Table Of ContentAliBaklouti
DeformationTheoryofDiscontinuousGroups
De Gruyter Expositions in
Mathematics
|
Editedby
LevBirbrair,Fortaleza,Brazil
VictorP.Maslov,Moscow,Russia
WalterD.Neumann,NewYorkCity,NewYork,USA
MarkusJ.Pflaum,Boulder,Colorado,USA
DierkSchleicher,Bremen,Germany
KatrinWendland,Freiburg,Germany
Volume 72
Ali Baklouti
Deformation Theory of
Discontinuous Groups
|
MathematicsSubjectClassification2020
Primary:22E25,22E27,22E40,22G15,32G05,57S30;Secondary:81S10,57M25,57M27,57S30
Author
Prof.AliBaklouti
FacultédesSciencesdeSfax
DépartementdeMathématiques
RoutedeSoukra
3038Sfax
Tunisia
[email protected]
ISBN978-3-11-076529-8
e-ISBN(PDF)978-3-11-076530-4
e-ISBN(EPUB)978-3-11-076539-7
ISSN0938-6572
LibraryofCongressControlNumber:2022934617
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2022WalterdeGruyterGmbH,Berlin/Boston
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Preface
Discontinuousactionsofgroupsplayanimportantroleinmanyfieldsofmathemat-
ics,especiallyinthestudyofRiemannsurfaces.Thisresearchaxisappearstobeasig-
nificantandindispensableframeworkbecauseofitscloserelationshipwithsomany
other fields in mathematics, such as geometry, topology, number theory, algebraic
geometry,differentialgeometryandwithdifferentfields,suchasphysicsandother
variousareas.ThestudyofKleiniangroups(discretegroupsoforientationpreserv-
ingisometriesofhyperbolicspaces),Fuchsiangroups,andthetheoryofautomorphic
formsareallrichareasofmathematicswithmanydeepresults.TheworkofThurston
on3-manifoldsandasageneralizationthedeformationsofKleiniangroupshavegiven
additionalfocustothisveryrichfieldofdiscontinuousgroupactions.
Whenitcomestothesettingofsolvablegroupsactions,theliteratureissome-
whatscarceinthisarea.Thisbookisdevotedmainlytostudyingvariousgeometric
andtopologicalconceptsrelatedtothedeformationandmodulispacesofdiscontin-
uousgroupactionsandbuildingsomeinterrelationshipsbetweentheseconcepts.It
containsthemostrecentdevelopmentsofthetheory,extendingfrombasicconcepts
toacomprehensiveexposition,andhighlightingthenewestapproachesandmethods
indeformationtheory.Itpresentsfullproofsofrecentresults,computesfundamental
examplesandservesasanintroductionandreferenceforstudentsandresearchers
inLietheory,discontinuousgroupsanddeformation(andmoduli)spaces.Italsoin-
cludesthemostrecentsolutionstomanyopenquestionsoverthelastdecadesand
bringsrelatednewestresearchresultsinthisarea.
Thefirstchapteraimstorecordsomemainbackgroundsonnilpotent,solvable
and exponential solvable Lie groups and some compact extensions. Fundamental
and basic examples, such as Heisenberg groups, threadlike groups, Euclidean mo-
tion groups and Heisenberg motion groups are treated with extensive details for
furtherdevelopmentsanduse.Asapreparationtodiscontinuousactions,anexplicit
descriptionofclosedanddiscretesubgroupsofthesegroupsisalsowelldeveloped.
The important notion of syndetic hull of closed subgroups is also introduced and
manyexistenceandunicityresultsareproved,includingtheextensionsofhomomor-
phisms of discrete subgroups to their syndetic hulls, which appears to be of major
roleinthecomputationoftheparameter,deformationandmodulispaces.
Thesecondchapterfocusesonthecharacterizationofproperactionofclosedsub-
groupsonsolvmanifoldsandonsomehomogeneousspacesofcompactextensions.
Inthecaseofm-stepnilpotentLiegroups,theproperactionofaclosedconnected
subgroupisshowntobeequivalenttoitsfreeactionsform ≤ 3.Suchafactfailsin
generaltoholdotherwise.Wealsogenerategeometriccriteriaoftheproperactionof
adiscontinuousgrouponanarbitraryhomogeneousspace,wherethegroupinques-
tionstandsforthesemidirectproductgroupK⋉ℝn,whereKisacompactsubgroup
ofGL(n,ℝ).InthecaseofHeisenbergmotiongroups,thesamerequirestheclassifi-
cationintothreecategoriesofalldiscretesubgroups.Asshown,thiswillbeacapital
https://doi.org/10.1515/9783110765304-201
VI | Preface
roleinthestudyofmanygeometricalconceptsrelatedtocorrespondingdeformation
andmodulispaces.
