sysM211 2000MathematicsSubjectClassi cationPrimary53C22Secondary05C38Keywordsandphrasessystoleseparatingsystolesystolicareasystolicratioasymptoticboundsurfacegraph1 2SSABOURAUwherethe ID: 188977
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ASYMPTOTICBOUNDSFORSEPARATINGSYSTOLESONSURFACESSTEPHANESABOURAUAbstract.TheseparatingsystoleonaclosedRiemanniansur-faceM,denotedbysys0(M),isde nedasthelengthoftheshort-estnoncontractibleloopswhicharehomologicallytrivial.Wean-swerpositivelyaquestionofM.Gromov[Gr96,2.C.2.(d)]abouttheasymptoticestimateontheseparatingsystole.Speci cally,weshowthattheseparatingsystoleofaclosedRiemanniansurfaceMofgenusandareagsatis esanupperboundsimilartoM.Gro-mov'sasymptoticestimateonthe(homotopy)systole.Thatis,sys0(M).logg.Contents1.Introduction12.Systolicinequalitiesongraphs43.Morsefunctionsand rstBettinumbers74.Constructionofgraphsonthesurface95.Fundamentalgroupsofgraphsandsurfaces126.Proofofthemaintheorem17References201.IntroductionLetMbeanonsimplyconnectedclosedRiemanniansurface.The(homotopy)systoleofM,denotedbysys1(M)orsys(M)forshort,isde nedasthelengthoftheshortestnoncontractibleloopsinM.Wede netheoptimalsystolicareaofanonsimplyconnectedclosedsurfaceMas(M)=infArea(M) sys(M)2(1.1) 2000MathematicsSubjectClassi cation.Primary53C22;Secondary05C38.Keywordsandphrases.systole,separatingsystole,systolicarea,systolicratio,asymptoticbound,surface,graph.1 2S.SABOURAUwherethein mumistakenoverthespaceofallthemetricsonM.Theoptimalsystolicareaisatopologicalinvariantofsurfaces.Theexactvalueoftheoptimalsystolicareaisknownforthetorus,cf.[Be93],theprojectiveplane[Pu52]andtheKleinbottle[Ba86].ForanotionofsystoleextendedtotheisometrygroupsofRiemannianmanifolds,theoptimalsystolicareahasalsobeencomputedforthe17cristallographicgroupsoftheplaneandthetrianglegroups[Ba93].Nootherexactvalueoftheoptimalsystolicareaisknown.However,nontriviallowerboundsontheoptimalsystolicareaofev-erynonsimplyconnectedclosedsurfacehavebeenestablished,cf.[Gr83],[Gr96],[KS06a],[KS05],[KS06b]and[Sa06a]forrecentdevelopments.Forinstance,wededucefrom[Pu52]and[Gr83,5.2.B]thateverynon-simplyconnectedclosedsurfaceMsatis es(M)2 (1.2)withequalityifandonlyifMishomeomorphictotheprojectiveplane.Suchaninequalityiscalledasystolicinequality.Inhigherdimension,nonoptimalsystolicinequalitiesexistfores-sentialmanifolds[Gr83]andoptimalsystolicinequalitiesexistforthenotionsofone-dimensionalstableandconformalsystoles,cf.[Gr99],[IK04],[BCIK].Werefertotheexpositorytexts[Be93],[Gr96],[Gr99]and[CK03],andthereferencesthereinforanaccountonthesubjectandothergeneralizationsinhigherdimensions.Thesystolicinequality(1.2)canbeimprovedbytakingintoaccountthetopologyofM.Forinstance,thefollowingresultofM.Gromovshowsthatclosedsurfacesoflargegenushavealargeoptimalsystolicarea.Theorem1.1([Gr83,6.4.D'],[Gr96,3.C.3]).ThereexistsapositiveconstantCsuchthateveryclosedsurfaceMofgenusgsatis es(M)Cg (lng)2:(1.3)Itisshownin[KS05],usingdi erenttechniques,that(1.3)holdsforeveryCprovidedthangislargeenough.Werefertotheendofthissectionforadiscussionaboutthedi erentproofsofthisresult.Seealso[Gr83],[Gr96]and[Sa06b]forgeneralizationsinhigherdimensions.AnupperboundontheoptimalsystolicareaofsurfacesoflargegenushasbeenfoundbyP.BuserandP.Sarnak[BS94].Namely,theyconstructhyperbolicsurfaces(M;hyp)ofgenusg,obtainedas SEPARATINGSYSTOLESONSURFACES3coveringsofanarithmeticRiemannsurface,suchthatsys(M;hyp)&lng(1.4)Otherconstructionsandhigherdimensiongeneralizationscanbefoundin[KSV].Combiningtheinequalities(1.3)and(1.4),weobtainCg (lng)2(M)C0g (lng)2(1.5)whereCandC0aretwouniversalpositiveconstants.Inparticular,thereisnolowerboundon(M)linearinthegenusg(see[BB05]forageneralizationinhigherdimension).GivenaclosedRiemanniansurfaceMofgenusg2,wede netheseparatingsystole,denotedbysys0(M),asthelengthoftheshortestnoncontractibleloopsinMwhicharehomologicallytrivial.Thatis,sys0(M)=infflength(\r)j\rinducesanontrivialclassin[1(M);1(M)]g:Inthisde nition,thehomologycoecientsareinZandthecom-mutatorofthefundamentalgroupofMisnoted[1(M);1(M)].Similartotheoptimalsystolicarea(M),wede ne0(M)=infArea(M) sys0(M)2wherethein mumistakenoverthespaceofallthemetricsonM.Weclearlyhavesys(M)sys0(M),cf.