Large Scale Geometry

203 Pages·2012·3.897 MB·English

by  Piotr W. Nowak, Guoliang Yu

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Markus Stroppel, Locally Compact Groups
Peter Kunkel and Volker Mehrmann, Differential-Algebraic Equations
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Andrzej Skowron´ski and Kunio Yamagata, Frobenius Algebras I. Basic Representation Theory
Piotr W. Nowak
Guoliang Yu
Large Scale
Geometry
Authors:
Piotr W. Nowak Guoliang Yu
Institute of Mathematics of the Department of Mathematics
Polish Academy of Sciences Texas A&M University
ul. S´niadeckich 8 College Station, TX 77843, USA
P. O. Box 21
E-mail: [email protected]
00-956 Warszawa, Poland
and
Faculty of Mathematics, Informatics, and Mechanics
University of Warsaw
Banacha 2
02-097 Warszawa, Poland
E-mail: [email protected]
2010 Mathematics Subject Classification: 51-01; 50F99, 20F69; 19K56, 57-01, 46L87, 58B34,
46L99, 53C20
Key words: Large scale geometry, quasi-isometry, coarse geometry, asymptotic dimension, finite
decomposition complexity, amenability, Property A, coarse embedding, expanders, a-T-menability,
coarse homology, uniformly finite homology, Baum–Connes conjecture, coarse Baum–Connes con-
jecture, Novikov conjecture, Borel conjecture
ISBN 978-3-03719-112-5
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,
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© European Mathematical Society 2012
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Typeset using the authors’ TEX files: I. Zimmermann, Freiburg
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Thisbookisdedicatedtoourfamilies
Franek,Ania,parentsandgrandparents;
Alexandria,Andrew,Jenny,parentsandgrandparents.