Wealsodefinethenotionsofweakandfiniteproperactionsandsubstantiatethat
theseareequivalenttofreeactionsofconnectedclosedsubgroupsoperatingonspe-
cialandmaximalsolvmanifolds.
WepayattentioninChapter3tothedeterminationoftheparameter,deformation
andmodulispacesoftheactionofadiscontinuousgroupΓ ⊂ Gonahomogeneous
spaceG/H innumeroussettings,GbeingaLiegroupandH aclosedsubgroupofG.
Thisissueisofmajorrelevancetounderstandthelocalgeometricstructuresofthese
spaces as many examples reveal. The strategy basically consists in building up ac-
curatecross-sectionsofadjointorbitsofdeformationparameters.Towardsuchgoal,
thefirststepconsistsingeneratinganalgebraiccharacterizationoftheabovespaces
makinguseoftheresultsontheexistenceofsyndetichullsdevelopedinthefirstchap-
ter.IntroducingtheGrassmanniantopology,wethenshowthattheparameterspace
isstratifiedintoG-invariantlayers,endowedwiththestructureofatotalspaceofa
principalfiberbundle.Thisallowstoexplicitlydetermine(toacertainextent)thepa-
rameteranddeformationspacesinmanyfundamentalcases.Forinstance,thesetting
ofHeisenberggroupsisextensivelypursuedinthefourthchapter,whereanecessary
andsufficientconditionforwhichthedeformationspaceisendowedwithasmooth
manifoldstructureisobtained.Thisfurtherallowstoextendthestudytothesetting
ofthedirectproductofHeisenberggroups.
Wealsodealwiththesettingofgeneralm-stepnilpotentLiegroupsinChapter4,
whereadescription ofthe parameterand deformationspacesarederived(m ≤ 3).
AnecessaryconditionfortheHausdorfnessofthedeformationspaceisalsoobtained.
Thesettingofthreadlikegroupsisalsostudiedandanexplicitdeterminationofthe
deformationspaceisprovided.Inthecaseofanon-Abeliandiscontinuousgroupof
rankk,thedeformationspaceisshowntobeendowedwithasmoothmanifoldstruc-
tureifandonlyifk >3.
Thefifthchapterisdevotedtostudythelocalrigiditypropertyofdeformationsin-
troducedbyA.WeilintheRiemanniancaseandgeneralizedfurtherbyT.Kobayashi.
Westatethelocalrigidityconjectureinthenilpotentsetting,whichassertsthatthe
localrigidityfailstoholdforanynontrivialdiscontinuousgroupactingonnilpotent
homogeneousspace.Wefurtherextendourstudytomanyexponentialandsolvable
settings.Namely,weshowthatthelocalrigidityfailswhentheLiealgebralofthesyn-
detichullofΓisnotcharacteristicallysolvableandintheexponentialsettingwhere
lisAbeliananddim(l)≥2.Besides,weprovetheexistenceofformalcoloreddiscon-
tinuousgroupsinthegeneralsolvablesetting.Thatis,theparameterspaceadmitsa
mixtureoflocallyrigidandformallynonrigiddeformations.InthecasewhereGisthe
diamondgroupandΓanontrivialfinitelygeneratedsubgroupofG(notnecessarily
discrete),thenthereisnoopenG-orbitsinHom(Γ,G).Inparticular,ifΓisadiscontin-
uousgroupforahomogeneousspaceG/H,thenthestronglocalrigiditypropertyfails
tohold.
Preface | VII
Wearealsoconcernedwithananalogueoftheso-calledSelberg–Weil–Kobayashi
localrigiditytheoreminthecontextofarealexponentialgroupGandH amaximal
subgroupofG,wherethelocalrigiditypropertyisshowntoholdifandonlyifthe
groupGisisomorphictoAff(ℝ),thegroupofaffinetransformationsoftherealline.
For more generality where G is a Lie group and Γ a finite group, we show that the
spaceHom(Γ,G)/Gisdiscreteandatmostcountable.Thisspaceisfiniteifinaddition
G hasfinitelymanyconnectedcomponents.Thishelpstoshow ananalogueof the
localrigidityconjectureholdsinbothcaseswhereGstandsforthecompactextension
K⋉ℝnandfortheHeisenbergmotiongroups.
