Automorphism groups of free groups surface groups and free abelian groups Martin R - PDF document

Automorphismgroupsoffreegroups,surfacegroupsandfreeabeliangroupsMartinR.BridsonandKarenVogtmannThegroupof22matriceswithintegerentriesanddeterminant1canbeidentiedeitherwiththegroupofouterautomorphismsofaranktwofreegrouporwiththegroupofisotopyclassesofhomeomorphismsofa2-dimensionaltorus.Thusthisgroupisthebeginningofthreenaturalsequencesofgroups,namelythegenerallineargroupsGL(n;Z),thegroupsOut(Fn)ofouterautomorphismsoffreegroupsofrankn2,andthemap-pingclassgroupsMod(Sg)oforientablesurfacesofgenusg1.Muchoftheworkonmappingclassgroupsandautomorphismsoffreegroupsismotivatedbytheideathatthesesequencesofgroupsarestronglyanalogous,andshouldhavemanypropertiesincommon.Thisprogramisoccasionallyderailedbyuncooperativefactsbuthasingeneralprovedtobeasuccess-fulstrategy,leadingtofundamentaldiscoveriesaboutthestructureofthesegroups.Inthisarticlewewillhighlightafewofthemoststrikingsimilar-itiesanddierencesbetweentheseseriesofgroupsandpresentsomeopenproblemsmotivatedbythisphilosophy.SimilaritiesamongthegroupsOut(Fn),GL(n;Z)andMod(Sg)beginwiththefactthatthesearetheouterautomorphismgroupsofthemostprim-itivetypesoftorsion-freediscretegroups,namelyfreegroups,freeabeliangroupsandthefundamentalgroupsofclosedorientablesurfaces1Sg.InthecaseofOut(Fn)andGL(n;Z)thisisobvious,inthecaseofMod(Sg)itisaclassicaltheoremofNielsen.Inallcasesthereisadeterminanthomomor-phismtoZ=2;thekernelofthismapisthegroupof\orientation-preserving"or\special"automorphisms,andisdenotedSOut(Fn);SL(n;Z)orMod(Sg)respectively.1GeometricandtopologicalmodelsAnaturalgeometriccontextforstudyingtheglobalstructureofGL(n;Z)isprovidedbythesymmetricspaceXofpositive-denite,realsymmetric1 matricesofdeterminant1(see[78]foraniceintroductiontothissubject).Thisisanon-positivelycurvedmanifolddieomorphictoRd,whered=1 2n(n+1)1.GL(n;Z)actsproperlybyisometriesonXwithaquotientofnitevolume.EachA2XdenesaninnerproductonRnandhenceaRiemannianmetricofconstantcurvatureandvolume1onthen-torusTn=Rn=Zn.OnecanrecoverAfromthemetricandanorderedbasisfor1Tn.ThusXishomeomorphictothespaceofequivalenceclassesofmarkedEuclideantori(Tn;)ofvolume1,whereamarkingisahomotopyclassofhomeomor-phisms:Tn!(Tn;)andtwomarkedtoriareconsideredequivalentifthereisanisometryi:(Tn1;1)!(Tn2;2)suchthat12i1ishomotopictotheidentity.ThenaturalactionofGL(n;Z)=Out(Zn)onTn=K(Zn;1)twiststhemarkingsontori,andwhenonetracesthroughtheidenticationsthisisthestandardactiononX.IfonereplacesTnbySgandfollowsexactlythisformalismwithmarkedmetricsofconstantcurvature1andxedvolume,thenonearrivesatthede-nitionofTeichmullerspaceandthenaturalactionofMod(Sg)=Out(1Sg)onit.TeichmullerspaceisagainhomeomorphictoaEuclideanspace,thistimeR6g6.InthecaseofOut(Fn)thereisnocanonicalchoiceofclassifyingspaceK(Fn;1)butratheranitecollectionofnaturalmodels,namelythenitegraphsofgenusnwithnoverticesofvalencelessthan3.Nevertheless,onecanproceedinessentiallythesameway:oneconsidersmetricsofxedvol-ume(sumofthelengthsofedges=1)onthevariousmodelsforK(Fn;1),eachequippedwithamarking,andonemakestheobviousidenticationsasthehomeomorphismtypeofagraphchangeswithasequenceofmetricsthatshrinkanedgetolengthzero.ThespaceofmarkedmetricstructuresobtainedinthiscaseisCullerandVogtmann'sOuterspace[27],whichisstratiedbymanifoldsubspacescorrespondingtothedierenthomeomor-phismtypesofgraphsthatarise.Thisspaceisnotamanifold,butitiscontractibleanditslocalhomotopicalstructureisanaturalgeneralizationofthatforamanifold(cf.