Foundations of Hyperbolic Manifolds

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This book provides a comprehensive exploration of the theoretical foundations of hyperbolic manifolds, serving as both a textbook and a reference. The third edition significantly expands on the second, introducing a wealth of new content, including a section on arithmetic hyperbolic groups, over 40 new lemmas, theorems, and corollaries, and more than 70 additional exercises. Enhanced with color figures, the book is structured into three parts. The first part focuses on hyperbolic geometry and discrete groups, highlighting the characterization of hyperbolic reflection groups and Euclidean crystallographic groups. The second part delves into the theory of hyperbolic manifolds, featuring key results like Mostow's rigidity theorem and the global geometry of finite volume hyperbolic manifolds. The third part integrates these concepts to develop the theory of hyperbolic orbifolds, culminating in Poincaré's fundamental polyhedron theorem. Aimed at second-year graduate students, the text emphasizes readability and thoroughness. Readers will gain a solid foundation for engaging with contemporary research on hyperbolic manifolds. Each chapter includes historical notes and references to both original ideas and modern interpretations, making it a valuable resource for the mathematical community.