Chapter 6 deals with the stability property, a different geometrical concept of
deformations, which measures in general the fact that in a neighborhood of φ ∈
Hom(Γ,G),thepropernesspropertyoftheactiononG/H ispreserved.Thedetermi-
nationofstablepointsisaverydifficultproblemingeneral,whichmainlyreduces
todescribeexplicitlytheinteriorofthesubsetofHom0(Γ,G)ofinjectivehomomor-
d
phisms with discrete image. We are then led to investigate about several kinds of
questionsofgeometricnaturerelatedtothestructureofthedeformationspaceandas
aresult,manystabilitytheoremswillbeestablishedinthenilpotentandexponential
casesandalsointhecontextofsomecompactextensions.
Ontheotherhand,itmaythenhappenthattheredoesnotexistaninfinitedis-
cretesubgroupΓofG,whichactsproperlydiscontinuouslyonG/H.Thisphenomenon
iscalledtheCalabi–Markusphenomenon.Basedonseveralupshotsprovedinprevi-
ouschapters,suchaphenomenontogetherwiththequestionofexistenceofcompact
Clifford–Kleinformsaresubjectofastudyinthecontextofsomecompactextensions
ofnilpotentLiegroups.
Theseventhchapterisdevotedtoresumesomeofthepreviousupshotsoncewe
removetheassumptiononthegroupsinquestiontobesimplyconnected.Thismeans
thatthecentermaybecompactandweshowinthiscasethatmanypreviouslyopen
questionsinthesimplyconnectedsettinggetanswered.Forinstance,inthecaseof
reducedHeisenberggroupsHr ,thedeformationspaceturnsouttobeaHausdorff
2n+1
space and even endowed with a smooth manifold structure for any arbitrary con-
nectedsubgroupHofGandanyarbitrarydiscontinuousgroupΓforG/Handthatthe
stabilitypropertyholds,whichisalsothecaseoftheproductLiegroupG = Hr ×
2n+1
Hr andH = Δ ,thediagonalsubgroupofG.Ontheotherhand,a(strong)local
2n+1 G
rigiditytheoremisobtainedforbothHr andHr ×Hr .Thatis,theparameter
2n+1 2n+1 2n+1
spaceadmitsa(strong)locallyrigidpointifandonlyifΓisfinite.
Thesettingofreducedthreadlikegroupsisalsoconsideredthroughsimilarques-
tions.WeshowthatalocalrigidityconjectureholdsforAbeliandiscontinuousgroups
andthatnon-Abeliandiscontinuousgroupsarestable.Wealsosingleoutthenotionof
stabilityonlayersandshowthatanyAbeliandiscontinuousgroupisstableonlayers.
Thepurposeofthelastchapteristodescribeadequantizationprocedurefortopo-
logicalmodulesoveradeformedalgebra.Wedefinethecharacteristicvarietyofatopo-
logicalmoduleasthecommonzeroesoftheannihilatoroftherepresentationobtained
VIII | Preface
bysettingthedeformationparametertozero.Ontheotherhand,thePoissoncharac-
teristicvarietyisdefinedasthecommonzeroesoftheidealobtainedbyconsidering
theannihilatorofthedeformedrepresentation,andthensettingthedeformationpa-
rametertozero.
Wenextapplysuchadequantizationproceduretothecaseofrepresentationsof
Liegroups.LetV =ℝdbealinearPoissonmanifold.ThenthedualV∗oflinearforms
onV formaLiesubalgebragofthealgebraS(gℂ)ofpolynomialsonV endowedwith
thePoissonbracket.WethenregardthePoissonmanifoldV asthedualg∗oftheLie
algebrag.
InthecasewhereG=expgisanexponentialsolvableLiegroup,theorbitmethod
appearstobeafundamentaltooltosmoothlylinktheirunitarydualswiththespace
ofcoadjointorbits.Wefirstbringexplicitcomputationsofthecharacteristicandthe
PoissoncharacteristicvarietiesinmanyfundamentalPoisson-linearexamples.Inthe
nilpotentcase,weshowthatanycoadjointorbitappearsasthePoissoncharacteristic
varietyofawell-chosentopologicalmodule.WethensubstantiatetheZariskiclosure
conjectureclaimingthatforanirreducibleunitaryrepresentationofG,associatedto
a coadjoint orbit Ω via the Kirillov orbit method, the Poisson characteristic variety
associatedtoatopologicalmodulewithanadequatewaycoincideswiththeZariski
closureing∗ oftheorbitΩ.Wealsoprovetheconjectureinmanyrestrictivecases,
notablyinthenilpotentsetting(withadifferentapproach)andinthecasewherethe
representationisinducedfromanormalpolarizingsubgoup.Wefinallyinvestigatethe
bicontinuityofKirillovandDixmiermapsinthelightofthisdequantizationprocess.
