Buildings Theory and Applications

Buildings Theory and Applications

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Hauptverfasser: Abramenko, Peter 1960- (VerfasserIn), Brown, Kenneth S. 1945- (VerfasserIn)
Format: Buch
Sprache:English
Veröffentlicht: New York, NY Springer 2008
Schriftenreihe:Graduate Texts in Mathematics 248
Schlagworte:
Immeubles (Théorie des groupes)
Buildings (Group theory)
Gebäude > Mathematik
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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
any_adam_object 1
author Abramenko, Peter 1960-
Brown, Kenneth S. 1945-
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author_facet Abramenko, Peter 1960-
Brown, Kenneth S. 1945-
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id DE-604.BV035030655
illustrated Illustrated
indexdate 2025-02-03T17:14:18Z
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
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owner_facet DE-384
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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
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_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

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