Buildings: Theory and Applications
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| Hauptverfasser: | , |
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| Format: | Buch |
| Sprache: | English |
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New York, NY
Springer
2008
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| Schriftenreihe: | Graduate Texts in Mathematics
248 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXI, 747 S. Ill., graph. Darst. 235 mm x 155 mm |
| ISBN: | 9780387788340 9780387788357 0387788344 |
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MARC
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| 020 | |a 9780387788340 |9 978-0-387-78834-0 | ||
| 020 | |a 9780387788357 |c Gb. : ca. EUR 58.80 (freier Pr.), ca. sfr 96.00 (freier Pr.) |9 978-0-387-78835-7 | ||
| 020 | |a 0387788344 |c Gb. : ca. EUR 58.80 (freier Pr.), ca. sfr 96.00 (freier Pr.) |9 0-387-78834-4 | ||
| 024 | 3 | |a 9780387788340 | |
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| 100 | 1 | |a Abramenko, Peter |d 1960- |e Verfasser |0 (DE-588)138001995 |4 aut | |
| 245 | 1 | 0 | |a Buildings |b Theory and Applications |c Peter Abramenko ; Kenneth S. Brown |
| 264 | 1 | |a New York, NY |b Springer |c 2008 | |
| 300 | |a XXI, 747 S. |b Ill., graph. Darst. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate Texts in Mathematics |v 248 | |
| 650 | 4 | |a Immeubles (Théorie des groupes) | |
| 650 | 4 | |a Buildings (Group theory) | |
| 650 | 0 | 7 | |a Gebäude |g Mathematik |0 (DE-588)4123258-6 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Gebäude |g Mathematik |0 (DE-588)4123258-6 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Brown, Kenneth S. |d 1945- |e Verfasser |0 (DE-588)136633293 |4 aut | |
| 830 | 0 | |a Graduate Texts in Mathematics |v 248 |w (DE-604)BV000000067 |9 248 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016699658&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016699658 | ||
Datensatz im Suchindex
| _version_ | 1804137960938930176 |
|---|---|
| adam_text | Contents
Preface
........................................................
vii
Introduction
................................................... 1
0.1
Coxeter Groups and Coxeter Complexes
................... 2
0.2
Buildings as Simplicial Complexes
........................ 4
0.3
Buildings as W-Metric Spaces
............................ 5
0.4
Buildings and Groups
................................... 6
0.5
The Moufang Property and the Classification Theorem
...... 6
0.6
Euclidean Buildings
..................................... 7
0.7
Buildings as Metric Spaces
............................... 7
0.8
Applications of Buildings
................................ 8
0.9
A Guide for the Reader
.................................. 8
1
Finite Reflection Groups
................................... 9
1.1
Definitions
............................................. 9
1.2
Examples
.............................................. 11
1.3
Classification
........................................... 15
1.4
Cell Decomposition
..................................... 17
1.4.1
Cells
........................................... 17
1.4.2
Closed Cells and the Face Relation
................. 19
1.4.3
Panels and Walls
................................ 21
1.4.4
Simplicial Cones
................................. 23
1.4.5
A Condition for a Chamber to Be Simplicial
........ 24
1.4.6
Semigroup Structure
............................. 25
1.4.7
Example: The Braid Arrangement
................. 28
1.4.8
Formal Properties of the
Poset
of Cells
............. 29
1.4.9
The Chamber Graph
............................. 30
1.5
The Simplicial Complex of a Reflection Group
.............. 35
1.5.1
The Action of
W
on E{W, V)
..................... 36
1.5.2
The Longest Element of
W
....................... 40
1.5.3
Examples
....................................... 41
Contents
1.5.4
The Chambers Are Simplicial
..................... 45
1.5.5
The Coxeter Matrix
.............................. 48
1.5.6
The Coxeter Diagram
............................ 49
1.5.7
Fundamental Domain and Stabilizers
............... 51
1.5.8
The
Poset
Σ
as a Simplicial Complex
.............. 52
1.5.9
A Group-Theoretic Description
oí
Σ...............
