Overview
- Expands on the second edition by including over 40 new lemmas, theorems, and corollaries, as well as a new section dedicated to arithmetic hyperbolic groups
- Offers a highly readable and self-contained exposition of the theoretical foundations of hyperbolic manifolds
- Provides readers with over 70 new exercises and features figures in color throughout
- Request lecturer material: sn.pub/lecturer-material
Part of the book series: Graduate Texts in Mathematics (GTM, volume 149)
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Table of contents (13 chapters)
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Front Matter
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Back Matter
Keywords
- Hyperbolic manifolds
- Euclidean geometry
- Spherical geometry
- Inversive geometry
- Isotopies of hyperbolic space
- Discrete groups
- Geometric manifolds
- Geometric surfaces
- Hyperbolic 3-manifolds
- Hyperbolic n-manifolds
- Geometrically finite n-manifolds
- Geometric orbifolds
- Low-dimensional geometry
- Low-dimensional topology
- Arithmetic hyperbolic groups
About this book
This book is an exposition of the theoretical foundations of hyperbolic manifolds. It is intended to be used both as a textbook and as a reference. This third edition greatly expands upon the second with an abundance of additional content, including a section dedicated to arithmetic hyperbolic groups. Over 40 new lemmas, theorems, and corollaries feature, along with more than 70 additional exercises. Color adds a new dimension to figures throughout.
The book is divided into three parts. The first part is concerned with hyperbolic geometry and discrete groups. The main results are the characterization of hyperbolic reflection groups and Euclidean crystallographic groups. The second part is devoted to the theory of hyperbolic manifolds. The main results are Mostow’s rigidity theorem and the determination of the global geometry of hyperbolic manifolds of finite volume. The third part integrates the first two parts in a development of the theory of hyperbolic orbifolds. The main result is Poincaré’s fundamental polyhedron theorem.The exposition is at the level of a second year graduate student with particular emphasis placed on readability and completeness of argument. After reading this book, the reader will have the necessary background to study the current research on hyperbolic manifolds.
From reviews of the second edition:
Designed to be useful as both textbook and a reference, this book renders a real service to the mathematical community by putting together the tools and prerequisites needed to enter the territory of Thurston’s formidable theory of hyperbolic 3-manifolds […] Every chapter is followed by historical notes, with attributions to the relevant literature, both of the originators of the idea present in the chapter and of modern presentation thereof. Victor V. Pambuccian, Zentralblatt MATH, Vol. 1106 (8), 2007
Authors and Affiliations
About the author
John G. Ratcliffe is Professor of Mathematics at Vanderbilt University. His research interests range from low-dimensional topology and hyperbolic manifolds to cosmology.
Bibliographic Information
Book Title: Foundations of Hyperbolic Manifolds
Authors: John G. Ratcliffe
Series Title: Graduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-3-030-31597-9
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2019
Hardcover ISBN: 978-3-030-31596-2Published: 07 November 2019
Softcover ISBN: 978-3-030-31599-3Published: 07 November 2020
eBook ISBN: 978-3-030-31597-9Published: 23 October 2019
Series ISSN: 0072-5285
Series E-ISSN: 2197-5612
Edition Number: 3
Number of Pages: XII, 800
Number of Illustrations: 8 b/w illustrations, 152 illustrations in colour
Topics: Geometry, Topology, Topological Groups, Lie Groups