Table Of ContentUniversitext
Springer-Verlag Berlin Heidelberg GmbH
Riccardo Benedetti • CarIo Petronio
Lectures an
Hyperbolic Geometry
With 175 Figures
Springer
Riccardo Benedetti
Carlo Petronio
UniversitA degli Studi di Pisa
Dipartimento di Matematica
Via F. Buonarroti 2
56127 Pisa
Italy
e-mail: benedetti/[email protected]
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ISBN 978-3-540-55534-6 ISBN 978-3-642-58158-8 (eBook)
DOI 10.1007/978-3-642-58158-8
Mathematics Subject Classification (1991): Primary: 51MlO,32GI5, 58B20
Secondary: 51M05,55NXX, 57M25, 55S40, 57R20
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Printed on acid-free paper 41/3142ck-54321
A Martina, Paolo, Clara e Stefania
R.B.
Ai miei genitori, Vitta e Luciano
C.P.
Table of Contents
Preface xi
Chapter A.
Hyperbolic Space I
A.I Models for Hyperbolic Space 1
A.2 Isometries of Hyperbolic Space: Hyperboloid Model 3
A.3 Conformal Geometry . . . . . . . . . . . . . 7
A.4 Isometries of Hyperbolic Space: Disc and Half-space Models 22
A.5 Geodesics, Hyperbolic Subspaces and Miscellaneo1ls Facts 25
A.6 Curvature of Hyperbolic Space 37
Chapter B.
Hyperbolic Manifolds
and the Compact Two-dimensional Case 45
B.1 Hyperbolic, Elliptic and Flat Manifolds 45
B.2 Topology of Compact Oriented Surfaces 55
B.3 Hyperbolic, Elliptic and Flat Surfaces 58
BA Teichmiiller Space 61
Chapter C.
The Rigidity Theorem (Compact Case) 83
C.1 First St.ep of the Proof: Extension of Pseudo-isometries 84
C.2 Second Step of the Proof: Volume of Ideal Simplices 94
C.3 Gromov Norm of a Compact Manifold .... 103
C.4 Third Step of the Proof:
the Gromov Norm and the Volume Are Proportional 105
C.5 Conclusion of the Proof, Corollaries and Generalizations 121
Chapter D.
Margulis' Lemma and its Applications 133
D.1 Margulis' Lemma . 133
Vlll Table of Contpnts
D.2 Local Geometry of a Hyperbolic Manifold 140
D.3 Ends of a Hyperbolic ~lanifold 143
Chapter E.
The Space of Hyperbolic Manifolds
and the Volume Function .... 159
E.1 The Chahauty and the Geometric Topology 160
E.2 Convergence in the Geometric Topology: Opening Cusps.
The Case of Dimension at kast Three . . . . . . . . . . . li4
E.3 The Case of Dimension Different from Three.
Conclusions and Examples ............ . 184
EA The Three-dimensional Case: Jorgensen's Part of the
So-called Jorgensen-Thurston Theory . . . . . . . . 100
E.5 The Three-dimensional Case. Thurston's Hyperbolic
Surgery Theorem: Statement and Preliminaries 106
E.5-i Definition and First Properties of T3 (Non-compact
Three-manifolds with "Triangulation" "'it hout Vertices) 108
E.5-ii Hyperbolic Structures on an Element of T3
and Realization of the Complete Structure . 201
E.5-iii Elements of T3 and Standard Spines 207
E.5-iv Some Links 'Whose Complements
are Realized as Elements of T3 . . . . 210
E.6 Proof of Thurston's Hyperbolic Surgery TIH'orem 223
E.6-i Algebraic Eqnations of 1{(M)
(Hyperbolic Structures Supported by MET3) 224
E.6-ii Dimension of 1{(AJ): General Case 234
E.6-iii The Case A/ is Complete Hyperbolic:
the Space of Deformations . . . . . . . . 251
E.6-iv Completion of the Deformed Hyperholic
Structures and Conclusion of the Proof 256
E.7 Applications to the Study of the Vohlme Function and
Complements about Three-dimensional Hyperholic Geometry 267
Chapter F.
Bounded Cohomology, a Rough Outline 273
F.1 Singular Cohomology 273
F.2 Bounded Singular Cohomology 277
F.3 Flat Fiber Bundles . . . . . 280
F.4 Euler Class of a Flat Vector Bundle 287
F.5 Flat Vector Bundles on Surfaces and the l\Iilnor-Sulliyan Theorem 294
Table of Contents lX
F.6 Sullivan's Conjecture and Amenable Groups 303
Subject Index 321
Notation Index 324
References . . 326
Preface
In recent years hyperbolic geometry was the object and the inspiration for
an extensive study which produced important and often amazing results and
open questions: it suffices to recall W. P. Thurston's works about the topology
and geometry of three-manifolds and the theory of the so-called hyperbolic (or
negatively curved) groups. However, it is still difficult to find graduate-level
textbooks in the theory of hyperbolic manifolds, starting "from the beginning"
and giving a rather complete and reasonably accessible treatise of some recent
results. The authors became aware of this difficulty while preparing a course
to be held at the Universita degli Studi di Pisa, and this book originated from
the first notes sketched on that occasion, widely modified and expanded by a
profitable collaboration during several months. The aim of this text is to give
a modest contribution to the filling of the gap mentioned above, and to the
knowledge of such a fascinating field of mathematics.
* * *
One of the main themes of this book is the conflict between the "flexibil
ity" and the "rigidity" properties of the hyperbolic manifolds: the first rad
ical difference arises between the case of dimension 2 and the case of higher
dimensions (as proved in chapters B and C), an elementary feature of this
phenomenon being the difference between the Riemann mapping theorem and
Liouville's theorem, as pointed out in chapter A. This chapter is rather el
ementary and most of its material may be the object of an undergraduate
course.
In chapter B we prove the existence of continuous moduli of hyperbolic
structures on a compact surface via a parametrization of the so-called Te
ichmiiller space: we chose the Fenchel-Nielsen parametrization as it is fully
placed in the realm of hyperbolic geometry.
Chapter C is devoted to the proof of G. D. Mostow's rigidity theorem (in
the compact case). We say two words about this theorem (and the proof we
present in the book) in order to point out the difficulties we mentioned above:
we chose the Gromov-Thurston proof of the rigidity theorem, as it makes use
of a machinery coming essentially from hyperbolic geometry (while Mostow's
original proof made extensive use of analysis). However if you follow the chief
references [Gro2] and [Thl] a problem arises: the core of the proof (where
the differences with Mostow's methods are sharper) consists in establishing a
