Hyperbolic Manifolds and Kleinian Groups [Hardcover]

A Kleinian group is a discrete subgroup of the isometry group of hyperbolic 3-space, which is also regarded as a subgroup of M??bius transformations in the complex plane. The present book is a comprehensive guide to theories of Kleinian groups from the viewpoints of hyperbolic geometry and complex analysis. After 1960, Ahlfors and Bers produced important work which helped make Kleinian groups an active area of complex analysis as a branch of Teichm??ller theory. Later, Thurston brought about a revolution in the field with his profound investigation of hyperbolic manifolds, and Sullivan developed an important complex dynamical approach. This book provides the fundamental results and key theorems necessary for access to the frontiers of the theory from a modern viewpoint.

0. Hyperbolic surfaces and Fuchsian groups: summary
1. Hyperbolic 3-manifolds
2. The basis of Kleinian group theory
3. Geometrically finite Kleinian groups
4. Finitely generated Kleinian groups
5. The sphere at infinity
6. Infinite ends of hyperbolic manifolds
7. Algebraic and geometric convergences
Appendix
References

The subject of this monograph is the modern theory of Kleinian groups as developed by Ahlfors, Bers, Thurston and their students. It prepares the reader for the current work on the subject. --Mathematical Reviews