PERMANENCE IN COARSE GEOMETRY ERIK GUENTNER BSTRACT - PDF document

2ERIKGUENTNERofferedmanyhelpfulcomments.ThissurveyisbasedonlecturesgivenattheconferencesGeo-metriclinearizationofgraphsandgroupsheldattheEPFLausanneandtheWorkshoponanalyticpropertiesofinnitegroupsheldattheUnivertityofGeneva.IamindebtedtoGoulnaraArzhant-sevaandAlainValetteforinvitationstothesemeetingsandfortheirencouragementinwritingthissurvey.2.COARSEMETRICSPACESInthissectionweshallestablishthebasicdenitionsandnotationsforthecategoryofcoarsemetricspaces.Throughout,weallowforpseudo-metricspaces–apseudo-metricisdenedbyallowingthepossibilitythatdistinctpointsareatdistancezero.LetXandYbemetricspaces.Afunctionf:X!Yisuniformlyexpansiveifthereexistsanon-decreasingfunction:[0;1)=infd(x;y)tfd(f(x);f(y))g;whicharenon-decreasingandsatisfytheinequalities(2.1)and(2.2),respectively.Assumingtheconditions(2.3)and(2.4)weobservethatisniteandisproper,respectively.Conversely,assumingfisuniformlyexpansiveoreffectivelyproper,thecondition(2.3)or(2.4)isreadilyveried.Afunctionwhichisbothuniformlyexpansiveandeffectivelyproperisacoarseembedding.AcoarseequivalenceisacoarseembeddingwhichiscoarselyontointhesensethatthereexistsC�0suchthattheC-neighborhoodoff(X)isallofY:8y2Y9x2XsuchthatdY(f(x);y)C; 4ERIKGUENTNERRemark.Althoughthelemmaiswidelyknown,wehaveprovidedsomedetailintheproofsoastomakethefollowingobservation–theassertionsdonotdependontheparticularfunctionsbutonlyontheassociatedquantitativedata.Forexample,ifacollectionoffunctionsf areall-uniformlyexpansive,-effectivelyproperandC-coarselyontothentheirinversesg ,denedasin(2.5),areall -uniformlyexpansiveandthecompositionsf g andg f areallmaxfC; Cg-closetotheidentity.Suchremarkswillplayanimportantroleforus,andweshallestablishaframeworkforworkingeffecientlywiththeminthenextsection.3.COARSEMETRICFAMILIESInwhatfollows,itwillbeconvenienttoworkwithfamiliesofmetricspaces,ratherthanasinglemetricspace.Following[17],ametricfamilyissimplyacollectionofmetricspacesX=fXg.TheindividualspacescomprisingXareitscomponents.Theindexsetispartofthedataofametricfamily.Typically,ametricfamilywillbeacollectionofsubspacesofagivencoarsemetricspace,eachequippedwiththesubspacemetric.Indeed,weshallseebelowthatifweallowextendedreal-valued`metrics'thisisalwaysthecase.LetX=fXgandY=fYgbemetricfamilies.AfunctionoffamiliesX!Yisacollectionoffunctionsff g,togetherwithastructuremap 7!(( ),( ))relatingthevariousindexsetssuchthatf :X( )!Y( );andsuchthateachXisthedomainofsomef .Theindexingsetsmaybedifferent,thesamespaceXmaybethedomainofmorethanonefunctionf ,etc.Wewritesimplyf:X!Y.Whencomposingfamiliesff gandfg gweassumetheindexingsetsarethesameandthatforeveryindex thedomanoff istherangeofg inwhichcasethecompositionisff g g.Remark.Toavoidclutteringthenotationweshallcontinuetobedeliberatelyvagueregarding`in-dexsets'and`structuremaps'.Tominimizeconfusion,weshallcontinuewiththeabovenotationthroughout:thefamilyXisalwaysindexedby,Yalwaysby,andthefunctionfisalwaysindexedby .Other,similarnotationwillhopefullybeclearfromthecontext.Asageneralrule,quantitativestatementsaboutmetricspaces,functions,etc.areapplied`uni-formly'tometricfamilies,theirmorphisms,etc.Thus,afunctionff