Table Of Content248
GraduateTextsinMathematics
EditorialBoard
S.Axler
K.A.Ribet
GraduateTextsinMathematics
1 Takeuti/Zaring.Introductionto 39 Arveson.AnInvitationtoC-Algebras.
AxiomaticSetTheory.2nded. 40 Kemeny/Snell/Knapp.Denumerable
2 Oxtoby.MeasureandCategory.2nded. MarkovChains.2nded.
3 Schaefer.TopologicalVectorSpaces. 41 Apostol.ModularFunctionsandDirichlet
2nded. SeriesinNumberTheory.2nded.
4 Hilton/Stammbach.ACoursein 42 J.-P.Serre.LinearRepresentationsof
HomologicalAlgebra.2nded. FiniteGroups.
5 MacLane.CategoriesfortheWorking 43 Gillman/Jerison.RingsofContinuous
Mathematician.2nded. Functions.
6 Hughes/Piper.ProjectivePlanes. 44 Kendig.ElementaryAlgebraicGeometry.
7 J.-P.Serre.ACourseinArithmetic. 45 Loève.ProbabilityTheoryI.4thed.
8 Takeuti/Zaring.AxiomaticSetTheory.
46 Loève.ProbabilityTheoryII.4thed.
9 Humphreys.IntroductiontoLieAlgebras
47 Moise.GeometricTopologyin
andRepresentationTheory.
Dimensions2and3.
10 Cohen.ACourseinSimpleHomotopy
48 Sachs/Wu.GeneralRelativityfor
Theory.
Mathematicians.
11 Conway.FunctionsofOneComplex
49 Gruenberg/Weir.LinearGeometry.
VariableI.2nded.
2nded.
12 Beals.AdvancedMathematicalAnalysis.
50 Edwards.Fermat’sLastTheorem.
13 Anderson/Fuller.RingsandCategories
ofModules.2nded. 51 Klingenberg.ACourseinDifferential
14 Golubitsky/Guillemin.StableMappings Geometry.
andTheirSingularities. 52 Hartshorne.AlgebraicGeometry.
15 Berberian.LecturesinFunctional 53 Manin.ACourseinMathematicalLogic.
AnalysisandOperatorTheory. 54 Graver/Watkins.Combinatoricswith
16 Winter.TheStructureofFields. EmphasisontheTheoryofGraphs.
17 Rosenblatt.RandomProcesses.2nded. 55 Brown/Pearcy.IntroductiontoOperator
18 Halmos.MeasureTheory. TheoryI:ElementsofFunctionalAnalysis.
19 Halmos.AHilbertSpaceProblemBook. 56 Massey.AlgebraicTopology:An
2nded. Introduction.
20 Husemoller.FibreBundles.3rded. 57 Crowell/Fox.IntroductiontoKnot
21 Humphreys.LinearAlgebraicGroups. Theory.
22 Barnes/Mack.AnAlgebraicIntroduction 58 Koblitz.p-adicNumbers,p-adic
toMathematicalLogic. Analysis,andZeta-Functions.2nded.
23 Greub.LinearAlgebra.4thed. 59 Lang.CyclotomicFields.
24 Holmes.GeometricFunctionalAnalysis 60 Arnold.MathematicalMethodsin
andItsApplications. ClassicalMechanics.2nded.
25 Hewitt/Stromberg.RealandAbstract 61 Whitehead.ElementsofHomotopy
Analysis. Theory.
26 Manes.AlgebraicTheories. 62 Kargapolov/Merizjakov.Fundamentals
27 Kelley.GeneralTopology. oftheTheoryofGroups.
28 Zariski/Samuel.CommutativeAlgebra. 63 Bollobas.GraphTheory.
Vol.I. 64 Edwards.FourierSeries.Vol.I.2nded.
29 Zariski/Samuel.CommutativeAlgebra. 65 Wells.DifferentialAnalysisonComplex
Vol.II.
Manifolds.2nded.
30 Jacobson.LecturesinAbstractAlgebraI.
66 Waterhouse.IntroductiontoAffine
BasicConcepts.
GroupSchemes.
31 Jacobson.LecturesinAbstractAlgebraII.
67 Serre.LocalFields.
LinearAlgebra.
68 Weidmann.LinearOperatorsinHilbert
32 Jacobson.LecturesinAbstractAlgebraIII.
Spaces.
TheoryofFieldsandGaloisTheory.
33 Hirsch.DifferentialTopology. 69 Lang.CyclotomicFieldsII.
34 Spitzer.PrinciplesofRandomWalk. 70 Massey.SingularHomologyTheory.
2nded. 71 Farkas/Kra.RiemannSurfaces.2nded.
35 Alexander/Wermer.SeveralComplex 72 Stillwell.ClassicalTopologyand
VariablesandBanachAlgebras.3rded. CombinatorialGroupTheory.2nded.
