Table Of ContentGraduate Texts in Mathematics
John G. Ratcliffe
Foundations
of Hyperbolic
Manifolds
Third Edition
Graduate Texts in Mathematics 149
Graduate Texts in Mathematics
Series Editors
Sheldon Axler
San Francisco State University, San Francisco, CA, USA
Kenneth Ribet
University of California, Berkeley, CA, USA
Advisory Board
Alejandro Adem, University of British Columbia
David Eisenbud, University of California, Berkeley & MSRI
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J. F. Jardine, University of Western Ontario
Jeffrey C. Lagarias, University of Michigan
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Jeremy Quastel, University of Toronto
Fadil Santosa, University of Minnesota
Barry Simon, California Institute of Technology
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Steven H. Weintraub, Lehigh University
Melanie Matchett Wood, University of California, Berkeley
Graduate Texts in Mathematics bridge the gap between passive study and
creative understanding,offering graduate-level introductions to advanced topics in
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characteristicfeaturesofthetheory.Althoughthesebooksarefrequentlyusedas
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John G. Ratcliffe
Foundations of Hyperbolic
Manifolds
Third Edition
John G. Ratcliffe
Department of Mathematics
Vanderbilt University
Nashville, TN, USA
ISSN 0072-5285 ISSN 2197-5612 (electronic)
Graduate Texts in Mathematics
ISBN 978-3-030-31596-2 ISBN 978-3-030-31597-9 (eB ook)
https://doi.org/10.1007/978-3-030-31597-9
Mathematics Subject Classification (2010): 57M50, 20H10, 30F40
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To Susan, Kimberly, Thomas, and Jack
Preface
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.
Particularemphasishasbeenplacedonreadabilityandcompletenessofar-
gument. Thetreatmentofthematerialisforthemostpartelementaryand
self-contained. Thereaderisassumedtohaveabasicknowledgeofalgebra
and topology at the (cid:12)rst-year graduate level of an American university.
Thebookisdividedintothreeparts. The(cid:12)rstpart, consistingofChap-
ters 1-7, is concerned with hyperbolic geometry and basic properties of
discrete groups of isometries of hyperbolic space. The main results are the
existence theorem for discrete re(cid:13)ection groups, the Bieberbach theorems,
and Selberg’s lemma. The second part, consisting of Chapters 8-12, is de-
votedtothetheoryofhyperbolicmanifolds. ThemainresultsareMostow’s
rigidity theorem and the determination of the structure of geometrically
(cid:12)nite hyperbolic manifolds. The third part, consisting of Chapter 13, in-
tegrates the (cid:12)rst two parts in a development of the theory of hyperbolic
orbifolds. The main results are the construction of the universal orbifold
covering space and Poincar(cid:19)e’s fundamental polyhedron theorem.
This book was written as a textbook for a one-year course. Chapters
1-7 can be covered in one semester, and selected topics from Chapters 8-
12 can be covered in the second semester. For a one-semester course on
hyperbolicmanifolds,the(cid:12)rsttwosectionsofChapter1andselectedtopics
fromChapters8-12arerecommended. Sincecompleteargumentsaregiven
in the text, the instructor should try to cover the material as quickly as
possiblebysummarizingthebasicideasanddrawinglotsofpictures. Ifall
the details are covered, there is probably enough material in this book for
a two-year sequence of courses.
Thereareover500exercisesinthisbook,whichshouldbereadaspartof
thetext. Theseexercisesrangeindi(cid:14)cultyfromelementarytomoderately
di(cid:14)cult, with the more di(cid:14)cult ones occurring toward the end of each set
of exercises. There is much to be gained by working on these exercises.
An honest e(cid:11)ort has been made to give references to the original pub-
lished sources of the material in this book. Most of these original papers
are well worth reading. The references are collected at the end of each
chapter in the section on historical notes.
Thisbookisacompleterevisionofmylecturenotesforaone-yearcourse
onhyperbolicmanifoldsthatIgaveattheUniversityofIllinoisduring1984.
vii
viii Preface
I wish to express my gratitude to:
(1) James Cannon for allowing me to attend his course on Kleinian
groups at the University of Wisconsin during the fall of 1980;
(2)WilliamThurstonforallowingmetoattendhiscourseonhyperbolic
3-manifolds at Princeton University during the academic year 1981-1982
and for allowing me to include his unpublished material on hyperbolic
Dehn surgery in Chapter 10;
(3) my colleagues at the University of Illinois who attended my course
on hyperbolic manifolds, Kenneth Appel, Richard Bishop, Robert Craggs,
George Francis, Mary-Elizabeth Hamstrom, and Joseph Rotman, for their
many valuable comments and observations;
(4) my colleagues at Vanderbilt University who attended my ongoing
seminar on hyperbolic geometry over the last seven years, Mark Baker,
Bruce Hughes, Christine Kinsey, Michael Mihalik, Efstratios Prassidis,
Barry Spieler, and Steven Tschantz, for their many valuable observations
and suggestions;
(5) my colleagues and friends, William Abiko(cid:11), Colin Adams, Boris
Apanasov, Richard Arenstorf, William Harvey, Linda Keen, Ruth Keller-
hals, Victor Klee, Bernard Maskit, Hans Munkholm, Walter Neumann,
Alan Reid, Robert Riley, Richard Skora, John Stillwell, Perry Susskind,
and Je(cid:11)rey Weeks, for their helpful conversations and correspondence;
(6) the library sta(cid:11) at Vanderbilt University for helping me (cid:12)nd the
references for this book;
(7) Ruby Moore for typing up my manuscript;
(8)theeditorialsta(cid:11)atSpringer-VerlagNewYorkforthecarefulediting
of this book.
