A primer on mapping class groups:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Princeton, NJ
Princeton University Press
2012
|
| Schriftenreihe: | Princeton mathematical series
49 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | Literaturverzeichnis Seite 447-463 |
| Beschreibung: | XIV, 472 Seiten Diagramme 23 cm |
| ISBN: | 9780691147949 0691147949 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804148385300611072 |
|---|---|
| adam_text | Contents
Preface
xi
Acknowledgments
xiii
Overview
1
PART
1.
MAPPING CLASS GROUPS
15
1.
Curves, Surfaces, and Hyperbolic Geometry
17
1.1
Surfaces and Hyperbolic Geometry
17
1.2
Simple Closed Curves
22
1.3
The Change of Coordinates Principle
36
1.4
Three Facts about Homeomorphisms
41
2.
Mapping Class Group Basics
44
2.1
Definition and First Examples
44
2.2
Computations of the Simplest Mapping Class Groups
47
2.3
The Alexander Method
58
3. Dehn Twists 64
3.1
Definition and Nontriviality
64
3.2 Dehn
Twists and Intersection Numbers
69
3.3
Basic Facts about
Dehn
Twists
73
3.4
The Center of the Mapping Class Group
75
3.5
Relations between Two
Dehn
Twists
77
3.6
Cutting, Capping, and Including
82
4.
Generating the Mapping Class Group
89
4.1
The Complex of Curves
92
4.2
The
Birman
Exact Sequence
96
4.3
Proof of Finite Generation
104
4.4
Explicit Sets of Generators
107
5.
Presentations and Low-dimensional Homology
116
5.1
The Lantern Relation and #i(Mod(S);Z)
116
5.2
Presentations for the Mapping Class Group
124
5.3
Proof of Finite Presentability
134
Viii CONTENTS
5.4 Hopfs
Formula
and #2(Mod(S);
Z)
140
5.5
The
Euler
Class
146
5.6
Surface Bundles and the Meyer Signature Cocycle
153
6.
The Symplectic Representation and the
Torelli
Group
162
6.1
Algebraic Intersection Number as a Symplectic Form
162
6.2
The Euclidean Algorithm for Simple Closed Curves
166
6.3
Mapping Classes as Symplectic Automorphisms
168
6.4
Congruence Subgroups, Torsion-free Subgroups, and Residual Finiteness
176
6.5
The
Torelli
Group
181
6.6
The Johnson Homomorphism
190
7.
Torsion
200
7.1
Finite-order Mapping Classes versus Finite-order Homeomorphisms
200
7.2
Orbifolds, the
84(5 - 1)
Theorem, and the Ag
+ 2
Theorem
203
7.3
Realizing Finite Groups as Isometry Groups
213
7.4
Conjugacy Classes of Finite Subgroups
215
7.5
Generating the Mapping Class Group with Torsion
216
8.
The Dehn-Nielsen-Baer Theorem
219
8.1
Statement of the Theorem
219
8.2
The Quasi-isometry Proof
222
8.3
Two Other Viewpoints
236
9.
Braid Groups
239
9.1
The Braid Group: Three Perspectives
239
9.2
Basic Algebraic Structure of the Braid Group
246
9.3
The Pure Braid Group
248
9.4
Braid Groups and Symmetric Mapping Class Groups
253
PART
2. TEICHMÜLLER
SPACE AND MODULI SPACE
261
10. Teichmüller
Space
263
10.1
Definition of
Teichmüller
Space
263
10.2 Teichmüller
Space of the Torus
265
10.3
The Algebraic Topology
269
10.4
Two Dimension Counts
272
10.5
The
Teichmüller
Space of a Pair of Pants
275
10.6 Fenchel-Nielsen
Coordinates
278
10.7
The 9g
- 9
Theorem
286
11. Teichmüller
Geometry
294
11.1
Quasiconformal Maps and an Extremal Problem
294
11.2
Measured Foliations
300
11.3
Holomorphic Quadratic Differentials
308
11.4 Teichmüller
Maps and
TeichmüUer s
Theorems
320
11.5 Grötzsch s
Problem
325
CONTENTS
ІХ
11.6
Proof of
Teichmüller s
Uniqueness Theorem
327
11.7
Proof of
Teichmüller s
Existence Theorem
330
11.8
The
Teichmüller
Metric
337
12.