(1.8)foramorepreciseesti-mate,hence(M)0(M).Lowerboundson0(M)canbededucedfrom[Gr83,5.4](indeed,thelengthofthecommutatoroftwoindependentloopsgivesanupperboundontheseparatingsystole).Inparticular,forsurfacesoflargegenus,M.Gromovshowedthefollowing.Theorem1.2([Gr83,5.4.B],[Gr96,2.C.2.(d)]).Forevery 1,thereexistsapositiveconstantC suchthateveryclosedsurfaceMofgenusg2satis es0(M)C g :Thehyperbolicsurfaces(M;hyp)constructedin[BS94]satisfysys0(M;hyp)&lng:(1.6)Therefore,Theorem1.2doesnotholdfor =1. 4S.SABOURAUM.Gromovaskedin[Gr96,2.C.2.(d)]whetheralowerboundon0(M)similarto(1.3)exits.Weanswerthisquestionpositively.Speci cally,weprovethefollowing.MainTheorem1.3.ThereexistsapositiveconstantCsuchthateveryclosedsurfaceMofgenusg2satis es0(M)Cg (lng)2:(1.7)Theestimate(1.6)showsthattheinequality(1.7)yieldstherightasymptoticbound.Moreprecisely,adoubleinequalitysimilarto(1.5)holdsfor0(M).SuchadoubleinequalityalsoholdsforH(M),whereH(M)isde nedbyreplacingin(1.1)thesystolewiththehomologysystole,cf.[Gr96].Recallthatthehomologysystole,denotedbysysH(M),isthelengthoftheshortesthomologicallynontrivialloopsinM.Weclearlyhavesys(M)sysH(M)butnolowerboundon0(M)canimmediatelybededucedfromalowerboundonH(M).Notealsothatsys(M)=minfsys0(M);sysH(M)g:(1.8)Thesystolicinequality(1.3)was rstprovedin[Gr83,p.74]byus-ingatechniqueknownas"di usionofchains".Itwasthenimprovedin[KS05],whereanupperboundfortheentropyofasystolicallyex-tremalsurfaceintermsofitssystoleisestablished.AsobservedbyF.Balache [B04],theinequality(1.3)canalsobederivedbycombin-ingtheworksofS.Kodani[Ko87],andB.Bollobas,E.SzemerediandA.Thomason[BT97,BS02]onsystolicinequalitiesofgraphs.WewillfollowthislatterapproachtoproveMainTheorem1.3.Moreprecisely,weconstructagraphonasurface(Section4),establishsys-tolicinequalitiesonthisgraph(Section2),relatethelengthofcyclesonthegraphtotheseparatingsystoleonthesurface(Section5)andde-ducealowerboundontheareaofthesurfacethroughacoreaformulaasin[Ko87](Section6).Acknowledgment.Theauthorwouldliketothanktherefereeforhisorhercomments,especiallyonsystolicinequalitiesongraphs,cf.Re-mark2.3.2.SystolicinequalitiesongraphsByde nition,ametricgraphisagraph(i.e.,a nite1-dimensionalsimplicialcomplex)endowedwithalengthstructure. SEPARATINGSYSTOLESONSURFACES5Thehomotopyclassofagraph ischaracterizedbyits rstBettinumberb( ),whichcanbecomputedasfollows:b( )=e( ) v( )+n( )(2.1)wheree( ),v( )andn( )arerespectivelythenumberofedges,ver-ticesandconnectedcomponentsof .De nition2.1.Thesystoleandthetotallengthofametricgraph ,denotedbysys( )andlength( ),arerespectivelyde nedasthelengthoftheshortestnoncontractibleloopof andthesumofthelengthsoftheedgesof .Thesetwometricinvariantsdonotdependonthesimplicialstructureofthegraph.Wede nethesystoliclengthof as0( )=length sys( ):Foreveryb1,wealsode neahomotopyinvariantofagraphas0(b)=inff0( )j metricgraphwith rstBettinumberbg:Weputa0whenwedealwithgraphs(asin0( )or0(b))andno0whenwedealwithsurfaces(asin(M)).In[BS02],B.BollobasandE.SzemerediimprovedthemultiplicativeconstantofasystolicinequalityforgraphsestablishedbyB.BollobasandA.Thomasson[BT97].Speci cally,theyprovedTheorem2.2([BS02]).Foreveryb3,wehave0(b)3 2b 1 log2(b 1)+log2log2(b 1)+4(2.2)wherelog2isthelogarithmtothebase2.Inparticular,foreveryb2,wehave0(b)1 4b lnb:(2.3)Strictlyspeaking(2.3)isaconsequenceof(2.2)onlyforb5.When2b4,thebound(2.3)stillholdsbecausethesystoliclengthofagraphisatleast1,thatis0(b)1.Remark2.3.Therefereeofthisarticlepointedouttotheauthorthataboundsimilarto(2.3)(withaslightlyworseconstantbutthisdoesnotmatterforourpurpose)canbeobtainedfromelementaryargu-mentsthatdonotrelyonthemainresultof[BS02].Indeed,theesti-mates(1)and(2)of[BS02,x2]allowustoextendtheinequality[BT97,Theorem5]statedfornon-weightedgraphstometricgraphs. 