53
1.5.10
Roots and Half-Spaces
............................ 55
1.6
Special Properties of
Σ
.................................. 58
1.6.1
Σ
Is a Flag Complex
............................. 59
1.6.2
Σ
Is a Colorable Chamber Complex
................ 59
1.6.3
Σ
Is Determined by Its Chamber System
........... 62
Coxeter Groups
............................................ 65
2.1
The Action on Roots
.................................... 65
2.2
Examples
.............................................. 67
2.2.1
Finite Reflection Groups
.......................... 67
2.2.2
The Infinite Dihedral Group
...................... 67
2.2.3
The Group PGL2(Z)
............................. 71
2.3
Consequences of the Deletion Condition
................... 78
2.3.1
Equivalent Forms of (D)
.......................... 78
2.3.2
Parabolic Subgroups and Cosets
................... 80
2.3.3
The Word Problem
.............................. 85
*2.3.4 Counting Cosets
................................. 88
2.4
Coxeter Groups
......................................... 91
2.5
The Canonical Linear Representation
...................... 92
2.5.1
Construction of the Representation
................ 93
2.5.2
The Dual Representation
......................... 95
2.5.3
Roots, Walls, and Chambers
...................... 96
2.5.4
Finite Coxeter Groups
............................ 97
2.5.5
Coxeter Groups and Geometry
.................... 99
2.5.6
Applications of the Canonical Linear Representation
. 100
*2.6 The Tits Cone
..........................................102
2.6.1
Cell Decomposition
..............................103
2.6.2
The Finite Subgroups of
W
.......................105
2.6.3
The Shape of X
.................................107
*2.7 Infinite
Hyperplane
Arrangements
........................107
Coxeter Complexes
........................................115
3.1
The Coxeter Complex
...................................115
3.2
Local Properties of Coxeter Complexes
....................119
3.3
Construction of Chamber Maps
...........................124
3.3.1
Generalities
.....................................124
3.3.2
Automorphisms
.................................125
3.3.3
Construction of Foldings
..........................126
3.4
Roots
.................................................128
Contents xi
3.4.1
Foldings
........................................129
3.4.2
Characterization of Coxeter Complexes
.............138
3.5
The Weyl Distance Function
.............................144
3.6
Products and Convexity
.................................146
3.6.1
Sign Sequences
..................................147
3.6.2
Convex Sets of Chambers
.........................148
3.6.3
Supports
.......................................149
3.6.4
Semigroup Structure
.............................150
3.6.5
Applications of Products
.........................156
3.6.6
Convex
Subcomplexes
............................158
3.6.7
The Support of a Vertex
..........................167
3.6.8
Links Revisited; Nested Roots
.....................169
Buildings as Chamber Complexes
..........................173
4.1
Definition and First Properties
...........................173
4.2
Examples
..............................................177
4.3
The Building Associated to a Vector Space
.................182
4.4
Retractions
............................................185
4.5
The Complete System of Apartments
......................191
4.6
Subbuildings
...........................................194
4.7
The Spherical Case
.....................................195
4.8
The Weyl Distance Function
.............................198
4.9
Projections (Products)
..................................202
4.10
Applications of Projections
...............................204
4.11
Convex
Subcomplexes
...................................207
4.11.1
Chamber
Subcomplexes
..........................208
*4.11.2
General
Subcomplexes............................
209
*4.12 The Homotopy Type of a Building
........................212
*4.13 The Axioms for a Thick Building
.........................214
Buildings as W-Metric Spaces
..............................217
5.1
Buildings of Type {W,S)
................................217
5.1.1
Definition and Basic Facts
........................218
5.1.2
Galleries and Words
..............................221
5.2
Buildings as Chamber Systems
...........................223
5.3
Residues and Projections
................................226
5.3.1
J-Residues
......................................226
5.3.2
Projections and the Gate Property
.................229
5.4
Convexity and
Subbuildings
..............................233
5.4.1
Convex Sets
.....................................233
5.4.2
Subbuildings
....................................235
*5.4.3 2-Convexity
.....................................237
5.5
Isometries and Apartments
...............................238
5.5.1
Isometries and
σ
-Isometries
.......................
238
5.5.2
Characterizations of Apartments
...................240
xii Contents
5.5.3
Existence
of Apartments
..........................242
5.5.4
Roots
..........................................245
5.6
W-Metric Spaces Versus Chamber Complexes
..............246
5.7
Spherical Buildings
......................................254
5.7.1
Opposition
......................................254
*5.7.2 A Metric Characterization of Opposition
............255
5.7.3
The Thin Case
..................................257
5.7.4
Computation of
σ0
...............................258
5.7.5
Projections
.....................................260
5.7.6
Apartments
.....................................262
5.7.7
The Dual of a Spherical Building
..................263
*5.8 Twin Buildings
.........................................266
5.8.1
Definition and First Examples
.....................266
5.8.2
Easy Consequences
..............................269
5.8.3
Projections and Convexity
........................270
5.8.4
Twin Apartments
................................275
5.8.5
Twin Roots
.....................................282
5.9
A Rigidity Theorem
.....................................286
5.10
An Extension Theorem
..................................289
*5.11 An Extension Theorem for Twin Buildings
.................290
*5.12 Covering Maps
.........................................291
6
Buildings and Groups
......................................295
6.1
Group Actions on Buildings
..............................295
6.1.1
Strong Transitivity
...............................296
6.1.2
Example
........................................298
6.1.3
Weyl Transitivity
................................300
6.1.4
The Bruhat Decomposition
.......................302
6.1.5
The Strongly Transitive Case
......................303
6.1.6
Group-Theoretic Consequences
....................304
6.1.7
The Thick Case
.................................306
6.1.8
Stabilizers
......................................307
6.2
Bruhat Decompositions, Tits Subgroups,
and BN-Pairs
..........................................307
6.2.1
Bruhat Decompositions
...........................307
