© 2019

Foundations of Hyperbolic Manifolds


Part of the Graduate Texts in Mathematics book series (GTM, volume 149)

Table of contents

  1. Front Matter
    Pages i-xii
  2. John G. Ratcliffe
    Pages 1-33
  3. John G. Ratcliffe
    Pages 34-51
  4. John G. Ratcliffe
    Pages 52-96
  5. John G. Ratcliffe
    Pages 97-141
  6. John G. Ratcliffe
    Pages 142-184
  7. John G. Ratcliffe
    Pages 185-259
  8. John G. Ratcliffe
    Pages 260-333
  9. John G. Ratcliffe
    Pages 334-374
  10. John G. Ratcliffe
    Pages 375-433
  11. John G. Ratcliffe
    Pages 434-505
  12. John G. Ratcliffe
    Pages 506-596
  13. John G. Ratcliffe
    Pages 597-697
  14. John G. Ratcliffe
    Pages 698-765
  15. Back Matter
    Pages 766-800

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


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

Authors and affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

About the authors

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
  • Series Abbreviated Title Graduate Texts Mathematics
  • DOI
  • Copyright Information Springer Nature Switzerland AG 2019
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-030-31596-2
  • Softcover ISBN 978-3-030-31599-3
  • eBook ISBN 978-3-030-31597-9
  • 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
    Topological Groups, Lie Groups
  • Buy this book on publisher's site