gofmetricfamiliesis-uniformlyexpansiveifthisistrueofeachf ;asimilardenitionappliesinthecaseof-effectivelyproperfunctions.Afunctionff gisC-coarselyontoifthisistrueofeachf andif,inaddition,eachYistherangeofsomef .Twofunctionsff gandfg gareC-closeif,foreach ,thefunctionsf andg havethesamedomainandareC-close.ThefollowinganalogofLemma2.2characterizingcoarseequivalenceoffamiliesholds(seetheremarkafterLemma2.2).3.1.Lemma.Letf:X!Y.Thefollowingareequivalent:(1)fisuniformlyexpansiveandthereexistsauniformlyexpansiveg:Y!XsuchthatthecompositionsfgandgfareclosetotheidentityofYandX,respectively;(2)fisacoarseembeddingandiscoarselyonto. 6ERIKGUENTNERRemark.Forsettheoreticconstructionswegenerallyrequirethatfamiliesaredenedwithrespecttothesameindexset,inwhichcaseoperationsare`componentwise'.Indeed,wehavealreadydonesowhendiscussingcompositionsofmapsoffamilies.Weshallintroducerelevantconstructionsaswego,butalwayswiththisgeneralprincipleinmind.Remark.Wehaveseenthatquantitativenotionsaretypicallyappliedtofamiliesina`uniform'way.Animportantexampleisthatofaboundedfamily–afamilyofmetricspacesofuniformlyboundeddiameter.4.COARSESTRUCTURESInhismonograph,Roeintroducedthenotionofacoarsestructure[22].AcoarsestructureonasetXisacollectionEofsubsetsofXXthatcontainsthediagonal,andisclosedunderthetakingofsubsets,inverses,productsandniteunions:(1)
2E(2)E2EandFE)F2E(3)E2E)E-12E(4)E,F2E)EF2E(5)E,F2E)E[F2EHere,theinverseandproductaredenedbyE-1=f(x;y):(y;x)2Eg;EF=f(x;y):9z2Xsuchthat(x;z)2Eand(z;y)2Fg:ThesubsetsinEarecalledentouragesorcontrolledsets.Asetequippedwithacoarsestructureisacoarsespace.1AcoarsespaceisconnectediftheunionofitsentouragesisallofXX.Whilewegenerallyassumeallcoarsespacesareconnected,weshalldiscussanexceptiontothisconventionbelow.Example.A(pseudo-)metricspaceadmitsanaturalcoarsestructure;entouragesforthemetriccoarsestructure2are(thesubsetsof)themetrictubesf(x;y):d(x;y)Cg:Coarsestructuresandcoarsespacesenjoyaphilosophicaladvantageovercoarsemetricspaces–forexample,weshallseebelowthatallleftinvariantboundedgeometrymetricsonacountablegroupinducethesamemetriccoarsestructurewhichisthereforetransparentlyuniquelydeterminedbythegroup.Ontheotherhand,theabsenceofanaturalguagecomplicatesthenotionofacoarsefamily–whileitisnaturaltospeakofsetsofuniformsizeindifferentmetricspacesitisnotpossibletodosoindifferentcoarsespaceswithoutimposingadditionalstructure.ThismotivatesourdenitionofacoarsefamilyasacollectionofcoarsespacesX=fXg,eachofwhichisacoarsesubspace 1Onoccasiontherequirementthatthediagonalbeanentourageisdroppedfromthedenitionofcoarsespace;inthiscase,acoursespaceinwhichthediagonalisanentourageiscalledunital.2Roecallsthistheboundedcoarsestructureassociatedtothemetric[22]. 