6.2.2
Axioms for Bruhat Decompositions
................309
6.2.3
The Thick Case
.................................312
6.2.4
Parabolic Subgroups
.............................315
6.2.5
Strongly Transitive Actions
.......................317
6.2.6
BN-Pairs
.......................................319
6.2.7
Simplicity Results
...............................322
*6.3 Twin BN-Pairs and Twin Buildings
.......................325
6.3.1
Group Actions on Twin Buildings
..................325
6.3.2
Group-Theoretic Consequences
....................328
6.3.3
Twin BN-Pairs
..................................330
6.4
Historical Remarks
......................................335
Contents xiii
6.5
Example: The General Linear Group
......................338
6.6
Example: The Symplectic Group
..........................340
6.7
Example: Orthogonal Groups
.............................344
6.7.1
The Standard Quadratic Form
....................344
6.7.2
More General Quadratic Forms
....................348
6.8
Example: Unitary Groups
................................349
6.9
Example: SLn over a Field with Discrete Valuation
..........351
6.9.1
Discrete Valuations
..............................351
6.9.2
The Group SLn(K)
..............................354
6.9.3
The Group SLn(K), Concluded
....................360
*6.10 Example: Weyl-Transitive Actions
........................364
6.10.1
Dense Subgroups
................................364
6.10.2
Dense Subgroups of SL2(QP)
......................364
*6.11 Example: Norm-1 Groups of Quaternion Algebras
...........365
6.11.1
Quaternion Algebras
.............................365
6.11.2
Density Lemmas
.................................368
6.11.3
Norm-1 Groups over
Q
and Buildings
..............369
*6.12 Example: A Twin BN-Pair
...............................371
Root Groups and the Moufang Property
...................375
7.1
Pre-Moufang Buildings and BN-Pairs
......................375
7.2
Calculation of Fixers
....................................380
7.2.1
Preliminaries: Convex Sets of Roots
................380
7.2.2
Fixers
..........................................382
7.3
Root Groups and Moufang Buildings
......................385
7.3.1
Definitions and Simple Consequences
...............385
7.3.2
Links
...........................................387
7.3.3
Subbuildings
....................................389
7.3.4
The Building Associated to a Vector Space
..........390
7.4
fc-Interiors of Roots
.....................................393
7.5
Consequences of the Rigidity Theorem
.....................400
7.6
Spherical Buildings of Rank at Least
3 ....................402
7.7
Group-Theoretic Consequences of the Moufang Property
.....403
7.7.1
The Groups U±, B±, and Uw
.....................403
7.7.2
Commutator Relations
...........................407
7.7.3
The Role of the Commutator Relations
.............408
7.7.4
The Structure of
[α, β]
...........................409
7.8
RGD Systems of Spherical Type
..........................411
7.8.1
The RGD Axioms
...............................411
7.8.2
Rank-1 Groups
..................................413
7.8.3
The Weyl Group
.................................417
7.8.4
The Groups Uw
.................................421
7.8.5
The BN-Pair and the Associated Moufang Building
.. 426
7.8.6
The Kernel of the Action
.........................430
7.8.7
Simplicity Results
...............................431
Contents
7.9
Examples of RGD Systems
...............................432
7.9.1
Classical Groups
.................................432
7.9.2
Chevalley Groups
................................440
*7.9.3 Nonsplit Algebraic Groups
........................443
Moufang Twin Buildings and RGD Systems
................449
8.1
Pre-Moufang Twin Buildings and Twin BN-Pairs
...........449
8.2
Calculation of Fixers
....................................452
8.2.1
Preliminaries: Convex Sets of Twin Roots
...........452
8.2.2
Fixers
..........................................453
8.3
Root Groups and Moufang Twin Buildings
.................454
8.3.1
Definitions and Simple Consequences
...............454
8.3.2
Example
........................................456
8.4
2-Spherical Twin Buildings of Rank at Least
3..............457
8.5
Group-Theoretic Consequences of the Moufang Property
.....459
8.5.1
The Groups U±, B±,andUw
.....................459
8.5.2
Commutator Relations
...........................462
8.5.3
Prenilpotent Pairs
...............................463
8.6
General RGD Systems
...................................466
8.6.1
The RGD Axioms
...............................466
8.6.2
The Weyl Group
.................................468
8.6.3
The Groups Uw
.................................471
8.6.4
The Building C{G, B+)
...........................473
8.7
A 2-Covering of C(G, B+)
................................476
8.7.1
The Chamber System
С
..........................477
8.7.2
The Morphism
к: С
->
C+
........................479
8.7.3
The Groups U w
.................................479
8.7.4
Spherical Residues
...............................481
8.7.5
The Main Result
................................483
8.8
Algebraic Consequences
.................................484
8.9
The Moufang Twin Building
.............................486
8.10
A Presentation of U+
....................................487
8.11
Groups of
Кас
-Moody Type
.............................491
8.11.1
Cartan Matrices
.................................492
8.11.2
Finite-Dimensional Lie Algebras
...................493
8.11.3
Кас
-Moody Algebras
............................494
8.11.4
The Weyl Group
.................................495
8.11.5
Кас
-Moody Groups
..............................495
8.11.6
The Simply Laced Case
...........................496
The Classification of Spherical Buildings
...................499
9.1
Introduction
...........................................499
9.2
Type An
...............................................501
9.3
Type Cn
...............................................502
9.4
Type Dn
...............................................503
Contents xv
9.5
Type En
...............................................504
9.6
Digression: Twisted Chevalley Groups
.....................504
9.7
Type F4
...............................................505
9.8
Type G2
...............................................506
9.9
Type I2(8)
.............................................506
9.10
Finite Simple Groups and Finite Buildings
.................506
*9.11 Remarks on the Simplified Proof
..........................508
*9.12 The Classification of Twin Buildings
......................509
10
Euclidean and Hyperbolic Reflection Groups
...............511
10.1
Euclidean Reflection Groups
.............................511
10.1.1 Affine
Concepts
.................................511
10.1.2
Formulas for
Affine
Reflections
....................513
10.1.3 Affine
Reflection Groups
..........................514
10.1.4
Finiteness Results
...............................516
10.1.5
The Structure of
С
..............................517
10.1.6
