Foundations of Hyperbolic Manifolds

793 Pages·2006·7.059 MB·English

by  John G. Ratcliffe

149
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(continued after index)
John G. Ratcliffe
Foundations of Hyperbolic
Manifolds
Second Edition
John G.Ratcliffe
Department of Mathematics
Stevenson Center 1326
Vanderbilt University
Nashville,Tennessee 37240
[email protected]
Editorial Board:
S.Axler K.A.Ribet
Department of Mathematics Department of Mathematics
San Francisco State University University of California,Berkeley
San Francisco,CA 94132 Berkeley,CA 94720-3840
USA USA
[email protected] [email protected]
Mathematics Subject Classification (2000):57M50,30F40,51M10,20H10
Library ofCongress Control Number:2006926460
ISBN-10:0-387-33197-2
ISBN-13:978-0387-33197-3
Printed on acid-free paper.
© 2006 Springer Science+Business Media,LLC
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Printed in the United States ofAmerica. (MVY)
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To Susan, Kimberly, and Thomas
Preface to the First Edition
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 first-year graduate level of an American university.
Thebookisdividedintothreeparts. Thefirstpart, 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 reflection 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
finite hyperbolic manifolds. The third part, consisting of Chapter 13, in-
tegrates the first 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´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,thefirsttwosectionsofChapter1andselectedtopics
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.
Thereareover500exercisesinthisbookwhichshouldbereadaspartof
thetext. Theseexercisesrangeindifficultyfromelementarytomoderately
difficult, with the more difficult ones occurring toward the end of each set
of exercises. There is much to be gained by working on these exercises.
An honest effort 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 to the First Edition
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 1981-82 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 Abikoff, 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 Jeffrey Weeks, for their helpful conversations and correspondence;
(6) the library staff at Vanderbilt University for helping me find the
references for this book;
(7) Ruby Moore for typing up my manuscript;
(8)theeditorialstaffatSpringer-VerlagNewYorkforthecarefulediting
of this book.
I especially wish to thank my colleague, Steven Tschantz, for helping
me prepare this book on my computer and for drawing most of the 3-
dimensional figures on his computer.
Finally, I would like to encourage the reader to send me your comments
and corrections concerning the text, exercises, and historical notes.
Nashville, June, 1994 John G. Ratcliffe
Preface to the Second Edition
Thesecondeditionisathoroughrevisionofthefirsteditionthatembodies
hundreds of changes, corrections, and additions, including over sixty new
lemmas, theorems, and corollaries. The following theorems are new in the
second edition: 1.4.1, 3.1.1, 4.7.3, 6.3.14, 6.5.14, 6.5.15, 6.7.3, 7.2.2, 7.2.3,
7.2.4, 7.3.1, 7.4.1, 7.4.2, 10.4.1, 10.4.2, 10.4.5, 10.5.3, 11.3.1, 11.3.2, 11.3.3,
11.3.4, 11.5.1, 11.5.2, 11.5.3, 11.5.4, 11.5.5, 12.1.4, 12.1.5, 12.2.6, 12.3.5,
12.5.5, 12.7.8, 13.2.6, 13.4.1. It is important to note that the numbering
oflemmas,theorems,corollaries,formulas,figures,examples,andexercises
may have changed from the numbering in the first edition.
The following are the major changes in the second edition. Section 6.3,
Convex Polyhedra, of the first edition has been reorganized into two sec-
tions, §6.3, Convex Polyhedra, and §6.4, Geometry of Convex Polyhedra.
Section 6.5, Polytopes, has been enlarged with a more thorough discussion
of regular polytopes. Section 7.2, Simplex Reflection Groups, has been
expanded to give a complete classification of the Gram matrices of spher-
ical, Euclidean, and hyperbolic n-simplices. Section 7.4, The Volume of a
Simplex, is a new section in which a derivation of Schla¨fli’s differential for-
mulaispresented. Section10.4, HyperbolicVolume, hasbeenexpandedto
includethecomputationofthevolumeofacompactorthotetrahedron. Sec-
tion 11.3, The Gauss-Bonnet Theorem, is a new section in which a proof
of the n-dimensional Gauss-Bonnet theorem is presented. Section 11.5,
Differential Forms, is a new section in which the volume form of a closed
orientablehyperbolicspace-formisderived. Section12.1,LimitSetsofDis-
crete Groups, of the first edition has been enhanced and subdivided into
two sections, §12.1, Limit Sets, and §12.2, Limit Sets of Discrete Groups.
The exercises have been thoroughly reworked, pruned, and upgraded.
There are over a hundred new exercises. Solutions to all the exercises in
the second edition will be made available in a solution manual.
Finally, I wish to express my gratitude to everyone that sent me correc-