6S.SABOURAUCombinedwiththeupperboundonthesystoliclengthofgraphsestablishedin[BB05],wehave,foreveryb2,1 4b lnb0(b)8ln(2)b lnb:De nition2.4.Ametricgraph issaidtobeadmissibleifthe rstBettinumberofeachofitsconnectedcomponentsisatleast2.Asinthecaseofsurfaces,wede netheseparatingsystoleofanadmissiblegraph assys0( )=infflength(\r)j\rinducesanontrivialclassin[1();1()]gwhererunsovertheconnectedcomponentsof .Aspreviously,wealsode ne00( )=length sys0( )(2.4)and,foreveryb2,00(b)=inff00( )j admissiblemetricgraphwith rstBettinumberbg:UsingTheorem2.2,weobtainalowerboundon00(b).Morepre-cisely,weshowProposition2.5.Foreveryb2,wehave00(b)1 64b lnb:Proof.Let beanadmissiblemetricgraphwhose rstBettinumberequalsb.Withoutlossofgenerality,wecanassumethat isconnectedandthatsys0( )=1.Denotebykthenumberof\smallcycles",i.e.,thenumberofsimpleloopsoflengthlessthan1 8.Supposethattwo\smallcycles",c1andc2,intersecteachother.Thehomotopyclassesofthesetwoloops(basedatthesamepoint)donotcommute.Therefore,theloopc1[c2[c 11[c 12,oflength2length(c1)+2length(c2)1 2;whichrepresentstheircommutator,isnotcontractiblein .Thus,sys0( )1 2,henceacontradiction.Therefore,the"smallcycles"of aredisjoint.Letcbeashortestpathbetweentwo\smallcycles"c1andc2(recallthat isconnected).Thecommutatorofthehomotopyclassesofc1andc[c2[c 1canberepresentedbyaloopoflengthatmost1 2+2length(c).Thus,sys0( )1 2+2length(c).Therefore,thelengthofcisatleast1 4. SEPARATINGSYSTOLESONSURFACES7Wededucethatthe1 8-(open)-neighborhoodsofthe\smallcycles"aredisjoint.Since isconnected,thelengthofthe1 8-neighborhoodofeach\smallcycle"ciisatleastlength(ci)+1 8.Therefore,length( )k 8.Ifkb 2,thenlength( )b 16.Thus,foreveryb2,wehave00(b)b 161 64b lnb:Ifkb 2,weremoveanedgefromeach\smallcycle".Thisgivesrisetoagraph 0 with rstBettinumberb0=b k(recallthatthe"smallcycles"aredisjoint).Byconstruction,thesystoleof 0isatleast1 8,i.e.,sys( 0)1 8.Further,length( )length( 0).Therefore,00( )1 80( 0):Sinceb 2b0b,theinequality(2.3)impliesthat00( )1 8b=2 4lnb:Hencetheresult.3.MorsefunctionsandfirstBettinumbersRecallthede nitionofatopologicalMorsefunction.Wereferto[Mo59]and[Mo75]forageneralstudyandapplications.De nition3.1.Letfbeacontinuousfunctiononaclosedn-manifoldM.ApointxofMissaidtoberegular(resp.criticalofindexp)ifthereexistsatopologicalchart atxsuchthatf 1 f(x)isalinearpro-jection(resp.therestrictionofaquadraticformofsignature(n p;p)).Inthesede nitions,thechart isnotnecessarilyadi eomorphism,onlyahomeomorphism.Furthermore,thesede nitionsdonotdependonthechoiceofthechart .ThefunctionfisatopologicalMorsefunctionifeverypointofMisregularorcriticalofindexpforsomeintegerp.NotethatatopologicalMorsefunctiononaclosedmanifoldhas nitelymanycriticalpoints.LetfbeatopologicalMorsefunctiononasurfaceMofgenusgwithonlyonecriticalpointoneachcriticallevel.Letxbeacriticalpointofindex1andy=f(x).SincefisatopologicalMorsefunctionwithonlyonecriticalpointoneachcriticallevel,theconnectedcomponentoff 1(y)containingthepointxisaunionZxoftwosimpleloopsintersectingonlyatx. 8S.SABOURAUDe nition3.2.WesaythatxisoftypeIiftheintersectionofeveryr-neighborhoodUr(Zx)ofZxwithf 1(]y r;y[)hasoneconnectedcomponent(onecylinder)forrsmallenough.Otherwise,wesaythatxisoftypeII.Inthiscase,theintersectionofeveryr-neighborhoodUr(Zx)ofZxwithf 1(]y r;y[)hastwocon-nectedcomponents(twocylinders)forrsmallenough.RecallthatMisorientable,fisatopologicalMorsefunctionandxisofindex1. \r0x \r00x xTypeII:twolegs \r0x \r00x xTypeI:onelegThefollowingresultcanbefoundin[Ko87,(4.9)].Weincludeaproofforthesakeofcompleteness.Lemma3.3.Supposethatfhasonlyonelocalminimum.Then,thefunctionfhasexactlygpointsoftypeII.Proof.LetNibethenumberofcriticalpointsofindexi.FromMorseformula[Mo59],[Mo75,Theorem10.1],wehave1 N1+N2=2 2g:(3.1)WealsohaveN1=NI+NII;(3.2)whereNIandNIIarethenumbersofcriticalpointsoftypeIorII.WecanlowerthecriticalpointsoftypeI(asin[Ma02,Theorem2.34]forinstance),preservingthetotalnumberofcriticalpoints,theirindexandtheirtype,sothatallthecriticalpointsoftypeIforthenewfunction,stilldenotedf,lieinf 1(] 1;t0[)andallthecriticalpointsoftypeIIorindex2lieinf 1(]t0;1[)forsomet0.Astincreases,everytimewepassacriticalvaluecorrespondingtoacriticalpointoftypeI,thenumberofholesoff 