The Structure of W, Part I
.......................521
10.1.7
Example
........................................523
П0.1.8
The Structure of W, Part II;
Affine Weyl
Groups
.... 526
10.2
Euclidean Coxeter Groups and Complexes
.................530
10.2.1
A Euclidean Metric on
Σ
........................530
*10.2.2 Connection with the Tits Cone
....................533
*10.3
Hyperbolic Reflection Groups
............................537
10.3.1
Hyperbolic Space;
Hyperplanes
and Reflections
......537
10.3.2
Reflection Groups in
ШР
..........................539
10.3.3
Example
........................................541
10.3.4
The
Poset
of Cells
...............................542
10.3.5
The Simplicial Case
..............................544
10.3.6
The General Case
................................545
*10.4 Hyperbolic Coxeter Groups and Complexes
................546
11
Euclidean Buildings
........................................549
11.1
CATCO) Spaces
.........................................549
11.2
Euclidean Buildings as Metric Spaces
......................554
11.3
The Bruhat-Tits Fixed-Point Theorem
....................558
11.4
Application: Bounded Subgroups
.........................561
11.5
Bounded Subsets of Apartments
..........................565
11.6
A Metric Characterization of the Apartments
...............569
11.7
Construction of Apartments
..............................573
11.8
The Spherical Building at Infinity
.........................579
11.8.1
Ideal Points and Ideal Simplices
...................580
11.8.2
Construction of the Building at Infinity
.............582
11.8.3
Type-Preserving Maps
...........................584
11.8.4
Incomplete Apartment Systems
....................585
11.8.5
Group-Theoretic Consequences
....................587
xvi Contents
11.8.6
Example
........................................590
11.9
Classification
...........................................592
*11.10 Moufang Euclidean Buildings
.............................593
12
Buildings as Metric Spaces
.................................597
12.1
Metric Realizations of Buildings
..........................598
12.1.1
The ^-Realization as a Set
........................598
12.1.2
A Metric on X
..................................601
12.1.3
From Apartments to Chambers and Back Again
.....604
12.1.4
The Effect of a Chamber Map
.....................605
12.1.5
The Carrier of a Point of X
.......................606
12.1.6
Chains and Galleries
.............................607
12.1.7
Existence of Geodesies
...........................610
12.1.8
Curvature
......................................611
12.2
Special Cases
...........................................615
12.3
The Dual Coxeter Complex
..............................617
12.3.1
Introduction
....................................617
12.3.2
Examples
.......................................618
12.3.3
Construction of the Dual Coxeter Complex
..........620
12.3.4
Properties
......................................622
12.3.5
Remarks on the Spherical Case
....................623
12.3.6
The Euclidean and Hyperbolic Cases
...............624
12.3.7
A Fundamental Domain
..........................624
12.3.8
A CAT(0) Metric on X
...........................626
12.3.9
The Gromov Hyperbolic Case: A CAT(-l) Metric
... 627
12.3.10
A Cubical Subdivision of Ed
......................628
12.4
The Davis Realization of a Building
.......................629
13
Applications to the Cohomology of Groups
.................633
13.1
Arithmetic Groups over the Rationals
.....................633
13.1.1
Definition
.......................................633
13.1.2
The Symmetric Space
............................634
13.1.3
The Cocompact Case
.............................635
13.1.4
The General Case
................................636
13.1.5
Virtual Notions
..................................638
13.2
S-Arithmetic Groups
....................................639
13.2.1
A p-adic Analogue of the Symmetric Space
..........640
13.2.2
Cohomology of S-Arithmetic Groups: Method
1.....641
13.2.3
Cohomology of S-Arithmetic Groups: Method
2.....642
13.2.4
The Nonreductive Case
...........................642
13.3
Discrete Subgroups of p-adic Groups
......................644
13.4
Cohomological Dimension of Linear Groups
................645
13.5
S-Arithmetic Groups over Function Fields
.................647
Contents xvii
14
Other
Applications.........................................651
14.1
Presentations of Groups
.................................651
14.1.1
Chamber-Transitive Actions
.......................652
14.1.2
Further Results for BN-Pairs
......................653
14.1.3
The Group U+
..................................654
14.1.4
S -Arithmetic Groups
.............................654
14.2
Finite Groups
..........................................655
14.3
Differential Geometry
...................................657
14.3.1
Mostów
Rigidity
.................................657
14.3.2
Further Rigidity Theorems
........................658
14.3.3
Isoparametric Submanifolds
.......................658
14.3.4
Singular Spaces and p-adic Groups
.................659
14.4
Representation Theory and Harmonic Analysis
.............660
A Cell Complexes
............................................661
A.I Simplicial Complexes
....................................661
A.I.I Definitions
......................................661
A.I.
2
Flag Complexes
.................................663
A.1.3 Chamber Complexes and Type Functions
...........664
A.1.4 Chamber Systems
................................668
*A.2 Regular Cell Complexes
.................................670
A.
2.1
Definitions and First Properties
...................670
A.2.2 Regular Cell Complexes from Polytopes
............673
A.
2.3
Regular Cell Complexes from Arrangements
.........674
A.3 Cubical Realizations of Posets
............................676
В
Root Systems
..............................................681
B.I Notation
...............................................681
B.2 Definition and First Properties
...........................682
B.3 The Dual Root System
..................................683
B.4 Examples
..............................................683
*B.5 Nonreduced Root Systems
...............................684
С
Algebraic Groups
..........................................685
C.I Group Schemes
.........................................685
C.2 The
Affine
Algebra of
G
.................................687
C.3 Extension of
Scalare
.....................................688
C.4 Group Schemes from Groups
.............................688
C.5 Linear Algebraic Groups
.................................689
C.6 Tori
...................................................690
C.7 Unipotent Groups
.......................................690
C.8 Connected Groups
......................................691
C.9 Reductive,
Semisimple,
and Simple Groups
.................691
CIO BN-Pairs and Spherical Buildings
.........................691
СИ
BN-Pairs and Euclidean Buildings
............:...........693
xviii Contents
С.