Preface
Large scale geometry is the study of geometric objects viewed from afar. In this
type of geometry two objects are considered to be the same, if they look roughly
thesamefromalargedistance. Forexample,whenviewedfromagreaterdistance
theEarthlookslikeapoint,whilethereallineisnotmuchdifferentfromthespace
ofintegers. Inthepastdecadesmathematicianshavediscoveredmanyinteresting
andbeautifullargescalegeometricproperties, whichhaveimportantapplications
intopology,analysis,computerscienceanddataanalysis.
Historically,largescalegeometricideasappearedinMostow’srigiditytheorem
[172], [173] and its generalization due to Margulis [155], [158], as well as in the
work of Švarc [227] and later Milnor [166] andWolf [243] on growth of groups.
TheimpetuscamewithGromov’spolynomialgrowththeorem[105]andsincethen
largescalegeometricideashaveenteredtheworldofgrouptheory,wheretheyhave
becomeafundamentaltool. Bynowitiswellknownthatmanyclassicalnotions
areinfactoflargescalegeometricnature,oneexamplebeingamenability,defined
byvonNeumanninhisworkontheBanach–Tarskiparadoxin1929.
Oneofthemotivationsforthisbookistheuseoflargescalegeometricmethods
in index theory. After theAtiyah–Singer index theorem was proved for compact
manifolds,anaturalquestionwaswhetheronecanextendsuchindextheoremsto
non-compact manifolds. Connes and Moscovici developed a higher index theory
forcoveringspaces[62]. Motivatedbytheirwork,Roeintroducedahigherindex
theory for general non-compact manifolds [207]. Powerful large scale geometric
methodshavebeenintroducedtocomputehigherindicesfornon-compactmanifolds
[249],[250]. Suchcomputationshaveallowedustoobtainsignificantprogresson
problemsliketheNovikovconjecture,theGromov–Lawson–Rosenbergconjecture,
orthezero-in-the-spectrumproblem,andthisareaisinastageofrapiddevelopment.
Morerecently, Guentner,TesseraandYu[117], [118]haveintroducedlargescale
geometricmethodstostudytopologicalrigidityofmanifolds. Theirmethodallows
themtoprovestrongresultsforthestableBorelconjectureandtheboundedBorel
conjecture.
At the same time, large scale geometric techniques are also of interest in ar-
eas such as Banach space geometry, where the coarse classification of Banach
spaces remains an interesting problem. Another area of application is theoretical
computerscience,whereembeddabilityproperties,suchascompressionofcoarse
embeddings of discrete metric spaces and graphs, allow one to obtain computa-
tional efficiency. Finally, we mention that large scale geometric methods have
foundinterestingapplicationsinlargedataanalysis[52],[236].
Our goal is to provide a gentle and fairly detailed introduction to large scale
geometric ideas that are used in the study of index problems. We hope that the
viii
text will be accessible to a broad audience, in particular to graduate students and
newcomerstothefield. Weprovidedetailedproofsformostofthetheoremsstated
in the text and every chapter has several exercises at the end. Additionally, some
stepsintheproofsareleftasexercisestothereader. Theseomissionsarealways
pointedoutinthetext.
Thebookisorganizedinthefollowingway. Inthefirstchapterwediscussbasic
properties of the coarse category, the geometric viewpoint on finitely generated
groups, and we conclude with a section on Gromov hyperbolicity. In the second
chapter we give a detailed overview of the notions of asymptotic dimension and
decomposition complexity. The third chapter covers amenability of groups. It
is slightly shorter, since amenability, as a classical notion, is already a subject
of several excellent monographs. In the fourth chapter we discuss the notion of
propertyAformetricspacesanditsconnectionstoamenability. Inthefifthchapter,
coarse embeddings into Banach spaces are studied. The sixth chapter is about
affine isometric actions of groups on Banach spaces, in particular a-T-menability
(alsoknownastheHaagerupproperty),Kazhdan’sproperty(T)andconstructions
of expanders. In the last chapter we introduce elements of large scale algebraic
topology: uniformlyfinitehomologyandcoarsehomologytheories. Theappendix
providesabriefsurveyofapplicationsofcoarsegeometricproperties,discussedin
thetext,tohigherindextheoryofellipticoperators,andtotopologicalandgeometric
rigidity.
WewouldliketothankWernerBallmann,FlorentBaudier,AlexanderDranish-
nikov,TomaszOdrzygóz´dz´,LinShan,RomainTesseraandYi-JunYaoforhelpful
commentsandcorrections. WeareparticularlygratefultoGrahamNiblo,Bogdan
Nica, Ján Špakula, AlainValette and Rufus Willett for carefully reading the text
andsuggestingnumerousimprovements. ItisapleasuretothankManfredKarbe
forhiswonderfulhelpandsupportforthisprojectandIreneZimmermannforher
beautifultypesetting.
Partofthisworkwasdoneduringthefirstauthor’sstaysatTexasA&MUniver-
sityandMSRI.Hewouldliketoexpresshisgratitudetobothinstitutionsfortheir
hospitality. BothauthorswouldliketothanktheNationalScienceFoundationfor
itsgeneroussupportduringthepreparationofthismanuscript.
Nashville/Warsaw,February2012 PiotrNowak,GuoliangYu
Contents
Preface vii
Notationandconventions xiii
1 Metricspacesandlargescalegeometry 1
1.1 Metricspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Groupsasmetricspaces. . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Quasi-isometries . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Coarseequivalences . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Hyperbolicspaces . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Notesandremarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Asymptoticdimensionanddecompositioncomplexity 26
2.1 Topologicaldimension . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Asymptoticdimension . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Dimensionofhyperbolicgroups . . . . . . . . . . . . . . . . . . 30
2.4 Upperboundsforasymptoticdimension . . . . . . . . . . . . . . 32
2.5 Asymptoticdimensionofsolvablegroups . . . . . . . . . . . . . 36
2.6 Groupswithinfiniteasymptoticdimension . . . . . . . . . . . . . 38
2.7 Decompositioncomplexity . . . . . . . . . . . . . . . . . . . . . 39
2.8 Invarianceandpermanence . . . . . . . . . . . . . . . . . . . . . 42
2.9 Groupswithfinitedecompositioncomplexity . . . . . . . . . . . 45
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Notesandremarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Amenability 49
3.1 Følnerconditions . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 TheHulanicki–Reitercondition . . . . . . . . . . . . . . . . . . 55
3.3 Invariantmeans . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Notesandremarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 PropertyA 63
4.1 Definitionandbasicproperties . . . . . . . . . . . . . . . . . . . 63
4.2 TheHigson–Roecondition . . . . . . . . . . . . . . . . . . . . . 66
4.3 FiniteasymptoticdimensionimpliespropertyA . . . . . . . . . . 70
x Contents
4.4 PropertyAandresiduallyfinitegroups . . . . . . . . . . . . . . . 73
4.5 Locallyfiniteexamples . . . . . . . . . . . . . . . . . . . . . . . 78
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Notesandremarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5 Coarseembeddings 83
5.1 Coarseembeddings . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2 EmbeddabilityintoHilbertspaces . . . . . . . . . . . . . . . . . 84
5.3 ExamplesofembeddablespaceswithoutpropertyA . . . . . . . . 89
5.4 Convexityandreflexivity . . . . . . . . . . . . . . . . . . . . . . 91
5.5 Coarseembeddingsandfinitesubsets . . . . . . . . . . . . . . . 96
5.6 Expanders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.7 Ageometriccharacterizationofnon-embeddability . . . . . . . . 101
5.8 Compressionofcoarseembeddings . . . . . . . . . . . . . . . . 108
5.9 Compression> 1 impliespropertyA . . . . . . . . . . . . . . . 110
2
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Notesandremarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6 GroupactionsonBanachspaces 119
6.1 Affineisometricactions . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 Metricallyproperactionsanda-T-menability . . . . . . . . . . . 121
6.3 Actionson` -spacesandreflexiveBanachspaces . . . . . . . . . 125
p
6.4 Kazhdan’sproperty(T) . . . . . . . . . . . . . . . . . . . . . . . 128
6.5 FixedpointsandKazhdan’sproperty(T) . . . . . . . . . . . . . . 130
6.6 Constructionofexpanders . . . . . . . . . . . . . . . . . . . . . 132
6.7 Property(T)andspectralconditions . . . . . . . . . . . . . . . . 134
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Notesandremarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7 Coarsehomology 143
7.1 Coarselocallyfinitehomology . . . . . . . . . . . . . . . . . . . 143
7.2 Uniformlyfinitehomology . . . . . . . . . . . . . . . . . . . . . 144
7.3 EilenbergswindlesandPonzischemes . . . . . . . . . . . . . . . 149
7.4 Aperiodictilesandnon-amenablespaces . . . . . . . . . . . . . . 154
7.5 Coarseninghomologytheories . . . . . . . . . . . . . . . . . . . 160
7.6 Thecoarseninghomomorphism . . . . . . . . . . . . . . . . . . 162
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Notesandremarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165