Moduli Space
342
12.1
Moduli Space as the Quotient of
Teichmüller
Space
342
12.2
Moduli Space of the Torus
345
12.3
Proper Discontinuity
349
12.4
Mumford s Compactness Criterion
353
12.5
The Topology at Infinity of Moduli Space
359
12.6
Moduli Space as a Classifying Space
362
PART
3.
THE CLASSIFICATION AND PSEUDO-ANOSOV THEORY
365
13.
The Nielsen-Thurston Classification
367
13.1
The Classification for the Torus
367
13.2
The Three Types of Mapping Classes
370
13.3
Statement of the Nielsen-Thurston Classification
376
13.4
Thurston s Geometric Classification of Mapping Tori
379
13.5
The Collar Lemma
380
13.6
Proof of the Classification Theorem
382
14.
Pseudo-Anosov Theory
390
14.1
Five Constructions
391
14.2
Pseudo-Anosov Stretch Factors
403
14.3
Properties of the Stable and Unstable Foliations
408
14.4
The Orbits of a Pseudo-Anosov Homeomorphism
414
14.5
Lengths and Intersection Numbers under Iteration
419
15.
Thurston s Proof
424
15.1
A Fundamental Example
424
15.2
A Sketch of the General Theory
434
15.3
Markov Partitions
442
15.4
Other Points of View
445
Bibliography
447
Index
465
The study of the mapping class group
Mod(S) is a classical topic that is experi¬
encing a renaissance. It lies at the juncture
of geometry, topology, and group theory.
This book explains as many important
theorems, examples, and techniques as
possible, quickly and directly, while at the
same time giving full details and keeping
the text nearly self-contained. The book
is suitable for graduate students.
A Primer on Mapping Class Groups
begins by explaining the main group-
theoretical properties of Mod(S). from
finite generation by
Dehn
twists and
low-dimensional homology to the Dehn-
Nielsen-Baer theorem. Along the way,
central objects and tools are introduced,
such as the
Birman
exact sequence,
the complex of curves, the braid group,
the symplectic representation, and the
Torelli
group. The book then introduces
Teichmüller
space and its geometry,
and uses the action of ModfS) on it to
prove the Nielsen-Thurston classifica¬
tion of surface homeomorphisms. Topics
include the topology of the moduli space
of Riemann surfaces, the connection
with surface bundles. pseudo-Anosov
theory, and Thurstons approach to the
classification.
Benson
Farb
is professor of math¬
ematics at the University of Chicago. He
is the editor of Problems on Mapping
Class Groups and Related Topics
and the coauthor of
Noncommutative
Algebra. Dan Margalit is assistant
professor of mathematics at Georgia
Institute of Technology.
PRINCETON MATHEMATICAL SERIES,
49
Phillip A. Griffiths, John
N.
Mather, and
Elias M.
Stein, Series Editors
To receive emails about new books in
your area of interest, sign up at
press.princeton.edu
PRINCETON UNIVERSITY PRESS
|
| any_adam_object | 1 |
| author | Farb, Benson 1967- Margalit, Dan 1976- |
| author_GND | (DE-588)1020817828 (DE-588)1020818212 |
| author_facet | Farb, Benson 1967- Margalit, Dan 1976- |
| author_role | aut aut |
| author_sort | Farb, Benson 1967- |
| author_variant | b f bf d m dm |
| building | Verbundindex |
| bvnumber | BV039560719 |
| classification_rvk | SK 260 |
| ctrlnum | (OCoLC)751023726 (DE-599)BSZ348294646 |
| dewey-full | 512.74 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.74 |
| dewey-search | 512.74 |
| dewey-sort | 3512.74 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV039560719 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T00:06:18Z |
| institution | BVB |
| isbn | 9780691147949 0691147949 |
| language | English |
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| oclc_num | 751023726 |
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| physical | XIV, 472 Seiten Diagramme 23 cm |
| publishDate | 2012 |
| publishDateSearch | 2012 |
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| publisher | Princeton University Press |
| record_format | marc |
| series | Princeton mathematical series |
| series2 | Princeton mathematical series |
| spelling | Farb, Benson 1967- (DE-588)1020817828 aut A primer on mapping class groups Benson Farb and Dan Margalit Princeton, NJ Princeton University Press 2012 © 2012 XIV, 472 Seiten Diagramme 23 cm txt rdacontent n rdamedia nc rdacarrier Princeton mathematical series 49 Literaturverzeichnis Seite 447-463 Klassengruppe (DE-588)4164018-4 gnd rswk-swf Homöomorphismus (DE-588)4352383-3 gnd rswk-swf Teichmüller-Modulgruppe (DE-588)4319741-3 gnd rswk-swf Teichmüller-Raum (DE-588)4131425-6 gnd rswk-swf Abbildung Mathematik (DE-588)4000044-8 gnd rswk-swf Klassifikation (DE-588)4030958-7 gnd rswk-swf Class groups (Mathematics) Abbildung Mathematik (DE-588)4000044-8 s Klassengruppe (DE-588)4164018-4 s DE-604 Teichmüller-Modulgruppe (DE-588)4319741-3 s Teichmüller-Raum (DE-588)4131425-6 s 1\p DE-604 Homöomorphismus (DE-588)4352383-3 s Klassifikation (DE-588)4030958-7 s 2\p DE-604 Margalit, Dan 1976- (DE-588)1020818212 aut Princeton mathematical series 49 (DE-604)BV000019035 49 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024412392&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024412392&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Farb, Benson 1967- Margalit, Dan 1976- A primer on mapping class groups Princeton mathematical series Klassengruppe (DE-588)4164018-4 gnd Homöomorphismus (DE-588)4352383-3 gnd Teichmüller-Modulgruppe (DE-588)4319741-3 gnd Teichmüller-Raum (DE-588)4131425-6 gnd Abbildung Mathematik (DE-588)4000044-8 gnd Klassifikation (DE-588)4030958-7 gnd |
| subject_GND | (DE-588)4164018-4 (DE-588)4352383-3 (DE-588)4319741-3 (DE-588)4131425-6 (DE-588)4000044-8 (DE-588)4030958-7 |
| title | A primer on mapping class groups |
| title_auth | A primer on mapping class groups |
| title_exact_search | A primer on mapping class groups |
| title_full | A primer on mapping class groups Benson Farb and Dan Margalit |
| title_fullStr | A primer on mapping class groups Benson Farb and Dan Margalit |
| title_full_unstemmed | A primer on mapping class groups Benson Farb and Dan Margalit |
| title_short | A primer on mapping class groups |
| title_sort | a primer on mapping class groups |
| topic | Klassengruppe (DE-588)4164018-4 gnd Homöomorphismus (DE-588)4352383-3 gnd Teichmüller-Modulgruppe (DE-588)4319741-3 gnd Teichmüller-Raum (DE-588)4131425-6 gnd Abbildung Mathematik (DE-588)4000044-8 gnd Klassifikation (DE-588)4030958-7 gnd |
| topic_facet | Klassengruppe Homöomorphismus Teichmüller-Modulgruppe Teichmüller-Raum Abbildung Mathematik Klassifikation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024412392&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024412392&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000019035 |
| work_keys_str_mv | AT farbbenson aprimeronmappingclassgroups AT margalitdan aprimeronmappingclassgroups |