1(] 1;t[)increasesbyone.Therefore,f 1(] 1;t0[)ishomeomorphictoaspherewithNI+1holes.Ontheotherhand,sincenocriticalpointoftypeIliesinf 1(]t0;1[),thespacef 1(]t0;1[)hasnohandle.Further,eachconnectedcomponentoff 1(]t0;1[)hasonlyonecriticalpointofindex2.Weconclude SEPARATINGSYSTOLESONSURFACES9aspreviouslythatf 1(]t0;1[)isaunionofpuncturedsphereswithNII+N2boundarycomponents.Therefore,NI+1=NII+N2:Combiningthisequationwith(3.1)and(3.2),wederivethedesiredresult,i.e.,NII=g.4.ConstructionofgraphsonthesurfaceLetMbearealanalyticRiemanniansurfaceofgenusg2.Fixapointx0lyinginasystolicloopofM.DenotebyFthedistancefunctionfromx0,i.e.,F(x)=dist(x0;x)foreveryx2M.Proposition4.1.ThefunctionFisatopologicalMorsefunction.Proof.Everypointlyingoutsidethecutlocusofx0isregular.There-fore,wewillonlyconsiderpointsinthecutlocusofx0.Nowtheresultfollowsfromthestudyofthecutlocusin[Be77,pp.194-198](seealso[Fi40]),where[Be77,Lemme3]isreplacedby[He82,Lemma1.2].Asobservedin[He82],theconditionofbisectionandthestructureofthecutlocus,cf.my35,my36,usedin[Be77]holdforeveryclosedanalyticsurface.Givenacriticalpointxofindex1ofF,thereexistsaunique(simple)geodesicloop \rxofindex0basedatx0formedoftwominimizingarcs, \r0xand \r00x,joiningx0tox.Conversely,ifa(simple)geodesicloopofindex0basedatx0isformedoftwominimizingarcsjoiningx0tox,thepointxisacriticalpointofindex1.Seethetwo guresinSection3.Fortechnicalreasons,itwouldbeconvenientifFhadonlyonecrit-icalpointoneachcriticallevel.Toachievethiscondition,wecanraisesomecriticalpointsofFasin[Ma02,Theorem2.34].Moreprecisely,forevery"2]0;inj(M)=2[smallenough,wecanapproximateFbyatopologicalMorsefunctionfonMpreservingthecriticalpointswithonlyonecriticalpointoneachcriticallevelandwithx0asitsonlylocalminimum,suchthatf(x0)=0;jjf Fjj"onM;fis(1+")-Lipschitz.Wecanalsoassumethatforeverycriticalpointxofindex1,withy:=f(x),theloop \rxsatis esthefollowing: \rxliesinf 1([0;y]); \rxcannotbedeformedintof 1([0;y[)inf 1([0;y]); 10S.SABOURAUaswellaslength( \rx\f 1([t;1[))2(y t)+2";(4.1)foreveryt2[0;y].Theheightof \rx,de nedasmaxf( \rx),isequaltof(x)=y.Notethatthereare nitelymanyloops \rx.De nition4.2.Usingthenotationsandde nitionsofSection3,wesaythattheloop \rxisoftypeIorIIifthecriticalpointxisoftypeIorII.Furthermore,ifxisoftypeII,thetwotrajectories \r0xand \r00xpassthroughthetwocylindersofUr(Zx)\f 1(]y r;y[)forrsmallenough.Seethetwo guresinSection3.Letusconstructbyinductiona\short"system ofloops \r1;:::; \rnbasedatx0.Theloop \r1isthenoncontractibleloopwiththeleastheightamongthe \rx's.Wede nebyinduction \riastheloopwiththeleastheightamongthe \rx'swhosehomotopyclassdoesnotlieinthesubgroupGi 1generatedby \r1;:::; \ri 1in1(M;x0).Here,byde nition,G0isthetrivialsubgroupof1(M;x0).Theloop \ripassesthroughtwocriticalpoints,namelyx0and xi.Itsheightmaxf( \ri)isequalto yi:=f( xi).Proposition4.3.i)Thesystem generates1(M;x0).Inparticular, containsatleast2gloops;ii)Thesystem containsatleastgloopsoftypeI;iii)Ifthehomotopyclassofapiecewisesmoothloop\rbasedatx0doesnotlieinGi 1,thenmaxf(\r)maxf( \ri).Proof.Supposethatthereexistsapiecewisesmoothloop\rbasedatx0whosehomotopyclassdoesnotlieinGi 1.Usingaheightdecreasingdeformationof\rifnecessary,wecanassumethat\rpassesthroughacriticalpointxofindex1suchthatmaxf(\r)=f(x)=y:(4.2)Thus,wecantake\r(passingthroughacriticalpointxofindex1sat-isfying(4.2))sothatitsheightisminimal.Theloop \rxpassingthroughthecriticalpointxofindex1suchthatf(x)agreeswiththeheightof\rsatis esthefollowing.Lemma4.4.Thehomotopyclassof \rxdoesnotlieinGi 1. SEPARATINGSYSTOLESONSURFACES11Proof.Theloop\rpasses nitelymanytimesthroughx.Thus,wecantake\rsothatitpassesaminimalnumberoftimesthroughx.De net0=infftj\r(t)=xg.Either\rj[0;t0][ \r0xor\rj[0;t0][ \r00xcanbedeformedintoaloop off([0;y[).Supposethattheformercaseoccurs(similarargumentsworkinthelattercase).Theheightof islessthan\r's.Sincetheheightof\risminimal,thehomotopyclassof liesinGi 1.Therefore,wecanassumethat\rj[0;t0]agreeswith \r0x.Since\rpassesaminimalnumberoftimesthroughx,theloop \r00x[\rj[t0;1]canbedeformedintoaloopofheightatmostf(x),whichpassesfewertimesthroughxthan\r(oneshouldexamineseparatelythetypeIandIIcases;seethetwo guresinSection3).Therefore,thehomotopyclassoftheloop \r00x[\rj[t0;1]liesinGi 1.Thus,wecanassumethat\rj[t0;1]agreeswith \r00x.Thatis,\ragreeswith \rx.Hencethedesiredresult.Letusnowprovei).Recallthatnisthenumberofelementsof .Supposethatthereexistsapiecewisesmoothloop\rbasedatx0whosehomotopyclassdoesnotlieinthesubgroupGngeneratedby .ByLemma4.4,thereexistsaloop \rxwhosehomotopyclassdoesnotlieinGn.Therefore, \rn+1exists(andagreeswith \rx),whichisabsurd.Hencei).Toestablishiii),wearguebycontradictionagain.ByLemma4.4,thereexistsaloop \rxwithy=f(x)maxf( \ri)whosehomotopyclassdoesnotlieinGi 1.Thisyieldsacontradictionwiththede nitionof \ri.Henceiii).FromLemma3.3,thenumberofloopsoftypeIIin isatmostg.Since hasexactly2gloops(eachofthemoftypeIorII),thesystem containsatleastgloopsoftypeI.Let bethesystemofthegshortestloops\r1;:::;\rgoftypeIin .Denotebyyi=maxf(\ri)theheightof\ri.Permutingtheindicesifnecessary,wecanassumethaty1y2yg.Lett0=0,t1=y1andtk=y1+(k 1) fork2,where :=1 6 00(g 1) g 1sys0(M):(4.3)and 00(t):=1 64t ln(2+t)(4.4)foreveryrealt0.Remark4.5.Theloop\r1agreeswith \r1.Further,sincex0liesinasystolicloopofM,theheightof \r1isboundedby1 2sys(M)+".Thatis,t11 2sys(M)+". 12S.SABOURAUDenoteby\rk1;:::;\rkgktheloopsof suchthattkmaxf(\rki)tk+1andletyki=maxf(\rki)betheheightof\rki.LetKbetheminimalintegerksuchthatygtk+1.WehaveKXk=1gk=g:(4.5)Arealtissaidtobegenericifthepreimagef 1(t)isformedofa(possiblyempty) niteunionofdisjointcircles.SincefisatopologicalMorsefunction(with nitelymanycriticalpoints),almosteverytisgeneric.Foragenerictwithtkttk+1and1kK 1,wede nethegraph kt:=f 1(t)[gk+1[i=1 \rk+1i\f 1([t;+1[):(4.6)Thegraph ktisendowedwiththelengthstructureinducedbytheinclusioninM.Themetricgraphobtainedfrom ktbyremovingtheconnectedcomponentshomeomorphictocirclesisanadmissiblegraph,cf.De nition2.4,notedb kt(assumingthatitisnonempty).Wehaveb kt kt:Notethatthegraphs ktandb ktdi erfromthegraphsde nedin[Ko87,x6](theselattermaycontainarcsofloopsoftypeI).Inparticular,thefundamentalgroupofb ktisasubgroupof1(M),cf.Section5.Theverticesofb ktagreewiththeintersectionpointsoff 1(t)and[gk+1i=1\rk+1i.Thus,thegraphb ktis3-regular(i.e.,thevalenceofeachvertexequals3)with2gk+1verticesand3gk+1edges.Recallthat ktliesinf 1([t;+1[).Hence,the rstBettinumberofb ktisequaltogk+1plusthenumberofconnectedcomponentsofb kt,cf.(2.1).Inparticular,ifb ktisnonempty,its rstBettinumberisatleast1+gk+1.5.FundamentalgroupsofgraphsandsurfacesInthissection,whereweusethepreviousconstructionsandnota-tions,weshowthatthefundamentalgroupofb ktliesinthefundamentalgroupofM.First,weshowthefollowing. SEPARATINGSYSTOLESONSURFACES13Lemma5.1.Nosimpleloopofb ktiscontractibleinMforeverygenerictwithtkttk+1and1kK 1.Inparticular,sys(b kt)sys(M):Proof.Wearguebycontradiction.Let\rbeasimpleloopofb ktcon-tractibleinM.Theloop\rboundsadiskinM.Thefunctionfadmitsnolocalminimumintheinteriorof.Oth-erwise,suchalocalminimumagreeswiththeuniquelocalminimumoff,whichisx0.Inthiscase,containsf 1([0;t[)andso\r1sincemaxf(\r1)=t1t.Thus,\r1iscontractibleinM.Henceacontradic-tion.Now,twocasesmayoccur:CASEI:Theloop\risnotcontainedinf 1(t).Then,\rpassesthroughanarc\ri0\f 1(]t;+1[)ofsomeloop\ri0of .Changingtheindexifnecessary,wecanassumethat\rpassesthroughnoarcof\riwithyiyi0(recallthatfhasonlyonecriticalpointoneachcriticallevel).Byconstruction,thesimplearc\riscomposedofarcs\ri\f 1(]t;+1[)where\ri2 andofsubarcsoff 1(t).Since\riscon-tractibleinM,theloop\ri0ishomotopictoaloopbasedatx0lyinginf 1([0;yi0[).Thus,thehomotopyclassof\ri0liesinasubgroupgener-atedbyloopsofheightlessthan\ri0's.Thisisimpossiblebyde nitionof\ri0,cf..Proposition4.3.iii).CASEII:Theloop\riscontainedinf 1(t).Sincefhasnolocalminimumintheinteriorof,thediskliesinf 1([t;+1[).Thereexistsaloop\riof ,withyi&#x-292;&#x.374;t,whichintersects\r(recallthat\rliesinb ktandthatnoconnectedcomponentofb ktishomeomorphictoacircle).Thearc\ri\f 1([t;+1[)liesin.Therefore,theloop\riishomotopictoaloopbasedatx0lyinginf 1([0;t]).Aspreviously,wederiveacontradiction.Letusintroducesomenotations.Fixagenericrealtwithtkttk+1and1kK 1suchthatb ktisnonempty.EverysucientlysmallopentubularneighborhoodNofaconnectedcomponentofb ktdeformationretractsontothroughfrsgwiths2[0;1]suchthatr0istheidentitymaponNandr 11(p)\@Nhasthreeortwoelementsdependingwhetherthepointpofisavertexornot(recallthatb ktis3-regular;seethe gurebelow).Wesetr=r1. 