[80]).OnecanalsolearnagreatdealaboutthegroupGL(n;Z)byexaminingitsactionsontheBorel-SerrebordicationofthesymmetricspaceXandonthesphericalTitsbuilding,whichencodestheasymptoticgeometryofX.TeichmullerspaceandOuterspacebothadmitusefulbordicationsthatarecloselyanalogoustotheBorel-Serrebordication[44,53,2].AndinplaceofthesphericalTitsbuildingforGL(n;Z)onehasthecomplexofcurves 1ifg2thenthecurvaturewillbenegative2 [46]forMod(Sg),whichhasplayedanimportantroleinrecentadvancesconcerningthelargescalegeometryofMod(Sg).Forthemomentthiscomplexhasnowell-establishedcounterpartinthecontextofOut(Fn).Thesecloselyparalleldescriptionsofgeometriesforthethreefamiliesofgroupshaveledmathematicianstotrytopushtheanalogiesfurther,bothforthegeometryandtopologyofthe\symmetricspaces"andforpurelygroup-theoreticpropertiesthataremostnaturallyprovedusingthegeometryofthesymmetricspace.Forexample,thesymmetricspaceforGL(n;Z)admitsanaturalequivariantdeformationretractionontoann(n1)=2-dimensionalcocompactsubspace,thewell-roundedretract[1].Similarly,bothOuterspaceandtheTeichmullerspaceofapuncturedorboundedorientablesur-faceretractequivariantlyontococompactsimplicialspines[27,44].Inallthesecases,theretractshavedimensionequaltothevirtualcohomologicaldimensionoftherelevantgroup.Forclosedsurfaces,however,thequestionremainsopen:Question1DoestheTeichmullerspaceforSgadmitanequivariantdefor-mationretractionontoacocompactspinewhosedimensionisequalto4g5,thevirtualcohomologicaldimensionofMod(Sg)?Furtherquestionsofasimilarnaturearediscussedin(2.1).TheissuesinvolvedinusingthesesymmetricspaceanalogstoprovepurelygrouptheoreticpropertiesareillustratedintheproofoftheTitsalternative,whichholdsforallthreeclassesofgroups.AgroupissaidtosatisfytheTitsalternativeifeachofitssubgroupseithercontainsanon-abelianfreegrouporelseisvirtuallysolvable.Thestrategyforprovingthisissimilarineachofthethreefamiliesthatweareconsidering:inspiredbyTits'soriginalproofforlineargroups(suchasGL(n;Z)),oneattemptstouseaping-pongargumentonasuitableboundaryatinnityofthesymmetricspace.Thisstrategyultimatelysucceedsbutthedetailsvaryenormouslybetweenthethreecontexts,andinthecaseofOut(Fn)theyareparticularlyintricate([4,3]versus[9]).Onendsthatthisisoftenthecase:analogiesbetweenthethreeclassesofgroupscanbecarriedthroughtotheorems,andthearchitectureoftheexpectedproofisoftenagoodguide,butatamoredetailedlevelthetechniquesrequiredvaryinessentialwaysfromoneclasstothenextandcanbeofcompletelydierentordersofdiculty.Letusreturntoproblemsmoredirectlyphrasedintermsofthegeometryofthesymmetricspaces.ThesymmetricspaceforGL(n;Z)hasaleft-invariantmetricofnon-positivecurvature,thegeometryofwhichisrelevanttomanyareasofmathematicsbeyondgeometricgrouptheory.Teichmuller3 