Coarse Geometry of Topological Groups

308 Pages·2021·1.885 MB·English

by  Christian Rosendal

COARSE GEOMETRY OF TOPOLOGICAL GROUPS
Thisbookprovidesageneralframeworkfordoinggeometricgrouptheoryformany
non-locallycompacttopologicaltransformationgroupsthatariseinmathematical
practice,includinghomeomorphismanddiffeomorphismgroupsofmanifolds,
isometrygroupsofseparablemetricspacesandautomorphismgroupsofcountable
structures.UsingRoe’sframeworkofcoarsestructuresandspaces,theauthordefinesa
naturalcoarsegeometricstructureonalltopologicalgroups.Thisstructureis
accessibletoinvestigation,especiallyinthecaseofPolishgroups,andoftenhasan
explicitdescription,generalisingwell-knownstructuresinfamiliarcasesincluding
finitelygenerateddiscretegroups,compactlygeneratedlocallycompactgroupsand
Banachspaces.Inmostcases,thecoarsegeometricstructureismetrisableandmay
evenberefinedtoacanonicalquasi-metricstructureonthegroup.Thebookcontains
manyworkedexamplesandsufficientintroductorymaterialtobeaccessibleto
beginninggraduatestudents.Anappendixoutlinesseveralopenproblemsinthis
youngandrichtheory.
ChristianRosendalisProfessorofMathematicsattheUniversityofMaryland.
HereceivedaSimonsFellowshipinMathematicsin2012andisFellowofthe
AmericanMathematicalSociety.
CAMBRIDGE TRACTS IN MATHEMATICS
GENERAL EDITORS
J.BERTOIN,B.BOLLOBA´S,W.FULTON,B.KRA,I.MOERDIJK,C.PRAEGER,
P.SARNAK,B.SIMON,B.TOTARO
Acompletelistofbooksintheseriescanbefoundatwww.cambridge.org/mathematics.
Recenttitlesincludethefollowing:
188. ModernApproachestotheInvariantSubspaceProblem.ByI.Chalendarand
J.R.Partington
189. NonlinearPerron–FrobeniusTheory.ByB.LemmensandR.Nussbaum
190. JordanStructuresinGeometryandAnalysis.ByC.-H.Chu
191. MalliavinCalculusforLe´vyProcessesandInfinite-DimensionalBrownianMotion.
ByH.Osswald
192. NormalApproximationswithMalliavinCalculus.ByI.NourdinandG.Peccati
193. DistributionModuloOneandDiophantineApproximation.ByY.Bugeaud
194. MathematicsofTwo-DimensionalTurbulence.ByS.KuksinandA.Shirikyan
195. AUniversalConstructionforGroupsActingFreelyonRealTrees.ByI.Chiswelland
T.Mu¨ller
196. TheTheoryofHardy’sZ-Function.ByA.Ivic´
197. InducedRepresentationsofLocallyCompactGroups.ByE.KaniuthandK.F.Taylor
198. TopicsinCriticalPointTheory.ByK.PereraandM.Schechter
199. CombinatoricsofMinusculeRepresentations.ByR.M.Green
200. SingularitiesoftheMinimalModelProgram.ByJ.Kolla´r
201. CoherenceinThree-DimensionalCategoryTheory.ByN.Gurski
202. CanonicalRamseyTheoryonPolishSpaces.ByV.Kanovei,M.Sabok,andJ.Zapletal
203. APrimerontheDirichletSpace.ByO.El-Fallah,K.Kellay,J.Mashreghi,and
T.Ransford
204. GroupCohomologyandAlgebraicCycles.ByB.Totaro
205. RidgeFunctions.ByA.Pinkus
206. ProbabilityonRealLieAlgebras.ByU.FranzandN.Privault
207. AuxiliaryPolynomialsinNumberTheory.ByD.Masser
208. RepresentationsofElementaryAbelianp-GroupsandVectorBundles.ByD.J.Benson
209. Non-homogeneousRandomWalks.ByM.Menshikov,S.Popov,andA.Wade
210. FourierIntegralsinClassicalAnalysis(SecondEdition).ByC.D.Sogge
211. Eigenvalues,MultiplicitiesandGraphs.ByC.R.JohnsonandC.M.Saiago
212. ApplicationsofDiophantineApproximationtoIntegralPointsandTranscendence.
ByP.CorvajaandU.Zannier
213. VariationsonaThemeofBorel.ByS.Weinberger
214. TheMathieuGroups.ByA.A.Ivanov
215. SlendernessI:Foundations.ByR.Dimitric
216. JustificationLogic.ByS.ArtemovandM.Fitting
217. DefocusingNonlinearSchro¨dingerEquations.ByB.Dodson
218. TheRandomMatrixTheoryoftheClassicalCompactGroups.ByE.S.Meckes
219. OperatorAnalysis.ByJ.Agler,J.E.Mccarthy,andN.J.Young
220. LecturesonContact3-Manifolds,HolomorphicCurvesandIntersectionTheory.
ByC.Wendl
221. MatrixPositivity.ByC.R.Johnson,R.L.Smith,andM.J.Tsatsomeros
222. AssouadDimensionandFractalGeometry.ByJ.M.Fraser
223. CoarseGeometryofTopologicalGroups.ByC.Rosendal
Coarse Geometry of Topological Groups
CHRISTIAN ROSENDAL
UniversityofMaryland
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,
NewDelhi–110025,India
103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learning,andresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108842471
DOI:10.1017/9781108903547
©ChristianRosendal2022
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2022
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
ISBN978-1-108-84247-1Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Contents
Preface pagevii
Acknowledgements ix
1 Introduction 1
1.1 Motivation 1
1.2 AWordontheTerminology 5
1.3 SummaryofFindings 6
2 CoarseStructureandMetrisability 19
2.1 CoarseandUniformSpaces 19
2.2 CoarseandUniformStructuresonGroups 23
2.3 CoarselyBoundedSets 26
2.4 ComparisonwithOtherLeft-InvariantCoarseStructures 32
2.5 MetrisabilityandMonogenicity 37
2.6 BornologousMaps 43
2.7 CoarselyProperE´carts 49
2.8 Quasi-metricSpaces 53
2.9 MaximalE´carts 59
3 StructureTheory 68
3.1 TheRoelckeUniformity 68
3.2 ExamplesofPolishGroups 78
3.3 RigidityofCategories 83
3.4 ComparisonofLeft-andRight-CoarseStructures 88
3.5 CoarseGeometryofProductGroups 92
3.6 GroupsofFiniteAsymptoticDimension 104
3.7 DistortedElementsandSubgroups 106
3.8 GroupswithProperty(PL) 112
v
vi Contents
4 Sections,CocyclesandGroupExtensions 125
4.1 Quasi-morphismsandBoundedCocycles 126
4.2 LocalBoundednessofExtensions 130
4.3 RefinementsofTopologies 138
4.4 GroupExtensions 141
4.5 ExternalExtensions 145
4.6 InternalExtensionsofPolishGroups 148
4.7 AFurtherComputationforGeneralExtensions 155
4.8 CoarseStructureofCoveringMaps 156
5 PolishGroupsofBoundedGeometry 166
5.1 GaugesandGroupsofBoundedGeometry 166
5.2 DynamicandGeometricCharacterisationsofBoundedGeometry 170
5.3 Examples 176
5.4 TopologicalCouplings 185
5.5 CoarseCouplings 202
5.6 RepresentationsonReflexiveSpaces 204
5.7 CompactG-FlowsandUnitaryRepresentations 207
5.8 EfficientlyContractibleGroups 210
5.9 Entropy,GrowthRatesandMetricAmenability 220
5.10 NetsinPolishGroups 230
5.11 Two-EndedPolishGroups 234
6 AutomorphismGroupsofCountableStructures 244
6.1 Non-ArchimedeanPolishGroupsandLarge-ScaleGeometry 244
6.2 OrbitalTypesFormulation 245
6.3 HomogeneousandAtomicModels 252
6.4 OrbitalIndependenceRelations 259
6.5 ComputingQuasi-isometryTypesofAutomorphismGroups 269
7 Zappa–Sze´pProducts 272
7.1 TheTopologicalStructure 272
7.2 Examples 275
7.3 TheCoarseStructureofZappa–Sze´pProducts 277
Appendix OpenProblems 285
References 289
Index 295
Preface
Thepresentbookistheproductofresearchinitiatedduringanextendedsab-
batical financed in part by a fellowship by the Simons Foundation. Although
thebasicdiscoveryofanappropriatecoarsestructureonPolishgroupsreally
materialised around 2013, the initial seeds were sown in several prior works
and,inmyowncase,inthetwostudies[77]and[78],wherevariousconcepts
ofboundednessintopologicalgroupsweredeveloped.