14S.SABOURAU@N Theconnectedcomponentsoftheboundary@NofNareformedof nitelymany(disjoint)simpleloops.Furthermore,sinceMisori-entableandthe rstBettinumberofisatleasttwo,theset@Nhasatleastthreeconnectedcomponents.Thus,Nishomeomorphictoaspherewithatleastthreeholes.Inparticular,Nisnothomeomorphictoacylinder.ThefollowingresultwillbeusefulintheproofofProposition5.4.Lemma5.2.Thesimpleloopsformingtheboundary@NofNarenoncontractibleinM.Proof.Supposethatthereexistsasimpleloop\rof@Nwhichiscon-tractibleinM.Theimage \rof\rbythe(deformation)retractionrisnotsimple,otherwiseitwouldbenoncontractibleinMfromLemma5.1(andsowouldbe\r).Thus,rtakestwopointsof\rtothesameimage.Sincethegraphis3-regular,the(deformation)retractionrtakestwoarcsof\rtothesameedgecof.Remark5.3.Changingtheedgecifnecessary,wewillassumethattheheightofcismaximalamongtheedgesofontowhichrsendstwoarcsof\r.Removingcfrom \rdecomposestheloop \rintotwoloopsc1andc2with \r=c[c1[c 1[c2: cc1c2Theloopsc1andc2intersecttheedgeconlyatitsendpoints.Since \ristheboundaryofanopendiskinM,theloopsc1andc2aretheboundariesofanopencylinderCinMcontainingc.Inparticular,theboundary@CofCagreeswithc1[c2.Thearcscandcihavedi erentheightsfori=1;2.Otherwise,theircommonheightwouldbet(recallthatfhasatmostonecriticalpoint SEPARATINGSYSTOLESONSURFACES15oneachlevelset)andc[ciwouldbecontainedinf 1(t),whichisimpossiblesincef 1(t)isformedofaunionofdisjointsimpleloops.Thesameargumentalsoshowsthatc1andc2havedi erentheights,unlesstheybothlieinf 1(t).Now,weconsidertwocases.CASEI:Assumethattheheightofcisgreaterthanc1'sandc2's.Notethatyc:=maxf(c)�t.TheconnectedcomponentZoff 1(yc)intersectingcliesinthecylinderCandisdisjointfrom@C.Thus,cuttingCalongthetwononcontractiblesimpleloopsformingthecom-ponentZgivesrisetotwocylinders,atthebottom,andonedisk,atthetop(see gurebelow). c1c2ZCcMoreprecisely,f 1([0;yc[)\Ciscomposedoftwocylinderswhoseboundarycomponentsagreewiththeconnectedcomponentsof@CandthesimpleloopsformingZ.Wederivethattheloop\rcof withmaxf(\rc)=yc,whichagreeswithcintheneighborhoodoff 1(yc),cf.(4.6),isoftypeII.Henceacontradictionsincetheloopsof areoftypeI.CASEII:Assumethattheheightofc1isgreaterthanc'sandc2's.Inparticular,maxf(c1)�t.Thereexistsaloop\ri0of ,withthesameheightasc1,suchthatc1\\ri0agreeswiththearc\ri0\f 1([t;+1[).Theloop\ri0isformedoftwoarcs,\r0i0and\r00i0,ofthesameheight,arisingfromx0andendingatthesameendpoint.FromRemark5.3,theloop \rpassesonlyoncethroughc1\\ri0.Letc01andc001bethearcsofc1n\ri0(possiblyreducedtopoints)joining\r0i0tocand\r00i0toc. 16S.SABOURAU c001c2C\r0i0c01Theloop\ri0ishomotopicto(\r0i0nc1)[c01[c0 11[(c1\\ri0)[c001[c00 11[(\r00i0nc1);whichagreeswith(\r0i0nc1)[c01[c1[c00 11[(\r00i0nc1):Byassumption,theloopc1ishomotopictotheloopc[c2[c 1.There-fore,\ri0ishomotopictotheloop i0=(\r0i0nc1)[c01[c[c2[c 1[c001[(\r00i0nc1):WederiveacontradictionfromProposition4.3.iii)sincetheheightof i0islessthan\ri0's.Letusrecallsomefactsaboutthedisk\row.DenotebyM0thesurfaceMendowedwitha xedhyperbolicmetric.Thedisk\rowde nedin[HS94](seealso[Sa04]forasimilar\row)deformsapiecewisesmoothloop\rofM0through\rtwitht0.LetCbea nitecollectionofpiecewisesmoothloopsinM0.Throughoutthedisk\row,theloopsofCsatisfythefollowing:simpleloopsremainsimple;disjointloopsremaindisjoint;forevery\r2C,thefamily\rteitherdisappears(i.e.,\rtcon-vergestoapoint)in nitetimeorconvergestoa(unique)non-contractiblegeodesicloopofM0ast!1.Usingthis\rowandLemma5.2,wecanprovethemainresultofthissection.Proposition5.4.Theinclusioni:,!Mofeveryconnectedcompo-nentofb