second part (chapters 5 and 6) has a different character. It is an excellent survey of topics connected with the proof of three main conjectures in the field (the tameness conjecture, the density conjecture and the ending lamination conjecture). A special feature of the book is the 'exercises and explorations' section added at the end of each chapter, introducing many other complementary topics connected with the main theme of the book. The book contains a lot of material and can be very valuable for getting an overview of this very broad and important field of mathematics.