12 Group
Schemes
versus
Groups
............................693
Hints/Solutions/Answers to Selected Exercises
................695
References
.....................................................719
Notation Index
................................................737
Subject Index
.................................................741
|
| adam_txt |
Contents
Preface
.
vii
Introduction
. 1
0.1
Coxeter Groups and Coxeter Complexes
. 2
0.2
Buildings as Simplicial Complexes
. 4
0.3
Buildings as W-Metric Spaces
. 5
0.4
Buildings and Groups
. 6
0.5
The Moufang Property and the Classification Theorem
. 6
0.6
Euclidean Buildings
. 7
0.7
Buildings as Metric Spaces
. 7
0.8
Applications of Buildings
. 8
0.9
A Guide for the Reader
. 8
1
Finite Reflection Groups
. 9
1.1
Definitions
. 9
1.2
Examples
. 11
1.3
Classification
. 15
1.4
Cell Decomposition
. 17
1.4.1
Cells
. 17
1.4.2
Closed Cells and the Face Relation
. 19
1.4.3
Panels and Walls
. 21
1.4.4
Simplicial Cones
. 23
1.4.5
A Condition for a Chamber to Be Simplicial
. 24
1.4.6
Semigroup Structure
. 25
1.4.7
Example: The Braid Arrangement
. 28
1.4.8
Formal Properties of the
Poset
of Cells
. 29
1.4.9
The Chamber Graph
. 30
1.5
The Simplicial Complex of a Reflection Group
. 35
1.5.1
The Action of
W
on E{W, V)
. 36
1.5.2
The Longest Element of
W
. 40
1.5.3
Examples
. 41
Contents
1.5.4
The Chambers Are Simplicial
. 45
1.5.5
The Coxeter Matrix
. 48
1.5.6
The Coxeter Diagram
. 49
1.5.7
Fundamental Domain and Stabilizers
. 51
1.5.8
The
Poset
Σ
as a Simplicial Complex
. 52
1.5.9
A Group-Theoretic Description
oí
Σ.
53
1.5.10
Roots and Half-Spaces
. 55
1.6
Special Properties of
Σ
. 58
1.6.1
Σ
Is a Flag Complex
. 59
1.6.2
Σ
Is a Colorable Chamber Complex
. 59
1.6.3
Σ
Is Determined by Its Chamber System
. 62
Coxeter Groups
. 65
2.1
The Action on Roots
. 65
2.2
Examples
. 67
2.2.1
Finite Reflection Groups
. 67
2.2.2
The Infinite Dihedral Group
. 67
2.2.3
The Group PGL2(Z)
. 71
2.3
Consequences of the Deletion Condition
. 78
2.3.1
Equivalent Forms of (D)
. 78
2.3.2
Parabolic Subgroups and Cosets
. 80
2.3.3
The Word Problem
. 85
*2.3.4 Counting Cosets
. 88
2.4
Coxeter Groups
. 91
2.5
The Canonical Linear Representation
. 92
2.5.1
Construction of the Representation
. 93
2.5.2
The Dual Representation
. 95
2.5.3
Roots, Walls, and Chambers
. 96
2.5.4
Finite Coxeter Groups
. 97
2.5.5
Coxeter Groups and Geometry
. 99
2.5.6
Applications of the Canonical Linear Representation
. 100
*2.6 The Tits Cone
.102
2.6.1
Cell Decomposition
.103
2.6.2
The Finite Subgroups of
W
.105
2.6.3
The Shape of X
.107
*2.7 Infinite
Hyperplane
Arrangements
.107
Coxeter Complexes
.115
3.1
The Coxeter Complex
.115
3.2
Local Properties of Coxeter Complexes
.119
3.3
Construction of Chamber Maps
.124
3.3.1
Generalities
.124
3.3.2
Automorphisms
.125
3.3.3
Construction of Foldings
.126
3.4
Roots
.128
Contents xi
3.4.1
Foldings
.129
3.4.2
Characterization of Coxeter Complexes
.138
3.5
The Weyl Distance Function
.144
3.6
Products and Convexity
.146
3.6.1
Sign Sequences
.147
3.6.2
Convex Sets of Chambers
.148
3.6.3
Supports
.149
3.6.4
Semigroup Structure
.150
3.6.5
Applications of Products
.156
3.6.6
Convex
Subcomplexes
.158
3.6.7
The Support of a Vertex
.167
3.6.8
Links Revisited; Nested Roots
.169
Buildings as Chamber Complexes
.173
4.1
Definition and First Properties
.173
4.2
Examples
.177
4.3
The Building Associated to a Vector Space
.182
4.4
Retractions
.185
4.5
The Complete System of Apartments
.191
4.6
Subbuildings
.194
4.7
The Spherical Case
.195
4.8
The Weyl Distance Function
.198
4.9
Projections (Products)
.202
4.10
Applications of Projections
.204
4.11
Convex
Subcomplexes
.207
4.11.1
Chamber
Subcomplexes
.208
*4.11.2
General
Subcomplexes.
209
*4.12 The Homotopy Type of a Building
.212
*4.13 The Axioms for a Thick Building
.214
Buildings as W-Metric Spaces
.217
5.1
Buildings of Type {W,S)
.217
5.1.1
Definition and Basic Facts
.218
5.1.2
Galleries and Words
.221
5.2
Buildings as Chamber Systems
.223
5.3
Residues and Projections
.226
5.3.1
J-Residues
.226
5.3.2
Projections and the Gate Property
.229
5.4
Convexity and
Subbuildings
.233
5.4.1
Convex Sets
.233
5.4.2
Subbuildings
.235
*5.4.3 2-Convexity
.237
5.5
Isometries and Apartments
.238
5.5.1
Isometries and
σ
-Isometries
.