spacehastwonaturalmetrics,theTeichmullermetricandtheWeyl-Petersenmetric,andagainthestudyofeachisarichsubject.Incontrast,themetrictheoryofOuterspacehasnotbeendeveloped,andinfactthereisnoobviouscandidateforanaturalmetric.Thus,thefollowingquestionhasbeenleftdeliberatelyvague:Question2DevelopametrictheoryofOuterspace.TheelementsofinniteorderinGL(n;Z)thatarediagonalizableoverCactasloxodromicisometriesofX.Whenn=2,theseelementsarethehyperbolicmatrices;eachxestwopointsatinnityinX=H2,oneasourceandoneasink.TheanalogoustypeofelementinMod(Sg)isapseudo-Anosov,andinOut(Fn)itisaniwip(irreduciblewithirreduciblepowers).Inbothcases,suchelementshavetwoxedpointsatinnity(i.e.inthenaturalboundaryofthesymmetricspaceanalog),andtheactionofthecyclicsubgroupgeneratedbytheelementexhibitsthenorth-southdynamicsfamiliarfromtheactionofhyperbolicmatricesontheclosureofthePoincaredisc[62],[54].InthecaseofMod(Sg)thiscyclicsubgroupleavesinvariantauniquegeodesiclineinTeichmullerspace,i.e.pseudo-Anosov'sareaxiallikethesemi-simpleelementsofinniteorderinGL(n;Z).InitialworkofHandelandMosher[43]showsthatinthecaseofiwipsonecannothopetohaveauniqueaxisinthesamemetricsense,butleavesopenthepossibilitythattheremaybeareasonablenotionofaxisinaweakersense.(Wehighlightedthisprobleminanearlierversionofthecurrentarticle.)Inamorerecentpreprint[42]theyhaveaddressedthislastpointdirectly,deninganaxisbundleassociatedtoanyiwip,cf.[63].Nevertheless,manyinterestingquestionsremain(someofwhicharehighlightedbyHandelandMosher).Thusweretainamodiedversionofouroriginalquestion:Question3Describethegeometryoftheaxisbundle(andassociatedob-jects)foraniwipactingonOuterSpace.2ActionsofAut(Fn)andOut(Fn)onotherspacesSomeofthequestionsthatweshallpresentaremorenaturallystatedintermsofAut(Fn)ratherthanOut(Fn),whilesomearenaturalforboth.Toavoidredundancy,weshallstateonlyoneformofeachquestion.4 2.1Baum-ConnesandNovikovconjecturesTwofamousconjecturesrelatingtopology,geometryandfunctionalanalysisaretheNovikovandBaum-Connesconjectures.TheNovikovconjectureforclosedorientedmanifoldswithfundamentalgroupsaysthatcertainhighersignaturescomingfromH(;Q)arehomotopyinvariants.ItisimpliedbytheBaum-Connesconjecture,whichsaysthatacertainassemblymapbetweentwoK-theoreticobjectsassociatedtoisanisomorphism.Kasparov[57]provedtheNovikovconjectureforGL(n;Z),andGuenther,HigsonandWeinbergerproveditforalllineargroups[40].TheBaum-ConnesconjectureforGL(n;Z)isopenwhenn4(cf.[61]).RecentlyStorm[79]pointedoutthattheNovikovconjectureformappingclassgroupsfollowsfromresultsthathavebeenannouncedbyHamenstadt[41]andKato[59],leavingopenthefollowing:Question4DomappingclassgroupsorOut(Fn)satisfytheBaum-Connesconjecture?DoesOut(Fn)satisfytheNovikovconjecture?AnapproachtoprovingtheseconjecturesisgivenbyworkofRosenthal[75],generalizingresultsofCarlssonandPedersen[23].AcontractiblespaceonwhichagroupactsproperlyandforwhichthexedpointsetsofnitesubgroupsarecontractibleiscalledanE .Rosenthal'stheoremsaysthattheBaum-ConnesmapforissplitinjectiveifthereisacocompactE =EthatadmitsacompacticationX,suchthat1.the-actionextendstoX;2.Xismetrizable;3.XGiscontractibleforeverynitesubgroupGof4.EGisdenseinXGforeverynitesubgroupGof5.compactsubsetsofEbecomesmallnearY=XrEunderthe-action:foreverycompactKEandeveryneighborhoodUXofy2Y,thereexistsaneighborhoodVXofysuchthat\rK\V=;implies\rKU.TheexistenceofsuchaspaceEalsoimpliestheNovikovconjecturefor.ForOut(Fn)thespineofOuterspacementionedintheprevioussectionisareasonablecandidatefortherequiredE ,andthereisasimilarlydenedcandidateforAut(Fn).Formappingclassgroupsofpuncturedsurfacesthecomplexofarcsystemswhichllupthesurfaceisagoodcandidate(note5 thatthiscanbeidentiedwithasubcomplexofOuterspace,asin[47],section5).Question5DoesthereexistacompacticationofthespineofOuterspacesatisfyingRosenthal'sconditions?Samequestionforthecomplexofarcsystemsllingapuncturedsurface.Inallofthecasesmentionedabove,thecandidatespaceEhasdimensionequaltothevirtualcohomologicaldimensionofthegroup.G.Mislin[68]hasconstructedacocompactE Gforthemappingclassgroupofaclosedsurface,butithasmuchhigherdimension,equaltothedimensionoftheTeichmullerspace.ThisleadsustoaslightvariationonQuestion1.Question6CanoneconstructacocompactE Gwithdimensionequaltothevirtualcohomologicaldimensionofthemappingclassgroupofaclosedsurface?2.2Properties(T)andFAAgrouphasKazdhan'sProperty(T)ifanyactionofthegroupbyisometriesonaHilbertspacehasxedvectors.KazdhanprovedthatGL(n;Z)hasproperty(T)forn3.Question7Forn�3,doesAut(Fn)haveproperty(T)?Thecorrespondingquestionformappingclassgroupsisalsoopen.IfAut(Fn)weretohaveProperty(T),thenanargumentofLubotzkyandPak[64]wouldprovideaconceptualexplanationoftheapparently-unreasonableeectivenessofcertainalgorithmsincomputerscience,specicallytheProd-uctReplacementAlgorithmofLeedham-Greenetal.IfagrouphasProperty(T)thenithasSerre'spropertyFA:everyactionofthegrouponanR-treehasaxedpoint.Whenn3,GL(n;Z)haspropertyFA,asdoAut(Fn)andOut(Fn),andmappingclassgroupsingenus3(see[28]).Incontrast,McCool[67]hasshownthatAut(F3)hasasubgroupofnite-indexwithpositiverstbettinumber,i.e.asubgroupwhichmapsontoZ.InparticularthissubgroupactsbytranslationsonthelineandthereforedoesnothavepropertyFAor(T).Sinceproperty(T)passestonite-indexsubgroups,itfollowsthatAut(F3)doesnothaveproperty(T).Question8Forn�3,doesAut(Fn)haveasubgroupofniteindexwithpositiverstbettinumber?6 Anothernite-indexsubgroupofAut(F3)mappingontoZwascon-structedbyAlexLubotzky,andwasexplainedtousbyAndrewCasson.Re-gardF3asthefundamentalgroupofagraphRwithonevertex.Thesingle-edgeloopsprovideabasisfa;b;cgforF3.Considerthe2-sheetedcovering^R!Rwithfundamentalgroupha;b;c2;cac1;cbc1iandletGAut(F3)bethestabilizerofthissubgroup.GactsonH1(^R;Q)leavinginvarianttheeigenspacesoftheinvolutionthatgeneratestheGaloisgroupofthecover-ing.Theeigenspacecorrespondingtotheeigenvalue1istwodimensionalwithbasisfacac1;bcbc1g.TheactionofGwithrespecttothisbasisgivesanepimorphismG!GL(2;Z).SinceGL(2;Z)hasafreesubgroupofnite-index,weobtainasubgroupofniteindexinAut(F3)thatmapsontoanon-abelianfreegroup.Onecanimitatetheessentialfeaturesofthisconstructionwithvariousothernite-indexsubgroupsofFn,thusproducingsubgroupsofniteindexinAut(Fn)thatmapontoGL(m;Z).Ineachcaseonendsthatmn1.Question9IfthereisahomomorphismfromasubgroupofniteindexinAut(Fn)ontoasubgroupofniteindexinGL(m;Z),thenmustmn1?Indeedonemightask:Question10Ifmn1andHAut(Fn)isasubgroupofniteindex,thendoeseveryhomomorphismH!GL(m;Z)haveniteimage?Similarquestionsareinterestingfortheothergroupsinourfamilies(cf.section3).Forexample,ifmn1andHAut(Fn)isasubgroupofniteindex,thendoeseveryhomomorphismH!Aut(Fm)haveniteimage?ApositiveanswertothefollowingquestionwouldanswerQuestion8;anegativeanswerwouldshowthatAut(Fn)doesnothaveproperty(T).Question11Forn4,dosubgroupsofniteindexinAut(Fn)havePropertyFA?Apromisingapproachtothislastquestionbreaksdownbecausewedonotknowtheanswertothefollowingquestion.Question12FixabasisforFnandletAn1Aut(Fn)bethecopyofAut(Fn1)correspondingtotherstn1basiselements.Let:Aut(Fn)!Gbeahomomorphismofgroups.If(An1)isnite,musttheimageofbenite?7 