Finiteness Properties of Arithmetic Groups Acting on Twin Buildings

128 Pages·2014·1.815 MB·English

by  Stefan Witzel (auth.)

Lecture Notes in Mathematics 2109
Stefan Witzel
Finiteness
Properties of
Arithmetic Groups
Acting on Twin
Buildings
Lecture Notes in Mathematics 2109
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
AdvisoryBoard:
CamilloDeLellis(Zürich)
MariodiBernardo(Bristol)
AlessioFigalli(Austin)
DavarKhoshnevisan(SaltLakeCity)
IoannisKontoyiannis(Athens)
GaborLugosi(Barcelona)
MarkPodolskij(Aarhus)
SylviaSerfaty(ParisandNY)
CatharinaStroppel(Bonn)
AnnaWienhard(Heidelberg)
Forfurthervolumes:
http://www.springer.com/series/304

Stefan Witzel
Finiteness Properties
of Arithmetic Groups
Acting on Twin Buildings
123
StefanWitzel
MathematischesInstitut
WestfälischeWilhelms-UniversitätMünster
Münster,Nordrhein-Westfalen,Germany
ISBN978-3-319-06476-5 ISBN978-3-319-06477-2(eBook)
DOI10.1007/978-3-319-06477-2
SpringerChamHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2014943904
MathematicsSubjectClassification(2010):20G30,22E40,20E42,51E24,57M07,20F65
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Acknowledgements
Many people have helped me, directly or indirectly, in writing these notes or the
thesistheyarebasedon.
First I want to thank Ralf Köhl and Kai-Uwe Bux. Ralf has been an excellent
Doktorvater who always cared for his students. He was highly encouraging and
has taught me a lot about the organization of mathematics. He also brought me
in contact with beautiful mathematical areas. Kai-Uwe Bux greatly influenced
my mathematical thinking and in particular my geometric intuition. His view on
mathematics has been an inspiration. Many of the ideas in these notes originate
fromjointworkwithRalfandKai-Uwe.
As a student I have been shaped by Karl Heinrich Hofmann. I thank him for
sharingsomeofhisendlessknowledge.
Ialsowanttothankthepeoplewhohavehelpedmebyansweringmyquestions
ormakinghelpfulremarksaboutproblemsthatarosewhilewritingmythesis.They
areJanEssert,SvenHerrmann,andMichaelJoswig.
Apart fromconcretehelp on an isolated problemit hasbeen importantto have
people around with whom I could talk about mathematics. For those discussions
I want to thank David Ghatei, Alexander Kartzow, Andreas Mars, Julia Sponsel
(at the time), and Markus-Ludwig Wermer in Darmstadt; Linus Kramer, Petra
Schwer, and Daniel Skodlerack in Münster; and Marco Marschler and Henning
NiesdroyinBielefeld.
JanEssert,AlexanderKartzow,AndreasMars,HenningNiesdroy,JuliaSponsel,
Markus-LudwigWermer,andMattZaremskyreadpreviousversionsofthesenotes
andhelpedtoeliminatemanymistakes.Iwanttothankthemandatthesametime
apologizeforproducingsomanynewmistakes.Ialsothanktwoanonymousreferees
fortheirconstructivecriticism.
Finally I want to thank Julia (now Witzel) for being by my side. I don’tknow
whatIwoulddowithouthersupportandencouragement.
Münster,Germany StefanWitzel
vii

Contents
1 BasicDefinitionsandProperties ........................................... 1
1.1 MetricSpaces.......................................................... 1
1.1.1 Geodesics...................................................... 1
1.1.2 ProductsandJoins............................................. 2
1.1.3 ModelSpaces.................................................. 3
1.1.4 CAT((cid:2))-Spaces................................................ 3
1.1.5 PolyhedralComplexes ........................................ 4
1.1.6 Links........................................................... 6
1.1.7 VisualBoundary............................................... 7
1.2 SphericalGeometry.................................................... 7
1.2.1 SphericalTriangles............................................ 8
1.2.2 DecomposingSphericalSimplices ........................... 9
1.2.3 SphericalPolytopeswithNon-obtuseAngles................ 11
1.3 FinitenessProperties................................................... 13
1.4 SimplicialMorseTheory.............................................. 17
1.5 NumberTheory........................................................ 18
1.5.1 Valuations...................................................... 19
1.5.2 DiscreteValuations............................................ 20
1.5.3 LocalFields.................................................... 21
1.5.4 S-Integers...................................................... 22
1.6 AffineVarietiesandLinearAlgebraicGroups ....................... 23
1.6.1 AffineVarieties................................................ 23
1.6.2 LinearAlgebraicGroups...................................... 24
1.6.3 S-ArithmeticGroups.......................................... 29
1.7 Buildings............................................................... 29
1.7.1 SphericalCoxeterComplexes ................................ 30
1.7.2 EuclideanCoxeterComplexes................................ 32
1.7.3 GeneralCoxeterComplexes.................................. 34
1.7.4 Buildings....................................................... 34
1.7.5 TwinBuildings ................................................ 38
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