ThemainobjectsofthisstudyarePolishgroups,thatis,separableandcom-
pletelymetrisabletopologicalgroups.Althoughthesecanofcoursebetreated
abstractly, I am principally interested in them as they appear in applications,
namelyastransformationgroupsofvariousmathematicalstructuresandeven
as the additive groups of separable Banach spaces. This class of groups has
received a substantial amount of attention over the past two decades and the
results here are my attempt at grappling with possible geometric structures
on them. In particular, this is a response to the question of how to apply the
language and techniques of geometric group theory, abstract harmonic and
functionalanalysistotheirstudy.
Becausethisprojecthasbeenalongtimecoming,myideasonthesubject
have evolved over time and have been influenced by a number of people.
Evidently,theworkofMikhailGromovpervadesallofgeometricgrouptheory
andhencealsotheideaspresentedhere.ButanotherspecificreferenceisJohn
Roe’slecturesoncoarsegeometry[75]andwhoseframeworkofcoarsespaces
allowedmetoextendthedefinitionofageometricstructuretoalltopological
groupsandnotjusttheadmittedlymoreinterestingsubclassoflocallybounded
Polishgroups.
Althoughthisbookcontainsthefirstformalpresentationofthetheory,parts
of it have already found their way into other publications. In particular, in
[80] I made a systematic study of equivariant geometry of amenable Polish
vii
viii Preface
groups, including separable Banach spaces, and made use of some of the
theory presented here. Similarly, in a collaboration with Kathryn Mann [57],
Iinvestigatedthelarge-scalegeometryofhomeomorphismgroupsofcompact
manifoldswithinthepresentframework,whileotherauthors[19,38,99]have
donesoforothergroups.Finally,[81]containsacharacterisationofthesmall-
scalegeometryofPolishgroupsanditsconnectiontothelargescale.
Acknowledgements
Overtheyears,Ihavegreatlybenefittedfromallmyconversationsonthistopic
withanumberofpeople.AlthoughIamboundtoexcludemany,theseinclude
Uri Bader, Bruno Braga, Michael Cohen, Yves de Cornulier, Marc Culler,
Alexander Dranishnikov, Alexander Furman, William Herndon, Kathryn
Mann, Julien Melleray, Emmanuel Militon, Nicolas Monod, Justin Moore,
Vladimir Pestov, Konstantin Slutsky, Sławomir Solecki, Andreas Thom,
Simon Thomas, Todor Tsankov, Phillip Wesolek, Kevin Whyte, Takamitsu
YamauchiandJosephZielinski.
Ialsotakethisoccasiontothankmycolleaguesandthedepartmentatthe
University of Illinois at Chicago for giving me great flexibility in my work,
and Chris Laskowski and the mathematics department at the University of
Maryland for hosting me during the preparation of this book. My research
benefitted immensely from the continuous support of the National Science
FoundationandafellowshipfromtheSimonsFoundation.
Finally, I wish to thank Valentin Ferenczi, Gilles Godefroy, Alexander
Kechris, Alain Louveau and Stevo Todorcˇevic´ for their support over the
years and, most of all, my entire family for indulging and occasionally even
encouragingmyscientificinterests.
My research benefitted immensely from the continuous support of the
National Science Foundation (DMS 1201295, DMS 1464974 and DMS
1764247)andafellowshipfromtheSimonsFoundation.
ix