36 Kelley/Namiokaetal.Linear 73 Hungerford.Algebra.
TopologicalSpaces. 74 Davenport.MultiplicativeNumber
37 Monk.MathematicalLogic. Theory.3rded.
38 Grauert/Fritzsche.SeveralComplex 75 Hochschild.BasicTheoryofAlgebraic
Variables. GroupsandLieAlgebras.
(continuedafterindex)
Peter Abramenko
Kenneth S. Brown
Buildings
Theory and Applications
(cid:65)(cid:66)(cid:67)
PeterAbramenko KennethS.Brown
DepartmentofMathematics DepartmentofMathematics
UniversityofVirginia CornellUniversity
Charlottesville,VA22904 Ithaca,NY14853
USA USA
[email protected] [email protected]
EditorialBoard
S.Axler K.A.Ribet
MathematicsDepartment MathematicsDepartment
SanFranciscoStateUniversity UniversityofCaliforniaatBerkeley
SanFrancisco,CA94132 Berkeley,CA94720-3840
USA USA
[email protected] [email protected]
ISBN:978-0-387-78834-0 e-ISBN:978-0-387-78835-7
LibraryofCongressControlNumber:2008922933
Mathematics Subject Classification (2000): (Primary) 51E24, 20E42, 20F55, 51F15, (Secondary)
14L35,20F05,20F65,20G15,20J05,22E40,22E65
©2008SpringerScience+BusinessMedia,LLC
BasedinpartonBuildings,KennethS.Brown,1989,1998
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,
NY10013,USA)andtheauthor,exceptforbriefexcerptsinconnectionwithreviewsorscholarly
analysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,
computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedis
forbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey
arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare
subjecttoproprietaryrights.
Printedonacid-freepaper.
987654321
springer.com
To our wives, Monika and Susan
Preface
This text started out as a revised version of Buildings by the second-named
author [53], but it has grown into a much more voluminous book. The earlier
bookwasintendedtogiveashort,friendly,elementaryintroductiontothethe-
ory,accessibletoreaderswithaminimalbackground.Moreover,itapproached
buildings from only one point of view, sometimes called the “old-fashioned”
approach: A building is a simplicial complex with certain properties.
The current book includes all the material of the earlier one, but we have
added a lot. In particular, we have included the “modern” (or “W-metric”)
approach to buildings, which looks quite different from the old-fashioned ap-
proach but is equivalent to it. This has become increasingly important in the
theory and applications of buildings. We have also added a thorough treat-
ment of the Moufang property, which occupies two chapters. And we have
added many new exercises and illustrations. Some of the exercises have hints
orsolutionsinthebackofthebook.Amoreextensivesetofsolutionsisavail-
able in a separate solutions manual, which may be obtained from Springer’s
Mathematics Editorial Department.
Wehavetriedtoaddthenewmaterialinsuchawaythatreaderswhoare
contentwiththeold-fashionedapproachcanstillgetanelementarytreatment
of it by reading selected chapters or sections. In particular, many readers will
want to omit the optional sections (marked with a star). The introduction
below provides more detailed guidance to the reader.
In spite of the fact that the book has almost quadrupled in size, we were
still not able to cover all important aspects of the theory of buildings. For
example, we give very little detail concerning the connections with incidence
geometry. And we do not prove Tits’s fundamental classification theorems
for spherical and Euclidean buildings. Fortunately, the recent books of Weiss
[281,283] treat these classification theorems thoroughly.
Applicationsofbuildingstovariousaspectsofgrouptheoryoccurinseveral
chaptersofthebook,startinginChapter6.Inaddition,Chapter13isdevoted
toapplicationstothecohomologytheoryofgroups,whileChapter14sketches
a variety of other applications.
viii Preface
Most of the material in this book is due to Jacques Tits, who originated
the theory of buildings. It has been a pleasure studying Tits’s work. We were
especially pleased to learn, while this book was in the final stages of produc-
tion,thatTitswasnamedasacorecipientofthe2008Abelprize.Thecitation
states:
Titscreatedanewandhighlyinfluentialvisionofgroupsasgeomet-
ric objects. He introduced what is now known as a Tits building,
which encodes in geometric terms the algebraic structure of linear
groups. The theory of buildings is a central unifying principle with
an amazing range of applications....
We hope that our exposition helps make Tits’s beautiful ideas accessible to a
broad mathematical audience.
We are very grateful to Pierre-Emmanuel Caprace, Ralf Gramlich, Bill
Kantor, Bernhard Mu¨hlherr, Johannes Rauh, Hendrik Van Maldeghem, and
RichardWeissformanyhelpfulcommentsonapreliminarydraftofthisbook.
Wewouldalsoliketothankallthepeoplewhohelpeduswiththeapplications
of buildings that we discuss in Chapter 14; their names are mentioned in the
introduction to that chapter.
Charlottesville, VA, and Ithaca, NY Peter Abramenko
June 2008 Kenneth S. Brown
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 Σ(W,V) ..................... 36
1.5.2 The Longest Element of W ....................... 40
1.5.3 Examples....................................... 41
x 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 of 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
2 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 PGL (Z) ............................. 71
2
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
3 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
4 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
5 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
