Hyperbolic Manifolds and Kleinian Groups
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Author Description
A Kleinian group is a discrete subgroup of the isometry group of hyperbolic 3-space, which is also regarded as a subgroup of Möbius 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 Teichmüller 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.
Menrva's Summary
THE THESIS
This text investigates the intersection of hyperbolic geometry and complex analysis through the study of Kleinian groups and their associated hyperbolic manifolds. The authors, Katsuhiko Matsuzaki and Masahiko Taniguchi, synthesize foundational results from the mid-20th century to the present, focusing on the evolution of the field from Ahlfors and Bers to Thurston and Sullivan. The book serves as a rigorous mathematical framework for understanding the discrete subgroups of the isometry group of hyperbolic 3-space.
THE SCOPE MAP
What You Will Find
- Comprehensive coverage of Kleinian groups as subgroups of Möbius transformations.
- Theoretical integration of Teichmüller theory with hyperbolic 3-space geometry.
- Detailed examination of complex dynamical approaches to manifold theory.
- Fundamental theorems and modern perspectives on hyperbolic structures.
THE AUTHORITY PERSPECTIVE
Experts recognize this work as a standard reference for graduate-level students and researchers in geometric topology and complex analysis. Readers frequently note the high level of technical density, which requires a strong background in advanced mathematics to fully comprehend the proofs and derivations presented.
Page Count:
264
Publication Date:
1998-07-16
Publisher:
Clarendon Press
ISBN-10:
0198500629
ISBN-13:
9780198500629
Geometry
Topology
Mathematics
Complex analysis