I especially wish to thank my colleague, Steven Tschantz, for helping
me to prepare this book on my computer and for drawing most of the
3-dimensional (cid:12)gures on his computer.
Finally, I would like to encourage readers to send me their comments
and corrections concerning the text, exercises, and historical notes.
Nashville, June, 1994 John G. Ratcliffe
Preface to the Third Edition
The third edition includes hundreds of changes and additions, including
over 40 new lemmas, theorems, and corollaries. The following theorems
are new in the third edition: 7.5.3, 7.5.4, 7.5.7, 7.6.8, 12.3.7, 12.4.8, 12.4.9,
12.4.10, 12.4.11, 12.6.6, 12.6.7, 12.8.1, 12.8.2, 12.8.3, 12.8.4, 12.8.5, 12.8.6,
12.8.7,12.8.8,12.8.9,12.8.10,12.8.11,12.8.12,13.2.5,13.2.7,13.2.8,13.2.9,
13.2.10, and 13.2.11. Moreover, Theorems 6.8.7 and 8.1.5 have been en-
hanced, and Theorems 3.5.6, 4.4.3, 6.3.2, and 9.4.3 have new proofs.
The notation for the positive Lorentz group has changed from PO(n;1)
toO+(n;1)andPO(n;1)isnowde(cid:12)nedtobetheprojectiveLorentzgroup
O(n;1)=f(cid:6)Ig. Thenewnotationismoreconsistentwithstandardpractice.
The following are the major changes in the third edition. Almost all of
the (cid:12)gures have been rendered in color. The most important elements of a
(cid:12)gureareusuallycoloredred. AnintroductiontoenhancedCoxetergraphs
for hyperbolic Coxeter groups has been added to x7.1. A more thorough
development of the theory of crystallographic groups is now given in x7.5.
More(cid:12)nitenesspropertiesofgeometrically(cid:12)nitegroupshavebeenaddedto
Sections x12.4 and x12.6. The most signi(cid:12)cant change in the third edition
is the addition of x12.8 on arithmetic hyperbolic groups. This section was
originally planned for the (cid:12)rst edition. Finally, more theory on compact
geometric orbifolds has been added to x13.2, including an introduction to
2-dimensional geometric orbifolds.
Tomakeroomforallthenewmaterialinthethirdedition,thehistorical
noteshavebeencondensedbyremovingthetitlesofbooksandpapers. For
thereaderofane-versionofthethirdedition,thishardlymatters,sincethe
book is now hyperlinked. The references have been pruned and updated
and now have links back to the historical sections in which they are cited.
There are over 70 new exercises. The exercise sets for the important
Sections x3.5 and x4.7 have doubled in size. Solutions to all the exercises
in the third edition will be made available in a solution manual.
Finally, I wish to express my thanks to my colleague Steven Tschantz
for major help on technical aspects of the production of this book.
Nashville, August, 2019 John G. Ratcliffe
ix
Contents
Preface vii
Preface to the Third Edition ix
1 Euclidean Geometry 1
x1.1 Euclid’s Parallel Postulate............................. 1
x1.2 Independence of the Parallel Postulate .................. 7
x1.3 Euclidean n-Space.................................... 13
x1.4 Geodesics ........................................... 22
x1.5 Arc Length.......................................... 28
x1.6 Historical Notes...................................... 32
2 Spherical Geometry 34
x2.1 Spherical n-Space .................................... 34
x2.2 Elliptic n-Space ...................................... 40
x2.3 Spherical Arc Length ................................. 42
x2.4 Spherical Volume..................................... 43
x2.5 Spherical Trigonometry ............................... 46
x2.6 Historical Notes...................................... 51
3 Hyperbolic Geometry 52
x3.1 Lorentzian n-Space ................................... 52
x3.2 Hyperbolic n-Space................................... 59
x3.3 Hyperbolic Arc Length................................ 71
x3.4 Hyperbolic Volume ................................... 75
x3.5 Hyperbolic Trigonometry.............................. 78
x3.6 Historical Notes...................................... 95
4 Inversive Geometry 97
x4.1 Reflections .......................................... 97
x4.2 Stereographic Projection .............................. 104
x4.3 M(cid:127)obius Transformations .............................. 107
x4.4 Poincar(cid:19)e Extension ................................... 113
x4.5 The Conformal Ball Model ............................ 119
x4.6 The Upper Half-Space Model .......................... 128
x4.7 Classi(cid:12)cation of Transformations ....................... 133
x4.8 Historical Notes...................................... 140
x