238
5.5.2
Characterizations of Apartments
.240
xii Contents
5.5.3
Existence
of Apartments
.242
5.5.4
Roots
.245
5.6
W-Metric Spaces Versus Chamber Complexes
.246
5.7
Spherical Buildings
.254
5.7.1
Opposition
.254
*5.7.2 A Metric Characterization of Opposition
.255
5.7.3
The Thin Case
.257
5.7.4
Computation of
σ0
.258
5.7.5
Projections
.260
5.7.6
Apartments
.262
5.7.7
The Dual of a Spherical Building
.263
*5.8 Twin Buildings
.266
5.8.1
Definition and First Examples
.266
5.8.2
Easy Consequences
.269
5.8.3
Projections and Convexity
.270
5.8.4
Twin Apartments
.275
5.8.5
Twin Roots
.282
5.9
A Rigidity Theorem
.286
5.10
An Extension Theorem
.289
*5.11 An Extension Theorem for Twin Buildings
.290
*5.12 Covering Maps
.291
6
Buildings and Groups
.295
6.1
Group Actions on Buildings
.295
6.1.1
Strong Transitivity
.296
6.1.2
Example
.298
6.1.3
Weyl Transitivity
.300
6.1.4
The Bruhat Decomposition
.302
6.1.5
The Strongly Transitive Case
.303
6.1.6
Group-Theoretic Consequences
.304
6.1.7
The Thick Case
.306
6.1.8
Stabilizers
.307
6.2
Bruhat Decompositions, Tits Subgroups,
and BN-Pairs
.307
6.2.1
Bruhat Decompositions
.307
6.2.2
Axioms for Bruhat Decompositions
.309
6.2.3
The Thick Case
.312
6.2.4
Parabolic Subgroups
.315
6.2.5
Strongly Transitive Actions
.317
6.2.6
BN-Pairs
.319
6.2.7
Simplicity Results
.322
*6.3 Twin BN-Pairs and Twin Buildings
.325
6.3.1
Group Actions on Twin Buildings
.325
6.3.2
Group-Theoretic Consequences
.328
6.3.3
Twin BN-Pairs
.330
6.4
Historical Remarks
.335
Contents xiii
6.5
Example: The General Linear Group
.338
6.6
Example: The Symplectic Group
.340
6.7
Example: Orthogonal Groups
.344
6.7.1
The Standard Quadratic Form
.344
6.7.2
More General Quadratic Forms
.348
6.8
Example: Unitary Groups
.349
6.9
Example: SLn over a Field with Discrete Valuation
.351
6.9.1
Discrete Valuations
.351
6.9.2
The Group SLn(K)
.354
6.9.3
The Group SLn(K), Concluded
.360
*6.10 Example: Weyl-Transitive Actions
.364
6.10.1
Dense Subgroups
.364
6.10.2
Dense Subgroups of SL2(QP)
.364
*6.11 Example: Norm-1 Groups of Quaternion Algebras
.365
6.11.1
Quaternion Algebras
.365
6.11.2
Density Lemmas
.368
6.11.3
Norm-1 Groups over
Q
and Buildings
.369
*6.12 Example: A Twin BN-Pair
.371
Root Groups and the Moufang Property
.375
7.1
Pre-Moufang Buildings and BN-Pairs
.375
7.2
Calculation of Fixers
.380
7.2.1
Preliminaries: Convex Sets of Roots
.380
7.2.2
Fixers
.382
7.3
Root Groups and Moufang Buildings
.385
7.3.1
Definitions and Simple Consequences
.385
7.3.2
Links
.387
7.3.3
Subbuildings
.389
7.3.4
The Building Associated to a Vector Space
.390
7.4
fc-Interiors of Roots
.393
7.5
Consequences of the Rigidity Theorem
.400
7.6
Spherical Buildings of Rank at Least
3 .402
7.7
Group-Theoretic Consequences of the Moufang Property
.403
7.7.1
The Groups U±, B±, and Uw
.403
7.7.2
Commutator Relations
.407
7.7.3
The Role of the Commutator Relations
.408
7.7.4
The Structure of
[α, β]
.409
7.8
RGD Systems of Spherical Type
.411
7.8.1
The RGD Axioms
.411
7.8.2
Rank-1 Groups
.413
7.8.3
The Weyl Group
.417
7.8.4
The Groups Uw
.421
7.8.5
The BN-Pair and the Associated Moufang Building
. 426
7.8.6
The Kernel of the Action
.430
7.8.7
Simplicity Results
.431
Contents
7.9
Examples of RGD Systems
.432
7.9.1
Classical Groups
.432
7.9.2
Chevalley Groups
.440
*7.9.3 Nonsplit Algebraic Groups
.443
Moufang Twin Buildings and RGD Systems
.449
8.1
Pre-Moufang Twin Buildings and Twin BN-Pairs
.449
8.2
Calculation of Fixers
.452
8.2.1
Preliminaries: Convex Sets of Twin Roots
.452
8.2.2
Fixers
.453
8.3
Root Groups and Moufang Twin Buildings
.454
8.3.1
Definitions and Simple Consequences
.454
8.3.2
Example
.456
8.4
2-Spherical Twin Buildings of Rank at Least
3.457
8.5
Group-Theoretic Consequences of the Moufang Property
.459
8.5.1
The Groups U±, B±,andUw
.459
8.5.2
Commutator Relations
.462
8.5.3
Prenilpotent Pairs
.463
8.6
General RGD Systems
.466
8.6.1
The RGD Axioms
.466
8.6.2
The Weyl Group
.468
8.6.3
The Groups Uw
.471
8.6.4
The Building C{G, B+)
.473
8.7
A 2-Covering of C(G, B+)
.476
8.7.1
The Chamber System
С
.477
8.7.2
The Morphism
к: С'
->
C+
.479
8.7.3
The Groups U'w
.479
8.7.4
Spherical Residues
.481
8.7.5
The Main Result
.483
8.8
Algebraic Consequences