AliBaklouti
Contents
Preface|V
1 Structuretheory|1
1.1 SolvableLiegroups|1
1.1.1 SolvableandexponentialsolvableLiegroups|1
1.1.2 HeisenbergLiegroups|5
1.1.3 ThreadlikeLiegroups|6
1.1.4 MaximalsubgroupsofsolvableLiegroups|7
1.2 Euclideanmotiongroups|10
1.2.1 Onorthogonalmatrices|10
1.2.2 Somestructureresults|13
1.2.3 DiscretesubgroupsofI(n)|15
1.2.4 ClosedsubgroupsofI(n)|26
1.3 Heisenbergmotiongroups|35
1.3.1 Firstpreliminaryresults|35
1.3.2 DiscretesubgroupsofHeisenbergmotiongroups|36
1.4 Syndetichulls|43
1.4.1 ExistenceresultsforcompletelysolvableLiegroups|43
1.4.2 CaseofexponentialLiegroups|44
1.4.3 CaseofreducedexponentialLiegroups|48
2 Properactionsonhomogeneousspaces|51
2.1 Properandfixed-pointactions|51
2.1.1 Discontinuousgroups|52
2.1.2 Clifford–Kleinforms|52
2.1.3 Weakandfiniteproperactions|53
2.1.4 Campbell–Baker–Hausdorffseries|56
2.1.5 Properactionsandcoexponentialbases|57
2.2 Properactionsfor3-stepnilpotentLiegroups|58
2.3 SpecialnilpotentLiegroups|62
2.4 Properactionsonsolvablehomogeneousspaces|65
2.4.1 Properactionsonspecialsolvmanifolds|65
2.4.2 Weakandfiniteproperactionsonsolvmanifolds|67
2.4.3 Properactionsonmaximalsolvmanifolds|72
2.4.4 Connectedsubgroupsactingproperlyonmaximalsolvmanifolds|72
2.4.5 Fromcontinuoustodiscreteactions|74
2.5 ProperactionforthecompactextensionK⋉ℝn|78
X | Contents
2.5.1 Criterionforproperaction|80
2.6 ProperactionsforHeisenbergmotiongroups|84
3 Deformationandmodulispaces|87
3.1 Deformationandmodulispacesofdiscontinuousactions|87
3.1.1 Parameter,deformationandmodulispaces|87
3.1.2 Caseofeffectiveactions|89
3.1.3 Deformationof(G,X)-structures|90
3.2 Algebraiccharacterizationofthedeformationspace|91
3.2.1 Thedeformationandmodulispacesintheexponentialsetting|91
3.2.2 Onpairs(G,H)havingLipsman’sproperty|96
3.3 CaseofAbeliandiscontinuousgroups|97
3.3.1 AnalysisonGrassmannians|98
3.3.2 Theparameterspacefornormalsubgroups|102
3.3.3 Thedeformationspacefornormalsubgroups|103
3.3.4 Examples|107
3.4 Non-Abeliandiscontinuousgroups|109
3.4.1 Structureofaprincipalfiberbundle|109
3.4.2 Thecontextwhere[Γ,Γ]isuniformin[G,G]|113
4 ThedeformationspacefornilpotentLiegroups|117
4.1 DeformationandmodulispacesforHeisenberggroups|117
4.1.1 Acriterionoftheproperaction,continued|117
4.1.2 Thedeformationspacefornon-Abelianactions|118
4.1.3 DeformationandmodulispaceswhenHcontainsthecenter|123
4.1.4 ThecasewhenHdoesnotmeetthecenter|125
4.1.5 CaseofcompactClifford–Kleinforms|133
4.1.6 Examples|137
4.1.7 AsmoothmanifoldstructureonT(Γ,H2n+1,H)|140
4.1.8 ProofofTheorem4.1.26|151
4.1.9 FromH2n+1totheproductgroupH2n+1×H2n+1|152
4.2 Caseof2-stepnilpotentLiegroups|156
4.2.1 DescriptionofthedeformationspaceT(l,g,h)|159
4.2.2 DecompositionofHom1(l,g)|160
4.2.3 Hausdorffnessofthedeformationspace|163
4.3 The3-stepcase|166
4.3.1 Somepreliminaryresults|166
4.3.2 OnthequotientspaceHom(l,g)/G|170
4.3.3 Descriptionoftheparameterandthedeformationspaces|190
4.3.4 Hausdorffnessofthedeformationspace|195
4.3.5 Illustratingexamples|195
4.4 Deformationspaceofthreadlikenilmanifolds|200
