Potential Theory and Geometry on Lie Groups

625 Pages·2020·4.61 MB·English

by  N. Varopoulos

PotentialTheoryandGeometryonLieGroups
This book provides a complete and reasonably self-contained account of a new
classification of connected Lie groups into two classes. The first part describes the
use of tools from potential theory to establish the classification and to show that the
analytic and algebraic approaches to the classification are equivalent. Part II covers
geometric theory of the same classification and a proof that it is equivalent to the
algebraic approach. Part III is a new approach to the geometric classification that
requires more advanced geometric technology, namely homotopy, homology and the
theory of currents. Using these methods, a more direct, but also more sophisticated,
approachtotheequivalenceofthegeometricandalgebraicclassificationismade.
Backgroundmaterialisintroducedgraduallytofamiliarisereaderswithideasfrom
areas such as Lie groups, algebraic topology and probability, in particular, random
walksongroups.Numerousopenproblemsinspirestudentstoexplorefurther.
N. Th. Varopoulos wasformanyyearsaprofessoratUniversite´deParisVI.He
isanhonorarymemberoftheInstitutUniversitairedeFrance.
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38. N.Th.VaropoulosPotentialTheoryandGeometryonLieGroups
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Potential Theory and Geometry
on Lie Groups
N. TH. VAROPOULOS
Universite´ deParisVI(PierreetMarieCurie)
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Informationonthistitle:www.cambridge.org/9781107036499
DOI:10.1017/9781139567718
©N.Th.Varopoulos2021
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Firstpublished2021
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Names:Varopoulos,N.,1940–author.
Title:PotentialtheoryandgeometryonLiegroups/N.Th.Varopoulos.
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2020.|
Series:Newmathematicalmonographs;38|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2019038658|ISBN9781107036499(hardback)
Subjects:LCSH:Liegroups.|Potentialtheory(Mathematics)|Geometry.
Classification:LCCQA387.V3652020|DDC512/.482–dc23
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Contents
Preface pagexxv
1 Introduction 1
1.1 DistanceandVolumeGrowth 1
1.2 AClassificationofUnimodularLocallyCompactGroups 2
1.3 LieGroups 3
1.3.1 Convolutionpowersofmeasures 3
1.3.2 Theheatdiffusionsemigroup 5
1.4 Geometric B–NB Classification of Lie Groups. An
Example 6
1.4.1 Invariant Riemannian structures on G and
quasi-isometries 7
1.4.2 Animportantexample 7
1.4.3 Isoperimetricinequalities 8
1.4.4 Thepolynomialfillingproperty 9
1.5 ASpecialClassofGroupsandtheMetricClassification 9
1.5.1 Thegeometricclassificationformodels 9
1.5.2 Thecoarsequasi-isometries 10
1.5.3 Coarsequasi-isometricmodels 10
1.5.4 AgeneralconnectedLiegroup 11
1.5.5 ThegeneralmetricB–NBclassification 11
1.5.6 Thedrawbackofthismetricclassification 11
1.6 HomotopyRetracts 12
1.6.1 Theclassicalretract 12
1.6.2 Thepolynomialretract 12
1.6.3 The polynomialretractpropertyused in the
B–NBclassification 12
1.6.4 Theinvestment/returnratio;orwhatittakesto
provethe(B–NB;Ht)theorem 13
1.7 HomologyonLieGroups 14
vii
viii Contents
1.7.1 ThedeRhamcomplex 14
1.7.2 ThecaseofaLiegroup 14
1.7.3 Thehomologicalinvestment/returnratio[sic] 16
1.8 CˇechCohomologyonaLieGroup 16
1.8.1 AgoodcoverofaLiegroup 16
1.8.2 TheCˇechcomplex 17
1.8.3 Thepolynomialcomplex 18
1.9 TheRoleoftheAlgebraintheB–NBClassification 18
1.10 ABroaderOverviewandSuggestionsfortheReader 19
PARTI ANALYTICANDALGEBRAIC
CLASSIFICATION 21
2 TheClassificationandtheFirstMainTheorem 23
Part2.1:AlgebraicDefinitionsandConvolutionsofMeasures 24
2.1 SolubleAlgebrasandTheirRoots.TheLeviDecom-
position 24
2.1.1 Thenilradical 24
2.1.2 TheradicalandtheLevidecomposition 26
2.2 TheClassification 26
2.2.1 Solublealgebras 26
2.2.2 AmenabilityandtheR-condition 27
2.3 Equivalent Formulations of the Classification and
Examples 27
2.3.1 Affinegeometry 27
2.3.2 Examples 28
2.3.3 TheuseofLie’stheorem 29
2.3.4 Thecompositeroots 29
2.3.5 Anillustration:themodularfunction 30
2.4 Measures on Locally Compact Groups and the C-
Theorem 31
2.4.1 Aclassofmeasures 31
2.5 PreliminaryFacts 32
2.5.1 TheHarnackprincipleforconvolution 32
2.5.2 ApplicationsofHarnack 33
2.5.3 Atechnicalreduction 34
2.5.4 Unimodulargroups 34
2.6 Structure Theorems for Lie Groups and the Exact
Sequence 35
Contents ix
2.6.1 Theuseofstructuretheory 35
2.6.2 SpecialcaseoftheC-theorem 36
2.6.3 TheC-conditionontheexactsequence 38
2.7 Notation,HeuristicsandDisintegrationofMeasures 38
2.7.1 Probabilisticlanguage 38
2.7.2 Thedisintegrationofmeasures,andnotation 39
2.8 SpecialPropertiesoftheConvolutionsinH 41
2.9 TheReductiontotheRandomWalkEstimate 43
2.10 The RandomWalk and Proofof the TheoremWhen
G/H∼=Rd 45
2.11 TheRandomWalkandProofoftheC-Theoreminthe
GeneralCase 46
2.11.1 Theideaoftheproof 46
2.11.2 Theproof 46
2.11.3 AnexerciseinLiegroups 48
Part2.2:TheHeatDiffusionKernelandGaussianMeasures 49
2.12 TheHeatDiffusionSemigroup 49
2.12.1 HeatdiffusionkernelandtheHarnackprinciple 50
2.12.2 Gaussianmeasures 51
2.13 TheC-Theorem 51
2.13.1 TheGaussianC-theorem 51
2.14 DistancesonaGroupandtheGeometryofGaussian
Measures 52
2.14.1 Statementsofthefacts 52
2.14.2 Thedistancedistortion 54
2.14.3 Thevolumegrowth 55
2.14.4 The Gaussian estimates for the projected
measureμˇ 55
2.15 TheDisintegrationofGaussianMeasures 56
2.16 TheGaussianRandomWalkonG/H andProofofthe
C-Theorem 58
2A Appendix:ProbabilisticEstimates 59
2A.1 Thesamplingforboundedvariables 60
2A.2 Avariantinthesampling 61
2A.3 GaussianvariablesandtheC-condition 62
3 NC-Groups 65
Part3.1:TheHeartoftheMatter 66
3.1 Amenability 66
3.1.1 Preliminaries 66