.484
8.9
The Moufang Twin Building
.486
8.10
A Presentation of U+
.487
8.11
Groups of
Кас
-Moody Type
.491
8.11.1
Cartan Matrices
.492
8.11.2
Finite-Dimensional Lie Algebras
.493
8.11.3
Кас
-Moody Algebras
.494
8.11.4
The Weyl Group
.495
8.11.5
Кас
-Moody Groups
.495
8.11.6
The Simply Laced Case
.496
The Classification of Spherical Buildings
.499
9.1
Introduction
.499
9.2
Type An
.501
9.3
Type Cn
.502
9.4
Type Dn
.503
Contents xv
9.5
Type En
.504
9.6
Digression: Twisted Chevalley Groups
.504
9.7
Type F4
.505
9.8
Type G2
.506
9.9
Type I2(8)
.506
9.10
Finite Simple Groups and Finite Buildings
.506
*9.11 Remarks on the Simplified Proof
.508
*9.12 The Classification of Twin Buildings
.509
10
Euclidean and Hyperbolic Reflection Groups
.511
10.1
Euclidean Reflection Groups
.511
10.1.1 Affine
Concepts
.511
10.1.2
Formulas for
Affine
Reflections
.513
10.1.3 Affine
Reflection Groups
.514
10.1.4
Finiteness Results
.516
10.1.5
The Structure of
С
.517
10.1.6
The Structure of W, Part I
.521
10.1.7
Example
.523
П0.1.8
The Structure of W, Part II;
Affine Weyl
Groups
. 526
10.2
Euclidean Coxeter Groups and Complexes
.530
10.2.1
A Euclidean Metric on
\Σ\
.530
*10.2.2 Connection with the Tits Cone
.533
*10.3
Hyperbolic Reflection Groups
.537
10.3.1
Hyperbolic Space;
Hyperplanes
and Reflections
.537
10.3.2
Reflection Groups in
ШР
.539
10.3.3
Example
.541
10.3.4
The
Poset
of Cells
.542
10.3.5
The Simplicial Case
.544
10.3.6
The General Case
.545
*10.4 Hyperbolic Coxeter Groups and Complexes
.546
11
Euclidean Buildings
.549
11.1
CATCO) Spaces
.549
11.2
Euclidean Buildings as Metric Spaces
.554
11.3
The Bruhat-Tits Fixed-Point Theorem
.558
11.4
Application: Bounded Subgroups
.561
11.5
Bounded Subsets of Apartments
.565
11.6
A Metric Characterization of the Apartments
.569
11.7
Construction of Apartments
.573
11.8
The Spherical Building at Infinity
.579
11.8.1
Ideal Points and Ideal Simplices
.580
11.8.2
Construction of the Building at Infinity
.582
11.8.3
Type-Preserving Maps
.584
11.8.4
Incomplete Apartment Systems
.585
11.8.5
Group-Theoretic Consequences
.587
xvi Contents
11.8.6
Example
.590
11.9
Classification
.592
*11.10 Moufang Euclidean Buildings
.593
12
Buildings as Metric Spaces
.597
12.1
Metric Realizations of Buildings
.598
12.1.1
The ^-Realization as a Set
.598
12.1.2
A Metric on X
.601
12.1.3
From Apartments to Chambers and Back Again
.604
12.1.4
The Effect of a Chamber Map
.605
12.1.5
The Carrier of a Point of X
.606
12.1.6
Chains and Galleries
.607
12.1.7
Existence of Geodesies
.610
12.1.8
Curvature
.611
12.2
Special Cases
.615
12.3
The Dual Coxeter Complex
.617
12.3.1
Introduction
.617
12.3.2
Examples
.618
12.3.3
Construction of the Dual Coxeter Complex
.620
12.3.4
Properties
.622
12.3.5
Remarks on the Spherical Case
.623
12.3.6
The Euclidean and Hyperbolic Cases
.624
12.3.7
A Fundamental Domain
.624
12.3.8
A CAT(0) Metric on X
.626
12.3.9
The Gromov Hyperbolic Case: A CAT(-l) Metric
. 627
12.3.10
A Cubical Subdivision of Ed
.628
12.4
The Davis Realization of a Building
.629
13
Applications to the Cohomology of Groups
.633
13.1
Arithmetic Groups over the Rationals
.633
13.1.1
Definition
.633
13.1.2
The Symmetric Space
.634
13.1.3
The Cocompact Case
.635
13.1.4
The General Case
.636
13.1.5
Virtual Notions
.638
13.2
S-Arithmetic Groups
.639
13.2.1
A p-adic Analogue of the Symmetric Space
.640
13.2.2
Cohomology of S-Arithmetic Groups: Method
1.641
13.2.3
Cohomology of S-Arithmetic Groups: Method
2.642
13.2.4
The Nonreductive Case
.642
13.3
Discrete Subgroups of p-adic Groups
.644
13.4
Cohomological Dimension of Linear Groups
.645
13.5
S-Arithmetic Groups over Function Fields
.647
Contents xvii
14
Other
Applications.651
14.1
Presentations of Groups
.651
14.1.1
Chamber-Transitive Actions
.652
14.1.2
Further Results for BN-Pairs
.653
14.1.3
The Group U+
.654
14.1.4
S'-Arithmetic Groups
.654
14.2
Finite Groups
.655
14.3
Differential Geometry
.657
14.3.1
Mostów
Rigidity
.657
14.3.2
Further Rigidity Theorems
.658
14.3.3
Isoparametric Submanifolds
.658
14.3.4
Singular Spaces and p-adic Groups
.659
14.4
Representation Theory and Harmonic Analysis
.660
A Cell Complexes
.661
A.I Simplicial Complexes
.661
A.I.I Definitions
.661
A.I.