8ERIKGUENTNERRemark.AmetricfamilyXhasniteasymptoticdimensionifandonlyifthereexistsadsuchthatthecomponentsXofXhaveasymptoticdimensionatmostd`uniformly'inthesenseofBellandDranishnikov[2].Precisely,givenR,eachcomponentXadmitsacoverU(X)whichpartitionsintod+1colorsasabovebutwith(5.1)supfdiamU:U2U(X),allXg1:Remark.Thenotionofasymptoticdimensionforcoarsespaceswasexplicitlydescribedinthemonograph[22]andiseasilyadaptedtothesettingofcoarsefamilies.TheessentialdenitionisthatasubsetUofacoarsespaceXisboundedifUUisanentourage;acollectionofsubsetsUiofXisuniformlyboundedif[UiUiisanentourage.Inametriccoarsestructurethesearetheusualnotions.Seealso[4].FiniteasymptoticdimensionwasintroducedbyGromov[15].Foranup-to-datesurveyofthisproperty,itsapplication,andawealthofexampleswerecommendthesurveyofBellandDranish-nikov[4].5.2.PropertyAandexactness.Acoarsemetricspaceisexactifitsatisesthefollowingparti-tionofunitycondition:foreveryR&#x]TJ/;ྲྀ ;.9;Ւ ;&#xTf 1;.36; 0 ;&#xTd [;0andevery"&#x]TJ/;ྲྀ ;.9;Ւ ;&#xTf 1;.36; 0 ;&#xTd [;0thereexistsapartitionofunityf'UgsubordinatetoauniformlyboundedcoverUsuchthatforxandy2X(5.2)d(x;y)R)XU2UjU(x)-U(y)j":Ametricfamilyisexactifone,equivalentlyeach,ofitstotalspacesisexact.ToexpressthatacoarsemetricspaceormetricfamilyisexactwewriteX2EXorX2EX.Remark.AmetricfamilyXisexactpreciselywhenitscomponentsare`uniformly'exactinthesensethatgivenRand"eachcomponentXadmitsapartitionofunityfXUgsatisfying(5.2)andsubordinatetoacoverU(X)satisfying(5.1).3Remark.Thedenitionofexactnessiseasilyadaptedtothegeneralsettingofcoarsespaces,againusingthenotionofboundednessmentionedabove.DadarlatandGuentnerintroducedexactnessformetricspacesasasubstituteforPropertyA,believingitsdenitioneasiertomanipulatethanthestandardcharacterizationsofPropertyA[12].Further,theyprovedtheequivalenceofexactnessandPropertyAformetricspacesofboundedgeometry.4PropertyAitselfwasintroducedbyYuinthecourseofhisworkontheNovikovconjecture[25];hewasinterestedindeningapropertythatisbotheasytoverifyincasesofinterestandthatimpliescoarseembeddability.Recently,WilletthaswrittenanexcellentsurveyofPropertyA[24].ThegermofauniformversionofPropertyAwasintroducedbyBell[6];relatednotionsplayimportantrolesintheworkofNowak[18]andDadarlatandGuentner[12].TheequivalenceofexactnessandPropertyAworksalsoforfamilies–exactnessofafamilyisequivalentto`uniform'PropertyAforitscomponents. 3Inthelanguageof[12]thecomponentsofXare`equi-exact'.4NickWrighthasexplainedtomehowtoextendthisequivalencetothegeneralcase.See[24]. 10ERIKGUENTNERWhileitisessentiallyobviousthatourbasicpropertiesFADd,FADandEXsatisfysubspacepermanence,thefollowinglemmaallowsustotreatsubspacepermanenceandcoarseinvariancesimultaneously.6.1.Lemma.ApropertyPiscoarselyinvariantandsatisessubspacepermanenceifandonlyifwheneverY2PandXcoarselyembedsinYthenX2P.Proof.AcoarseembeddingX!Yfactorsasthecompositionofacoarseequivalenceandtheinclusionofafamilyofsubspaces.Conversely,theinclusionofafamilyofsubspacesisacoarseembeddingand,ifXandYarecoarselyequivalenttheneachcoarselyembedsintheother.6.2.Theorem.Ourbasicproperties,FADd,FADandEXarecoarselyinvariantandsatisfysub-spacepermanence.Precisely,letPbeoneoftheseproperties.IfY2PandXcoarselyembedsinYthenX2P.Proof.ForthepropertiesFADd,FADandEXtheresultforfamiliesfollowseitherdirectlyor,withaidofLemma3.2afterpassingtototalspaces,fromtheanalogous(andwell-known)resultforsinglemetricspaces.See[2,22]and[12,Rem.2.11].OurnextpermanencepropertyconcernstheattempttoconcludethatacoarsemetricspacehaspropertyPfromtheknowledgethatitiswrittenastheunionofsubspaceseachofwhichhavepropertyP.ClearlythesubspacesmusthavepropertyPuniformly.Further,ifnotevery(count-able)spacehaspropertyP,anadditionalhypothesis,typicallysomesortofexcisioncondition,isnecessaryingeneral.UnionPermanence.SupposeZ=X[Y.IfXandY2PthenZ2P.SupposeZ=[Xi.SupposefurtherthatforeveryR�0thereexistsasubspaceW=W(R)ZsuchthatW2PandsuchthatforeveryindexthecollectionfXi;-WgiisR-disjoint.IftheXi;havePuniformlythenZ2P.Inaccordwithourconventions,wewriteZ=[iXiwhenZandtheXisharethesameindexsetand,inaddition,foreachindexwehaveZ=[iXi;.Inparticular,allfamiliesinthestatementareindexedbythesameset.Remark.TheassumptionthattheXi;havePuniformlymeansthatthefamilyfXi;g,asbothandivary,satisesP.TheXi;-W,forxedandvaryingi,aresubspacesofZ,allowingustospeakofofR-disjointness.Remark.IfthepropertyPiscoarselyinvariantandsatisessubspacepermanencethentheniteas-sertionofunionpermanencefollowsfromtheinniteassertion.Indeed,excisetheR-neighborhoodoftheintersectionX\Yfromeachandobserve:(1)X-NR(X\Y)andY-NR(X\Y)areR-disjoint;(2)NR(X\Y)iscoarselyequivalenttoX\Y,whichisasubspaceofbothXandY.Theformulationforfamiliesislefttothereader.Thisobservationappliestoeachofourbasicproperties. 12ERIKGUENTNERProof.Asusual,adirectproofwithoutappealtototalspacesispossible,andamountstocarefulbookkeepingwiththeconstantsappearingintheproofforasinglespace.Theproofisacombina-tionofThm.3.1andCor.3.3of[12].Fiberingpermanenceissomewhatmoresubtlethantheotherpermanenceproperties,andcaremustbetakentoformulateitcorrectlyforourotherbasicproperties.Forasymptoticdimension,thefollowingresultisbothsimpleanduseful.6.6.Theorem.Letf:X!Ybeuniformlyexpansive.SupposethatY2FAD.SupposethereexistsdsuchthatforeveryboundedfamilyofsubspacesZYtheinverseimagef-1(Z)2FADd.ThenX2FAD.Remark.Inthecaseofsinglespaces,asopposedtofamilies,signicantlymorerenedresultsarepossible.Perhapstherstsuchresultis[5,Thm.1],whichisstatedforaLipschitzmapbetweengeodesicmetricspaces,andachievedaboundmuchbetterthanthatinherentinourproof.Anoptimalresultwasobtainedrecentlyin[7].Thereadermaywishtoadaptthesetothesettingoffamilies.Seethesurvey[4]formoredetails.Intheproof,andsubsequently,weshallworkwithanobviousreformulationofniteasymptoticdimensionintermsofcolorings.Forexample,theconclusionX2FADd00isrephrasedasfollows:foreveryRthereexistsanSsuchthatforeverythereexistsacoverUofXandacoloringc:U!f0;:::;d00gsatisfying(1)ifU2Uthendiam(U)S;(2)ifU6=V2Uandc(U)=c(V)thend(U;V)R.Forbrevityweshallexpress(1)bysayingthatUisS-bounded,and(2)bysayingthatcisa(d00;R)-coloringofU.Proof.Forvariety,andtoillustratethealternate`quantitative'pointofview,weshallgiveadirectproofwithoutappealtototalspaces.Thestatementisathinlydisguisedversionofthesimplesttypeofresultconcerningasymptoticdimensionofaproduct–andisproventhesameway.Compare[11].Somotivated,weassumeY2FADd0andshallprovethatX2FADd00,whered00=(d+1)(d0+1)-1.Further,werestrictattentiontothe(equivalent)caseinwhichf,XandYarealldenedoverthesameindexset;thuswehaveafamilyof-uniformlyexpansivemapsf:X!Y:Asanalpreparationwerephrasethepreimageconditioninthestatement:foreveryS0andRthereexistsanSsuchthatforeverysubsetUofeveryYsatisfyingdiam(U)S0thereexistsacoverWoff-1(U)withthefollowingtwoproperties:(1)WisS-bounded;(2)Wadmitsa(d;R)-coloring.