tions and suggestions for improvements. I especially wish to thank Keith
Conrad, Hans-Christoph Im Hof, Peter Landweber, Tim Marshall, Mark
Meyerson, Igor Mineyev, and Kim Ruane for their suggestions.
Nashville, November, 2005 John G. Ratcliffe
ix
Contents
Preface to the First Edition vii
Preface to the Second Edition ix
1 Euclidean Geometry 1
§1.1. Euclid’s Parallel Postulate . . . . . . . . . . . . . . . . . . 1
§1.2. Independence of the Parallel Postulate . . . . . . . . . . . 7
§1.3. Euclidean n-Space . . . . . . . . . . . . . . . . . . . . . . . 13
§1.4. Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
§1.5. Arc Length. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
§1.6. Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Spherical Geometry 35
§2.1. Spherical n-Space . . . . . . . . . . . . . . . . . . . . . . . 35
§2.2. Elliptic n-Space . . . . . . . . . . . . . . . . . . . . . . . . 41
§2.3. Spherical Arc Length . . . . . . . . . . . . . . . . . . . . . 43
§2.4. Spherical Volume . . . . . . . . . . . . . . . . . . . . . . . 44
§2.5. Spherical Trigonometry . . . . . . . . . . . . . . . . . . . . 47
§2.6. Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Hyperbolic Geometry 54
§3.1. Lorentzian n-Space . . . . . . . . . . . . . . . . . . . . . . 54
§3.2. Hyperbolic n-Space . . . . . . . . . . . . . . . . . . . . . . 61
§3.3. Hyperbolic Arc Length . . . . . . . . . . . . . . . . . . . . 73
§3.4. Hyperbolic Volume . . . . . . . . . . . . . . . . . . . . . . 77
§3.5. Hyperbolic Trigonometry . . . . . . . . . . . . . . . . . . . 80
§3.6. Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . 98
4 Inversive Geometry 100
§4.1. Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
§4.2. Stereographic Projection . . . . . . . . . . . . . . . . . . . 107
§4.3. Mo¨bius Transformations . . . . . . . . . . . . . . . . . . . 110
§4.4. Poincar´e Extension . . . . . . . . . . . . . . . . . . . . . . 116
§4.5. The Conformal Ball Model . . . . . . . . . . . . . . . . . . 122
§4.6. The Upper Half-Space Model . . . . . . . . . . . . . . . . 131
§4.7. Classification of Transformations . . . . . . . . . . . . . . 136
§4.8. Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . 142
x
Contents xi
5 Isometries of Hyperbolic Space 144
§5.1. Topological Groups . . . . . . . . . . . . . . . . . . . . . . 144
§5.2. Groups of Isometries . . . . . . . . . . . . . . . . . . . . . 150
§5.3. Discrete Groups . . . . . . . . . . . . . . . . . . . . . . . . 157
§5.4. Discrete Euclidean Groups . . . . . . . . . . . . . . . . . . 165
§5.5. Elementary Groups . . . . . . . . . . . . . . . . . . . . . . 176
§5.6. Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . 185
6 Geometry of Discrete Groups 188
§6.1. The Projective Disk Model . . . . . . . . . . . . . . . . . . 188
§6.2. Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 194
§6.3. Convex Polyhedra . . . . . . . . . . . . . . . . . . . . . . . 201
§6.4. Geometry of Convex Polyhedra . . . . . . . . . . . . . . . 212
§6.5. Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
§6.6. Fundamental Domains . . . . . . . . . . . . . . . . . . . . 234
§6.7. Convex Fundamental Polyhedra . . . . . . . . . . . . . . . 246
§6.8. Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . 253
§6.9. Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . 261
7 Classical Discrete Groups 263
§7.1. Reflection Groups . . . . . . . . . . . . . . . . . . . . . . . 263
§7.2. Simplex Reflection Groups . . . . . . . . . . . . . . . . . . 276
§7.3. Generalized Simplex Reflection Groups . . . . . . . . . . . 296
§7.4. The Volume of a Simplex . . . . . . . . . . . . . . . . . . . 303
§7.5. Crystallographic Groups . . . . . . . . . . . . . . . . . . . 310
§7.6. Torsion-Free Linear Groups . . . . . . . . . . . . . . . . . 322
§7.7. Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . 332
8 Geometric Manifolds 334
§8.1. Geometric Spaces . . . . . . . . . . . . . . . . . . . . . . . 334
§8.2. Clifford-Klein Space-Forms . . . . . . . . . . . . . . . . . . 341
§8.3. (X,G)-Manifolds . . . . . . . . . . . . . . . . . . . . . . . 347
§8.4. Developing . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
§8.5. Completeness . . . . . . . . . . . . . . . . . . . . . . . . . 361
§8.6. Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
§8.7. Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . 373
9 Geometric Surfaces 375
§9.1. Compact Surfaces . . . . . . . . . . . . . . . . . . . . . . . 375
§9.2. Gluing Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 378
§9.3. The Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . 390
§9.4. Moduli Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 391
§9.5. Closed Euclidean Surfaces . . . . . . . . . . . . . . . . . . 401
§9.6. Closed Geodesics . . . . . . . . . . . . . . . . . . . . . . . 404
§9.7. Closed Hyperbolic Surfaces . . . . . . . . . . . . . . . . . . 411