NotethattheobviousanalogofthisquestionforGL(n;Z)hasapositiveanswerandplaysaroleinthefoundationsofalgebraicK-theory.AdierentapproachtoestablishingProperty(T)wasdevelopedbyZuk[85].HeestablishedacombinatorialcriteriononthelinksofverticesinasimplyconnectedG-complexwhich,ifsatised,impliesthatGhasprop-erty(T):onemustshowthatthesmallestpositiveeigenvalueofthediscreteLaplacianonlinksissucientlylarge.Onemighthopetoapplythiscri-teriontooneofthenaturalcomplexesonwhichAut(Fn)andOut(Fn)act,suchasthespineofOuterspace.ButDavidFisherhaspointedouttousthattheresultsofIzekiandNatayani[55](alternatively,SchoenandWang{unpublished)implythatsuchastrategycannotsucceed.2.3ActionsonCAT(0)spacesAnR-treemaybedenedasacompleteCAT(0)spaceofdimension21.ThusonemightgeneralizepropertyFAbyasking,foreachd2N,whichgroupsmustxapointwhenevertheyactbyisometriesonacompleteCAT(0)spaceofdimensiond.Question13WhatistheleastintegersuchthatOut(Fn)actswithoutaglobalxedpointonacompleteCAT(0)spaceofdimension?AndwhatistheleastdimensionforthemappingclassgroupMod(Sg)?TheactionofOut(Fn)onthersthomologyofFndenesamapfromOut(Fn)toGL(n;Z)andhenceanactionofOut(Fn)onthesymmetricspaceofdimension1 2n(n+1)1.Thisactiondoesnothaveaglobalxedpointandhenceweobtainanupperboundon.Ontheotherhand,sinceOut(Fn)haspropertyFA,2.Infact,motivatedbyworkofFarbonGL(n;Z),Bridson[14]hasshownthatusingaHelly-typetheoremandthestructureofnitesubgroupsinOut(Fn),onecanobtainalowerboundonthatgrowsasalinearfunctionofn.Notethatalowerboundof3n3onwouldimplythatOuterSpacedidnotsupportacompleteOut(Fn)-equivariantmetricofnon-positivecurvature.IfXisaCAT(0)polyhedralcomplexwithonlynitelymanyisometrytypesofcells(e.g.anitedimensionalcubecomplex),theneachisometryofXiseitherelliptic(xesapoint)orhyperbolic(hasanaxisoftranslation)[15].Ifn4thenavariationonanargumentofGersten[36]showsthatinanyactionofOut(Fn)onX,noNielsengeneratorcanactasahyperbolicisometry. 2topologicalcoveringdimension8 Question14Ifn4,thencanOut(Fn)actwithoutaglobalxedpointonanite-dimensionalCAT(0)cubecomplex?2.4LinearityFormanekandProcesi[33]provedthatAut(Fn)isnotlinearforn3byshowingthatAut(F3)containsa\poisonsubgroup",i.e.asubgroupwhichhasnofaithfullinearrepresentation.SinceAut(Fn)embedsinOut(Fn+1),thissettlesthequestionoflinearityforOut(Fn)aswell,exceptwhenn=3.Question15DoesOut(F3)haveafaithfulrepresentationintoGL(m;C)forsomem2N?Notethatbraidgroupsarelinear[8]butitisunknownifmappingclassgroupsofclosedsurfacesare.BrendleandHamidi-Tehrani[13]showedthattheapproachofFormanekandProcesicannotbeadapteddirectlytothemappingclassgroups.Moreprecisely,theyprovethatthetypeof\poisonsubgroup"describedabovedoesnotariseinmappingclassgroups.ThefactthattheabovequestionremainsopenisanindicationthatOut(F3)canbehavedierentlyfromOut(Fn)fornlarge;theexistenceofniteindexsubgroupsmappingontoZwasanotherinstanceofthis,andweshallseeanotherinourdiscussionofautomaticstructuresandisoperimetricinequalities.3