ktinducesamonomorphismbetweenthefundamentalgroups.Thatis,i:1() !1(M)isinjective. SEPARATINGSYSTOLESONSURFACES17Proof.Theboundary@NofNM0decomposesintosimpleloopsc1;:::;ck.Wecanassumethatthenormalvectorsofthetwo-sidedloopscipointtowardN.Considerapiecewisesmoothloop\rofNcontractibleinMandapplythedisk\rowtoc1;:::;ckand\r.TheloopciconvergestotheuniquenoncontractiblegeodesicloopofM0freelyhomotopictocithroughthedisk\rownotedcti.Forev-eryt0,theloopscti,whicharesimpleanddisjointasaretheloopsci,boundanopensetNt'Ntowardwhichpointthenormalvectorsofthetwo-sidedloopscti.Ontheotherhand,theloops\rtremaindisjointfromtheloopsctithroughthedisk\row.Inparticular,theloops\rtlieinNt.NotethatthereexistsahomeomorphismbetweenNtandNwhichtakesctitociandthefreehomotopyclassof\rtto\r's.Since\riscontractibleinM,the\row\rtNtdisappearsin nitetime.Therefore,\rtiscontractibleinNtforsomet.Thisimpliesthat\riscontractibleinNandsoinsinceNdeformationretractsonto.Thus,thehomomorphismi:1() !1(M)isinjective.Weimmediatelydeducethefollowing.Corollary5.5.Wehavesys0(b kt)sys0(M):(5.1)Remark5.6.Contrarytob kt,thefundamentalgroupsofthegraphsde nedin[Ko87]arenotnecessarilyisomorphicsubgroupsof1(M).Withoutthisproperty,itisstillpossibletoboundthesystoleofthesegraphs,cf.Lemma5.1,butaboundsimilarto(5.1)doesnotholdingeneral.6.ProofofthemaintheoremUsingthepreviousnotationsandresults,weshowthemainresultofthisarticlefollowing[Ko87].ProofofMainTheorem1.3.Sinceeverysmoothmetriccanbeapprox-imatedbyarealanalyticoneandsincetheareaandthesystolearecon-tinuousonthespaceofallmetrics,wecanassumethatMisarealana-lyticRiemanniansurfaceofgenusg2.Keepingthesamenotationsasintheprevioussections,wecanalsoassumethatsys(M)1 4sys0(M),otherwisetheinequality(1.3)yieldstheresult.Let\r11and\r21betwoloopsof basedatx0suchthatmaxf(\r11)=t11 2sys(M)+"andmaxf(\r21)t1+ ,cf.Remark4.5.Thecommutatorofthehomotopyclassesof\r11and\r21canberepresentedbyaloopcoflengthlessthan4t1+2 .Ast13 4sys(M)3 16sys0(M) 18S.SABOURAUand 1 8sys0(M),thelengthofcislessthansys0(M).Thus,ciscontractibleandthehomotopyclassesof\r11and\r21commute.Sincethecentralizerofeverynontrivialelementof1(M)isisomorphictoZ(recallthatg2),thehomotopyclassesof\r11and\r21areproportional.Furthermore,thehomotopyclassofasimpleloopontheorientablesurfaceMisindivisible.Therefore,thetwononcontractibleloops\r11and\r21of agree.Thus,theindexg1de nedinSection4isequalto1,i.e.,g1=1:Lettbegenericwithtkttk+1and1kK 1.Ifb ktisnonempty,i.e.,gk+1&#x-282;&#x.344;0,thenb ktisanadmissiblegraph,cf.De nition2.4andSection4,with rstBettinumberatleast1+gk+12.Thus,since00isnondecreasing,cf.(2.4),wehavelength(b kt)00(1+gk+1)sys0(b kt):FromProposition2.5,wederivethat00(1+gk+1) 00(gk+1),cf.(4.4),where 00(gk+1)=1 64gk+1 ln(2+gk+1):(6.1)Hence,fromtheinclusionb kt ktandCorollary5.5,wehavelength( kt) 00(gk+1)sys0(M):Furthermore,thisinequalityholdswhengk+1vanishes.Thus,fromthede nitionof kt,cf.(4.6),wehavelengthf 1(t)+gk+1Xi=1length \rk+1i\f 1([t;+1[) 00(gk+1)sys0(M)Combinedwiththeestimate(4.1),wederivelengthf 1(t)+gk+1Xi=12(tk+1i t)+2"gk+1 00(gk+1)sys0(M)wheretk+1i=maxf(\rk+1i).Notethattkttk+1tk+1itk+2.Integratingfromtktotk+1for1kK 1,weobtainZtk+1tklengthf 1(t)dt+2gk+1Xi=1Ztk+1tk(tk+1i t)dt+2"gk+1 00(gk+1)sys0(M) SinceZtk+1tktk+1i tdt3 2 2,wehaveZtk+1tklengthf 1(t)dt 00(gk+1)sys0(M) 3gk+1 2 2"gk+1 : SEPARATINGSYSTOLESONSURFACES19Now,applythecoareaformulatothe(1+")-Lipschitzfunctionf,cf.[Fe69,3.2.11].OneobtainsArea(M)1 1+"Z10lengthf 1(t)dt:Hence,(1+")Area(M)K 1Xk=1Ztk+1tklengthf 1(t)dt K 1Xk=1 00(gk+1)!sys0(M) 3 K 1Xk=1gk+1! 2 2" K 1Xk=1gk+1! KXk=2 00(gk)!sys0(M) 3 KXk=2gk! 2 2" KXk=2gk! Sinceg1=1,theinequality(4.5)yieldsKXk=2gk=g 1:Furthermore,foreveryp;q2N, 00(p+q)=1 64p ln(2+p+q)+q ln(2+p+q)1 64p ln(2+p)+1 64q ln(2+q) 00(p)+ 00(q)Inparticular,KXk=2 00(gk) 00 KXk=2gk!= 00(g 1):Hence,(1+")Area(M) 00(g 1)sys0(M) 3(g 1) 2 2"(g 1) :Passingtothelimitas"!0,weobtainArea(M) 00(g 1)sys0(M) 3(g 1) 21 12 00(g 1)2 g 1sys0(M)2 20S.SABOURAUsince =1 6 00(g 1) g 1sys0(M),cf.