(NotethatthecovercomprisedofallsubsetsoftheYofdiameterS0isbounded.) 14ERIKGUENTNERProof.LetPbeasinthestatement,andletXbeametricfamily.Theconversebeingimmediate,weshowthatifthecollectionofallboundedsubspacesofXsatisesPunifromlythenXitselfsatisesP.FixabasepointxineachcomponentXofX.Deneacontractive,inparticularuniformlyexpansivemapf:X!fRgusingthedistancefunction:f(x)=d(x;x);f:X!R:IfnowZisaboundedfamilyofsubspacesofRthenf-1(Z)isafamilyofboundedsubspacesofX,whichsatisesPbyhypothesis.Thus,beringpermanenceapplies.Remark.Intheproof,thedistinctionbetween`familyofboundedsubspaces'and`boundedfamilyofsubspaces'ispurposeful.Thetheoremassertsaformof`locality'forP,andmayberephrasedinseveralsuggestiveways.WeworkwithasinglespaceX.First,assumingPsatisessubspacepermanencewemayrephrasebyassertingequivalenceofthefollowing:(1)X2P;(2)thefamilyfB(x;R)g,asbothx2XandR2Rvary,satisesP;(3)foreachxedx2XthefamilyfB(x;R)g,asR2Rvaries,satisesP.Second,foralocallynitespaceXwehave:X2PpreciselywhenthecollectionofitsnitesubsetssatisesPuniformly.InthisformtheresultwasknownforpropertyCEforquitesometime[14].Morerecently,aversionwasformulatedandprovedforPropertyA[8];seealso[24]foranothervariant.7.DERIVEDPERMANENCERESULTSFORGROUPSMotivatedbythewealthofapplicationswefocusonthecaseof(countable)groups.Thees-sentialideaistoprovethatagroupGhaspropertyPbyobservingthatitactsbyisometriesonaspacehavingpropertyPinsuchawaythatthestabilizersoftheactionhavepropertyP–anap-plicationofberingpermanence.AcloselyrelatedquestionconcernsprovingthatGhaspropertyPassumingitisbuiltfromgroupshavingpropertyPbyfamiliarconstructionsfromgrouptheory.Henceforth,weshallworkwithsinglespacesandsinglegroups,leavingtothereadertoformulateappropriate`family'versions.7.1.Groupsascoarsemetricspaces.LetGbeagroup.AlengthfunctiononGisafunction`:G![0;1)satisfying(1)`(1)=0(2)`(s-1)=`(s)(3)`(st)`(s)+`(t)Weallowforthepossibilitythatsomenon-identityelementsofGhavelengthzero.Alengthfunctionisproperif,foreveryC0,thesetfs2G:`(s)Cg 16ERIKGUENTNERRemark.TheanalogforapossiblyuncountablegroupwithitscanonicalcoarsestructureistreatedbyDranishnikovandSmith[13].Weturntogroupactions,whichwealwaysassumetobebyisometries.ThesimplestsituationoccurswhenagroupGactsmetricallyproperlyonametricspaceY.Inthiscasetheorbitmapf:G!Y;f(s)=s
yisacoarseembedding;theeasiestwaytoseethisistoobservethat`(s)=d(y;s
y)denesaproperlengthfunctiononGandtoappealtoProposition7.1.1.Thus,anypropertysatisfyingsubspacepermanencewillpassfromYtoG.Thisdiscussionapplies,inparticular,whenYislocallyniteandtheactionisfree,orhasnitestabilizers.Weareinterestedinthecomplementarycasewheretheactionhasinnitestabilizersand/orthemetricspaceisnotlocallynite.7.2.3.Theorem.LetPbeapropertysatisfyingsubspace,niteunionandberingpermanence.IfGactsonalocallynitespaceY2Pandthereexistsay2YforwhichthestabilizerGy2PthenG2P.Proof.Thisisabasicapplicationofberingpermanence.Beginbyobservingthatitsufcestoconsiderthecaseofatransitiveaction–simplychooseyasinthestatementandrestrictthegivenactiontotheorbitGywhich,asasubspaceofY,satisesP.Fixabasepointy2Y.Weemploytheorbitmapandbeginbycheckingthatitisuniformlyexpansive.Indeed,thisishintedintheparagraphjustbeforethestatement:ifd(s;t)Athend(f(s);f(t))=d(y;s-1t
