Equidistribution of Dense Subgroups on Nilpotent Lie Groups

arXiv:0710.4489v1 [math.GR] 24 Oct 2007EQUIDISTRIBUTIONOFDENSESUBGROUPSONNILPOTENTLIEGROUPS

EMMANUELBREUILLARDAbstract.LetΓbeadensesubgroupofasimplyconnectednilpotentLiegroupGgeneratedbyafinitesymmetricsetS.Weconsiderthen-ballSnforthewordmetricinducedbySonΓ.WeshowthatSn(withuniformmeasure)becomesequidistributedonGwithrespecttotheHaarmeasureasntendstoinfinity.Wealsoprovetheanalogousresultforrandomwalkaverages.

1.IntroductionInthispaperweareconcernedwithequidistributionpropertiesofdensesubgroupsofLiegroups.LetGbeaLiegroupandS={1,a1,...,as,a−11,...,a−1s}afinitesymmetricsetofelementsinG,whichweassumetogenerateadensesubgroupΓinG.LetSnbethen-ballintheCayleygraphofΓinducedbyS,i.e.thesetofelementsofΓthatcanbewrittenasaproductofatmostnelementsfromS.WegiveourselvestwoopensubsetsUandVinGwithLebesgue-negligibleboundary.Considertheratios

Rn(U,V)=|Sn∩U|

m(V)VariouskindsofaveragesarepossibleinsteadoftheuniformaveragesoverSnabove.InparticularonemayconsiderrandomwalkaveragesoraveragesovertheabstractfreegroupgeneratedbyS.Limitssuchas(1)andresultsofthiskindareusuallycalledratiolimittheorems(see[19],[16]).Arnol’dandKrylovwereamongthefirsttoconsidersuchaquestionandtheyshowedin[2]howtousetherepresentationtheoryofGtoquicklygiveapositiveanswertothisproblemwhenGisacompactLiegroupandSisthesetoffreegeneratorsofadensefreesubgroupofranksinG.Then(1)holdswithmthenormalizedHaarmeasureonG.Inthispaperweprovethat(1)holdswithmaHaarmeasurewhenGisanyclosedsubgroupofasimplyconnectednilpotentLiegroup(Corollary1.2below).Thecaseofsemi-simpleLiegroups(evenSL2(R))remainsanopenquestionuptonow.In[17],Kazhdan2EMMANUELBREUILLARDconsideredthecasewhenG=Isom(Rd)◦ford=2andobtainedananalogousratiolimittheoremforrandomwalkaverages.HisargumentwascorrectedandtheresultextentedbyGuivarc’hin[16](seealso[7]forthepreciseasymptotics)butthecasewhend>2isawell-knownopenproblem.Letusfixsomenotationandthenstatethemainresultofthispaper.ForagroupG,weletvolGbealeftHaarmeasureonG.RecallthataccordingtoatheoremofGuivarc’h(see[15]and[8]),ifGisnilpotent,locallycompactandcompactlygenerated,thenthereisanintegerd(G)∈NsuchthatvolG(Ωn)≈nd(G)foranycompactgeneratingsetΩ.WhenGisconnectedandsimplyconnected,theintegerd(G)isgivenbytheBass-Guivarc’hformula(9)below.Weshow:

Theorem1.1.(Densesubgroupsareequidistributed)LetΓbeafinitelygeneratednilpotentgroupandS={1,a1,...,as,a−11,...,a−1s}afinitegeneratingset.LetGaclosedsubgroupofasimplyconnectednilpotentLiegroup.Supposeφ:Γ→Gisahomomorphismwithdenseimage.ThenthereisapositiveconstantC>0suchthat

(2)limn→+∞

|Sn∩φ−1(B)|

2n|{k∈[−n,n],kα∈(a,b)mod1}|=b−a.ForabelianGthetheoremcanbededucedfromWeyl’sequidistribution.However,aswewillseebelow,whenGisnilpotent,itrequiresdifferenttechniques.CombiningAlexopoulos’theorem[1]ontheasymptoticsofthereturnprobabilityofrandomwalksonfinitelygeneratednilpotentgroupswithTheorem1.1allowstoprovetheanalogouslimittheoremforrandomwalkaverages,namely:

Corollary1.3.(LocalLimitTheorem)LetGbeasimplyconnectednilpotentLiegroupandµasymmetricandfinitelysupportedandprobabilitymeasureonGwhosesupportgeneratesadensesubgroup,thenthereisc(µ)>0suchthatforanyBorelsetBwithLebesgue-negligibleboundary,

(3)limn→+∞n−d(G)/2µ∗n(B)=c(µ)·volG(B)

whereµ∗ndenotesthen-thfoldconvolutionpowerofµ.TheproofofTheorem1.1makesuseoftwocrucialingredients.First,weneedpreciseinformationontheshapeofthen-ballsSninΓ,andthisisessentiallyprovidedbyPansu’sEQUIDISTRIBUTIONOFDENSESUBGROUPSONNILPOTENTLIEGROUPS3theoremfrom[20].Second,weuseanowwell-knownprinciplefromergodictheory(see[10],[18])accordingtowhichtheergodicpropertiesoftheactionofΓonahomogeneousspaceN/MaredualtothoseoftheactionofMonN/Γ.Theuniqueergodicityofunipotentflowsonnilmanifolds,whichisan“ancestor”ofRatner’stheorem,allowsthentoreducetheequidistributionstatementtoacomputationoftheasymptoticvolumeofcosetsofMinsidelargeballsinN.ThegoalofSection2istoprovethisvolumeestimate(Proposition2.14)andgivesomebackgroundonhomogeneousquasi-normsonnilpotentLiegroups.InSection3,wecompletetheproofofTheorem1.1.

N.B.:a.Weprove(seeTheorem3.1and3.5)thatthelimit(2)existsalsoformoregeneralaveragesoftheformB(n)∩φ−1(U)inplaceofSn∩φ−1(U),whereB(n)isthen-ballforaquasi-normonΓ,oranyleft-invariantcoarselygeodesicdistance.b.Thetechniquesofthispaperallowtogetuniformityofconvergencein(2)(resp.(3)),whenBisallowedtovaryamongtranslatesxBsuchthatthedistancebetweenxand1isao(n)(resp.o(√