(4.3).Therefore,Area(M)1 216g 1 (ln(1+g))2sys0(M)2:Thus,theinequality(1.7)holdsforg2withC=2 18.NotethatthevalueoftheconstantCcanbeimproved,especiallyforlargevaluesofg.References[BB05]Babenko,I.;Balache ,F.:Geometriesystoliquedessommesconnexesetdesrev^etementscycliques,Math.Ann.333(2005)157{180.[B04]Balache ,F.:Surdesproblemesdelageometriesystolique,Semin.Theor.Spectr.Geom.Grenoble22(2004)71{82.[BCIK]Bangert,V.;Croke,C.;Ivanov,I.;Katz,M.:Boundarycaseofequal-ityinoptimalLoewner-typeinequalities,Trans.Amer.Math.Soc.359(2007)1-17.[Ba86]Bavard,C.:InegaliteisosystoliquepourlabouteilledeKlein,Math.Ann.274(1986)439{441.[Ba93]Bavard,C.:L'airesystoliqueconformedesgroupescristallographiquesduplan,Ann.Inst.Fourier43(1993)815{842.[Be77]Berger,M.:Volumeetrayond'injectivitedanslesvarietesriemanniennesdedimension3,OsakaJ.Math.14(1977)191{200.[Be93]Berger,M.:SystolesetapplicationsselonGromov,SeminaireBourbaki,Exp.711,Asterisque216(1993)279{310.[BS02]Bollobas,B.;Szemeredi,E.:Girthofsparsegraphs,J.GraphTheory39(2002)194{200.[BT97]Bollobas,B.;Thomason,A.:OnthegirthofHamiltonianweaklypan-cyclicgraphs,J.GraphTheory26(1997)165{173.[BS94]Buser,P.;Sarnak,P.:OntheperiodmatrixofaRiemannsurfaceoflargegenus.WithanappendixbyJ.H.ConwayandN.J.A.Sloane.Invent.Math.117(1994),no.1,27{56.[CK03]Croke,C.;Katz,M.:UniversalvolumeboundsinRiemannianmani-folds,SurveysinDi erentialGeometryVIII,LecturesonGeometryandTopologyheldinhonorofCalabi,Lawson,Siu,andUhlenbeckatHarvardUniversity,May3-5,2002,editedbyS.T.Yau(Somerville,MA:InternationalPress,2003.)pp.109{137.SeearXiv:math.DG/0302248[Fe69]Federer,H.:Geometricmeasuretheory.Grundlehrendermathematis-chenWissenschaften,153.Springer{Verlag,Berlin,1969.[Fi40]Fiala,F.:Leproblemedesisoperimetressurlessurfacesouvertesacour-burepositive,Comment.Math.Helv.13(1941)293{346.[Gr83]Gromov,M.:FillingRiemannianmanifolds,J.Di .Geom.18(1983)1{147.[Gr96]Gromov,M.:Systolesandintersystolicinequalities.ActesdelaTableRondedeGeometrieDi erentielle(Luminy,1992),291{362,Semin.Congr.,vol.1,Soc.Math.France,Paris,1996.www.emis.de/journals/SC/1996/1/ps/smf sem-cong 1 291-362.ps.gz SEPARATINGSYSTOLESONSURFACES21[Gr99]Gromov,M.:MetricstructuresforRiemannianandnon-Riemannianspaces.Progr.inMathematics152,Birkhauser,Boston,1999.[HS94]Hass,J.;Scott,P.:Shorteningcurvesonsurfaces,Topology33(1994)25{43.[He82]Hebda,J:Somelowerboundsfortheareaofsurfaces,Invent.Math.65(1982)485{490.[IK04]Ivanov,S.;Katz,M.:GeneralizeddegreeandoptimalLoewner-typeinequalities,IsraelJ.Math.141(2004)221{233.[KSV]Katz,M.;Schaps,M.;Vishne,U.:Logarithmicgrowthofsystoleofarith-meticRiemannsurfacesalongcongruencesubgroups,preprintavailableatarXiv:math.DG/0505007[KS05]Katz,M.;Sabourau,S.:Entropyofsystolicallyextremalsurfacesandasymptoticbounds,Ergod.Th.Dynam.Sys.,25(2005),1209{1220.[KS06a]Katz,M.;Sabourau,S.:HyperellipticsurfacesareLoewner,Proc.Amer.Math.Soc.134(2006),no.4,1189-1195.[KS06b]Katz,M.;Sabourau,S.:AnoptimalsystolicinequalityforCAT(0)met-ricsingenustwo,Paci cJ.Math.227(2006),no.1,155-176.[Ko87]Kodani,S.:Ontwo-dimensionalisosystolicinequalities,KodaiMath.J.10(1987),no.3,314{327.[Ma02]Matsumoto,Y.:AnintroductiontoMorsetheory,TranslationsofMath-ematicalMonographs,208,Amer.Math.Soc.,Providence,RI,2002[Mo59]Morse,M.:Topologicallynon-degeneratefunctionsonacompactn-manifoldM,J.AnalyseMath.7(1959)189{208.[Mo75]Morse,M.:Topologicallynondegeneratefunctions,Fund.Math.88(1975)17{52.[My35]Myers,S.B.:Connectionsbetweendi erentialgeometryandtopology,I.Simplyconnectedsurfaces,DukeMath.1(1935),no.3,376{391.[My36]Myers,S.B.:Connectionsbetweendi erentialgeometryandtopology,II.Closedsurfaces,DukeMath.2(1936),no.1,96{102.[Pu52]Pu,P.M.:SomeinequalitiesincertainnonorientableRiemannianman-ifolds,Paci cJ.Math.2(1952)55{71.[Sa04]Sabourau,S.:Fillingradiusandshortclosedgeodesicsofthe2-sphere,Bull.Soc.Math.France132(2004)105{136.[Sa06a]Sabourau,S.:Entropyandsystolesonsurfaces,Ergod.Th.Dynam.Sys.,26(2006),no.5,1653{1669.[Sa06b]Sabourau,S.:Systolicvolumeandminimalentropyofasphericalmani-folds,J.Di .Geom.,74(2006),no.1,155{176.LaboratoiredeMathematiquesetPhysiqueTheorique,UniversitedeTours,ParcdeGrandmont,37400Tours,FranceE-mailaddress:sabourau@lmpt.univ-tours.fr