MapstoandfromOut(Fn)AparticularlyintriguingaspectoftheanalogybetweenGL(n;Z)andthetwootherclassesofgroupsistheextenttowhichthecelebratedrigidityphenomenaforlatticesinhigherranksemisimplegroupstransfertomappingclassgroupsandOut(Fn).Manyofthequestionsinthissectionconcernaspectsofthisrigidity;questions9to11shouldalsobeviewedinthislight.BridsonandVogtmann[21]showedthatanyhomomorphismfromAut(Fn)toagroupGhasniteimageifGdoesnotcontainthesymmetricgroupn+1;inparticular,anyhomomorphismAut(Fn)!Aut(Fn1)hasimageoforderatmost2.Question16Ifn4andg1,doeseveryhomomorphismfromAut(Fn)toMod(Sg)haveniteimage?9 By[21],onecannotobtainhomomorphismswithinniteimageunlessMod(Sg)containsthesymmetricgroupn+1.Forlargeenoughgenus,youcanrealizeanysymmetricgroup;buttheorderofanitegroupofsymmetriesisatmost84g-6,sohereoneneeds84g6(n+1)!.TherearenoinjectivemapsfromAut(Fn)tomappingclassgroups.ThisfollowsfromtheresultofBrendleandHamidi-Tehranithatwequotedear-lier.ForcertaingonecanconstructhomomorphismsAut(F3)!Mod(Sg)withinniteimage,butwedonotknowtheminimalsuchg.Question17LetbeanirreduciblelatticeinasemisimpleLiegroupofR-rankatleast2.DoeseveryhomomorphismfromtoOut(Fn)haveniteimage?Thisisknownfornon-uniformlattices(see[16];itfollowseasilyfromtheKazdhan-MargulisnitenesstheoremandthefactthatsolvablesubgroupsofOut(Fn)arevirtuallyabelian[5]).FarbandMasurprovidedapositiveanswertotheanalogousquestionformapstomappingclassgroups[32].TheproofoftheirtheoremwasbasedonresultsofKaimanovichandMasur[56]concerningrandomwalksonTeichmullerspace.(See[54]and,foranalternativeapproach,[6].)Question18IsthereatheoryofrandomwalksonOuterspacesimilartothatofKaimanovichandMasurforTeichmullerspace?PerhapsthemostpromisingapproachtoQuestion17isviaboundedco-homology,followingthetemplateofBestvinaandFujiwara'sworkonsub-groupsofthemappingclassgroup[6].Question19IfasubgroupGOut(Fn)isnotvirtuallyabelian,thenisH2b(G;R)innitedimensional?IfmnthenthereareobviousembeddingsGL(n;Z)!GL(m;Z)andAut(Fn)!Aut(Fm),buttherearenoobviousembeddingsOut(Fn)!Out(Fm).BogopolskiandPuga[10]haveshownthat,form=1+(n1)kn,wherekisanarbitrarynaturalnumbercoprimeton1,thereisinfactanembedding,byrestrictingautomorphismstoasuitablecharacteristicsubgroupofFm.Question20ForwhichvaluesofmdoesOut(Fn)embedinOut(Fm)?Whatistheminimalsuchm,andisittrueforallsucientlylargem?10 Ithasbeenshownthatwhennissucientlylargewithrespecttoi,thehomologygroupHi(Out(Fn);Z)isindependentofn[50,51].Question21IsthereamapOut(Fn)!Out(Fm)thatinducesanisomor-phismonhomologyinthestablerange?Anumberofthequestionsinthissectionand(2.2)askwhethercertainquotientsofOut(Fn)orAut(Fn)arenecessarilynite.Thefollowingquo-tientsarisenaturallyinthissetting:deneQ(n;m)tobethequotientofAut(Fn)bythenormalclosureofm,whereistheNielsenmovedenedonabasisfa1;:::;angbya17!a2a1.