y)B=supfd(y;r
y):`(r)Ag1:ApplyLemma2.1.Itremainstocheckthattheorbitmapsatisestheinverseimageconditioninthestatementofberingpermanence.Formally,letZYbeaboundedfamilyofsubspacesofY,saywithuniformboundS.Asweareassumingtheactionistransitive,wendforeverycomponentZofZanelements2Gforwhichs
ZB(y;S).Usingtheequivarianceoftheorbitmapweconcludethatf-1(Z)iscoarselyequivalent(evenisometric)asafamily,toafamilyofsubspacesofthesinglemetricspace(7.1)f-1(B(y;S))=fs2G:d(y;s
y)Sg:Thus,byanapplicationofsubspacepermanence,itremainsonlytoseethatthiscoarsestabilizersatisesP.But,bylocalniteness,itistheniteunionofcosetssGy,eachofwhichisisometrictoGy.Anapplicationofniteunionpermanencecompletestheproof.7.2.4.Corollary.ApropertyPsatisfyingsubspace,niteunionandberingpermanenceisclosedundergroupextensions.Remark.InthecorollaryweassumePsatisesniteunionpermanence.Thisisnecessaryonlybecausewededucethecorollaryfromtheprecedingtheorem–thecorollaryremainstruewithoutthisassumption.Indeed,niteunionpermanenceisusedonlyneartheendoftheprooftoconcludethattheunionofcosets(7.1)satisesP.Inthecaseofanextension1!H!G!G=H!1 18ERIKGUENTNER FIGURE1.Bass-Serretree.herevisavertexintheBass-Serretree,viewedasanAorB-coset.Thus,XisthedisjointunionoftheAandB-cosetsand,aseachelementofGliesinpreciselyonA-cosetandoneB-coset,weseethatXcomprisestwo(disjoint)copiesofG.Anadjacencyoccursbetweenxvandxw,wherex2Gisviewedasanelementoftwoadjacentverticesvandw.Atransitionoccursbetweenxvandyv,wherexandy2GareelementsofacommonAorB-cosetv. FIGURE2.Adjacencyandtransition:b2B.ThemetriconXisthemaximalmetricsatisfyingtherequirementsthatanadjacencyhasdistanceatmostone,andthatatransitionhasdistanceatmostthedistancemeasuredintheambientgroupGor,whatisthesame,inthecosetvGequippedwiththesubspacemetric.AformulaforthemetriconXisgivenin[11].Remark.Itisimportanttousethesubspacemetricontheindividualcosetsatthisstage–thischoiceeliminatesdistortion,sothatthemetriconXbehavesasonewouldhope.InFigure2,forexample,d(xv;yv)willactuallyequalthedistanceintheambientgroup–thereisnopossibilityofndingashortcutintheneighboringcosetw.Similarly,d(zw;yv)=inffd(y;x)+1+d(x;z):x2v\w=bCg;wherethedistancesd(y;x)andd(x;z)aremeasuredintherespectivecosetsvandw.Thisisaspecialcaseoftheformulafrom[11].(InFigure2thesmallrectangleinsidethecosetBrepresentsthecopyofbCB,andsimilarlythesmallsemidiskinsidethecosetbA.)ThegroupGactsbyisometriesontheBass-SerretreeTbypermutingcosets.Further,GactsbyisometriesonXaccordingtotheformulas
xv=(sx)s
v.And,f(xv)=vdenesanequivariantcontractionX!T–comparingFigures1and3themapisevident.Weshouldliketoapply