2
Flag Complexes
.663
A.1.3 Chamber Complexes and Type Functions
.664
A.1.4 Chamber Systems
.668
*A.2 Regular Cell Complexes
.670
A.
2.1
Definitions and First Properties
.670
A.2.2 Regular Cell Complexes from Polytopes
.673
A.
2.3
Regular Cell Complexes from Arrangements
.674
A.3 Cubical Realizations of Posets
.676
В
Root Systems
.681
B.I Notation
.681
B.2 Definition and First Properties
.682
B.3 The Dual Root System
.683
B.4 Examples
.683
*B.5 Nonreduced Root Systems
.684
С
Algebraic Groups
.685
C.I Group Schemes
.685
C.2 The
Affine
Algebra of
G
.687
C.3 Extension of
Scalare
.688
C.4 Group Schemes from Groups
.688
C.5 Linear Algebraic Groups
.689
C.6 Tori
.690
C.7 Unipotent Groups
.690
C.8 Connected Groups
.691
C.9 Reductive,
Semisimple,
and Simple Groups
.691
CIO BN-Pairs and Spherical Buildings
.691
СИ
BN-Pairs and Euclidean Buildings
.:.693
xviii Contents
С.
12 Group
Schemes
versus
Groups
.693
Hints/Solutions/Answers to Selected Exercises
.695
References
.719
Notation Index
.737
Subject Index
.741 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Abramenko, Peter 1960- Brown, Kenneth S. 1945- |
| author_GND | (DE-588)138001995 (DE-588)136633293 |
| author_facet | Abramenko, Peter 1960- Brown, Kenneth S. 1945- |
| author_role | aut aut |
| author_sort | Abramenko, Peter 1960- |
| author_variant | p a pa k s b ks ksb |
| building | Verbundindex |
| bvnumber | BV035030655 |
| callnumber-first | Q - Science |
| callnumber-label | QA174 |
| callnumber-raw | QA174.2 |
| callnumber-search | QA174.2 |
| callnumber-sort | QA 3174.2 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 260 |
| classification_tum | MAT 514f |
| ctrlnum | (OCoLC)226280663 (DE-599)DNB987972812 |
| dewey-full | 512.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.2 |
| dewey-search | 512.2 |
| dewey-sort | 3512.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV035030655 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:49:37Z |
| indexdate | 2024-07-09T21:20:36Z |
| institution | BVB |
| isbn | 9780387788340 9780387788357 0387788344 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016699658 |
| oclc_num | 226280663 |
| open_access_boolean | |
| owner | DE-384 DE-20 DE-703 DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-706 DE-11 DE-355 DE-BY-UBR DE-188 DE-83 |
| owner_facet | DE-384 DE-20 DE-703 DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-706 DE-11 DE-355 DE-BY-UBR DE-188 DE-83 |
| physical | XXI, 747 S. Ill., graph. Darst. 235 mm x 155 mm |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Graduate Texts in Mathematics |
| series2 | Graduate Texts in Mathematics |
| spelling | Abramenko, Peter 1960- Verfasser (DE-588)138001995 aut Buildings Theory and Applications Peter Abramenko ; Kenneth S. Brown New York, NY Springer 2008 XXI, 747 S. Ill., graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Graduate Texts in Mathematics 248 Immeubles (Théorie des groupes) Buildings (Group theory) Gebäude Mathematik (DE-588)4123258-6 gnd rswk-swf Gebäude Mathematik (DE-588)4123258-6 s DE-604 Brown, Kenneth S. 1945- Verfasser (DE-588)136633293 aut Graduate Texts in Mathematics 248 (DE-604)BV000000067 248 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016699658&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Abramenko, Peter 1960- Brown, Kenneth S. 1945- Buildings Theory and Applications Graduate Texts in Mathematics Immeubles (Théorie des groupes) Buildings (Group theory) Gebäude Mathematik (DE-588)4123258-6 gnd |
| subject_GND | (DE-588)4123258-6 |
| title | Buildings Theory and Applications |
| title_auth | Buildings Theory and Applications |
| title_exact_search | Buildings Theory and Applications |
| title_exact_search_txtP | Buildings Theory and Applications |
| title_full | Buildings Theory and Applications Peter Abramenko ; Kenneth S. Brown |
| title_fullStr | Buildings Theory and Applications Peter Abramenko ; Kenneth S. Brown |
| title_full_unstemmed | Buildings Theory and Applications Peter Abramenko ; Kenneth S. Brown |
| title_short | Buildings |
| title_sort | buildings theory and applications |
| title_sub | Theory and Applications |
| topic | Immeubles (Théorie des groupes) Buildings (Group theory) Gebäude Mathematik (DE-588)4123258-6 gnd |
| topic_facet | Immeubles (Théorie des groupes) Buildings (Group theory) Gebäude Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016699658&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT abramenkopeter buildingstheoryandapplications AT brownkenneths buildingstheoryandapplications |