PERMANENCEINCOARSEGEOMETRY218.2.Theorem.Letf:X!Ybeuniformlyexpansive.SupposeY2EXandthatforeveryboundedfamilyofsubspacesZYtheinverseimagef-1(Z)2CE.ThenX2CE.ThefailureofthegeneralformofberingpermanenceforCEhasinterestingconsequencesinthecaseofgroupextensions.Indeed,followingthetreatmentintheprevioussection,werecoverabasicresultofDadarlatandGuentner[11],originallyprovendirectly.8.3.Theorem.Anextensionwithcoarselyembeddablekernelandexactquotientisitselfcoarselyembeddable.Precisely,ifanormalsubgroupHofGsatisesCEandG=HsatisesEXthenGsatisesCE.Apaprtfromacoupleofessentiallyobviousremarks,therewaslittleprogressonthegeneralproblemofextensionsuntilrecently.Buildingontheirworkregardingstabilityofa-T-menabilityunderwreathproducts,deCornulier,StalderandValetteobtainedthefollowingveryinterestingresult[10].8.4.Theorem.IfthecountablediscretegroupsGandHarecoarselyembeddablethensoistheirwreathproductHoG=H(G)oG.Here,H(G)denotesthegroupofnitelysupportedH-valuedfunctionsonG,onwhichGactsbytranslation.Weshallnotenterintothedetailsoftheproof,whichrestsonacharacterizationofCEintermsofembeddingsinspaceswithmeasuredwalls(basicallycontainedin[21])andaningeniousconstructionwithsuchspaces.See[10].Remark.Asisclearfromtheconstructionofthewreathproduct,apropertyPsatisfyingourprim-itivepermanenceresultswillbeclosedunderformationofwreathproducts.Indeed,itfollowsfromberingpermanencethatnitesumsofHsatisfyP,fromlimitpermanencethatH(G)satis-esP,andfromanotherapplicationofberingpermanence(anextension)thatHoGsatisesP.Thus,propertyEXisclosedunderformationofwreathproductsand,inkeepingwiththeabove,ifH2CEandG2EXthenHoG2CE.Thus,thecontentinthetheoremisintheweakenedhypothesisonG.WeconcludeourdiscussionofCEwithremarksconcerninglimits,coarsespaces,andacaveat.AcountablediscretegroupsatisesCEpreciselywhenitsnitelygeneratedsubgroupsdo.ThisfollowsfromTheorems8.1and7.2.1;seealso[11]fortheoriginalproof.Generalizedlimitperma-nenceresultsvalidinthesettingofcoarsespaceswouldbeapplicabletonon-necessarilycountablegroups.InthecaseofFADdageneralizedformoflimitpermanencewasobtainedbyDranish-nikovandSmith[13,4]:G2FADdpreciselywheneachofitsnitelygeneratedsubgroupssatisesFADd.But,thegeneralizationfailsforpropertyCE.AnexampleisG=R.EquipGwiththecanonicalcoarsestructuredescribedearlier.Asthiscoarsestructureisnotmetrizable,GcannotsatisfyCE.Nevertheless,thenitelygeneratedgeneratedsubgroupofGare(uptoisomorphism)preciselythegroupsZn,forsomenandeachofthesesatisesCE.Finally,wecompleteourcollectionofmiscellaneousremarksbymentioningworkofOzawa,whichprovidesanalternateapproachtothepermanenceofPropertyAwithrespectto(amalga-mated)freeproducts,andrelativehyperbolicity[20]. PERMANENCEINCOARSEGEOMETRY23[20]NarutakaOzawa.Boundaryamenabilityofrelativelyhyperbolicgroups.TopologyAppl.,153(14):2624–2630,2006.[21]GuyanRobertsonandTimSteger.Negativedenitekernelsandadynamicalcharacterizationofproperty(T)forcountablegroups.ErgodicTheoryDynam.Systems,18(1):247–253,1998.[22]J.Roe.LecturesonCoarseGeometry,volume31ofUniversityLectureSeries.AmericanMathematicalSociety,Providence,RI,2003.[23]J.P.Serre.Trees.Springer,NewYork,1980.TranslationfromFrenchof“Arbres,Amalgames,SL2”,Ast´erisqueno.46.[24]RufusWillett.SomenotesonpropertyA.InLimitsofgraphsingrouptheoryandcomputerscience,pages191–281.EPFLPress,Lausanne,2009.[25]G.Yu.TheCoarseBaum-ConnesconjectureforspaceswhichadmitauniformembeddingintoHilbertspace.InventionesMath.,139:201–240,2000.UNIVERSITYOFHAWAI`IATM¯ANOA,DEPARTMENTOFMATHEMATICS,2565MCCARTHYMALL,HON-OLULU,HI96822E-mailaddress:erik@math.hawaii.edu