(AllsuchNielsenmovesareconjugateinAut(Fn),sothechoiceofbasisdoesnotalterthequotient.)TheimageofaNielsenmoveinGL(n;Z)isanelementarymatrixandthequotientofGL(n;Z)bythenormalsubgroupgeneratedbythem-thpowersoftheelementarymatricesisthenitegroupGL(n;Z=m).ButBridsonandVogtmann[21]showedthatifmissucientlylargethenQ(n;m)isinnitebecauseithasaquotientthatcontainsacopyofthefreeBurnsidegroupB(n1;m).SomefurtherinformationcanbegainedbyreplacingB(n1;m)withthequotientsofFnconsideredinsubsection39.3ofA.Yu.Ol'shanskii'sbook[73].ButweknowverylittleaboutthegroupsQ(n;m).Forexample:Question22ForwhichvaluesofnandmisQ(n;m)innite?IsQ(3;5)innite?Question23CanQ(n;m)haveinnitelymanynitequotients?Isitresiduallynite?4IndividualelementsandmappingtoriIndividualelements2GL(n;Z)canberealizedasdieomorphisms^ofthen-torus,whileindividualelements 2Mod(Sg)canberealizedasdieomorphisms^ ofthesurfaceSg.ThusonecanstudyviathegeometryofthetorusbundleoverS1withholonomy^andonecanstudy viathegeometryofthe3-manifoldthatbresoverS1withholonomy^ .(Ineachcasethemanifolddependsonlyontheconjugacyclassoftheelement.)ThesituationforAut(Fn)andOut(Fn)ismorecomplicated:thenaturalchoicesofclassifyingspaceY=K(Fn;1)arenitegraphsofgenusn,andnoelementofinniteorder2Out(Fn)isinducedbytheactionon1(Y)ofahomeomorphismofY.ThusthebestthatonecanhopeforinthissituationistoidentifyagraphYthatadmitsahomotopyequivalenceinducing11 andthathasadditionalstructurewell-adaptedto.OnewouldthenformthemappingtorusofthishomotopyequivalencetogetagoodclassifyingspaceforthealgebraicmappingtorusFnoZ.ThetraintracktechnologyofBestvina,FeighnandHandel[7,4,3]isamajorpieceofworkthatderivessuitablegraphsYwithadditionalstructureencodingkeypropertiesof.ThisresultsinadecompositiontheoryforelementsofOut(Fn)thatiscloselyanalogousto(butmorecomplicatedthan)theNielsen-Thurstontheoryforsurfaceautomorphisms.Manyoftheresultsmentionedinthissectionarepremisedonadetailedknowledgeofthistechnologyandoneexpectsthataresolutionofthequestionswillbetoo.ThereareseveralnaturalwaystodenethegrowthofanautomorphismofagroupGwithnitegeneratingsetA;inthecaseoffree,free-abelian,andsurfacegroupstheseareallasymptoticallyequivalent.Themosteasilydenedgrowthfunctionis\r(k)where\r(k):=maxfd(1;k(a)ja2Ag.IfG=Znthen\r(k)'kdforsomeintegerdn1,orelse\r(k)growsexponentially.IfGisasurfacegroup,theNielsen-Thurstontheoryshowsthatonlybounded,linearandexponentialgrowthcanoccur.IfG=Fnand2Aut(Fn)then,asintheabeliancase,\r(k)'kdforsomeintegerdn1orelse\r(k)growsexponentially.Question24Canonedetectthegrowthofasurfaceorfree-grouphomo-morphismbyitsactiononthehomologyofacharacteristicsubgroupofniteindex?Noticethatonehastopasstoasubgroupofniteindexinordertohaveanyhopebecauseautomorphismsofexponentialgrowthcanacttriviallyonhomology.A.Piggott[74]hasansweredtheabovequestionforfree-groupautomorphismsofpolynomialgrowth,andlinear-growthautomorphismsofsurfacesareeasilydealtwith,buttheexponentialcaseremainsopeninbothsettings.Finerquestionsconcerninggrowthareaddressedintheon-goingworkofHandelandMosher[43].Theyexplore,forexample,theimplicationsofthefollowingcontrastinbehaviourbetweensurfaceautomorphismsandfree-groupautomorphisms:inthesurfacecasetheexponentialgrowthrateofapseudo-Anosovautomorphismisthesameasthatofitsinverse,butthisisnotthecaseforiwipfree-groupautomorphisms.FormappingtoriofautomorphismsoffreeabeliangroupsG=ZnoZ,thefollowingconditionsareequivalent(see[17]):Gisautomatic;Gisa12 CAT(0)group3;Gsatisesaquadraticisoperimetricinequality.Inthecaseofmappingtoriofsurfaceautomorphisms,allmappingtorisatisfytherstandlastoftheseconditionsandoneunderstandsexactlywhichSgoZareCAT(0)groups.Brady,BridsonandReeves[12]showthatthereexistmappingtorioffree-groupautomorphismsFoZthatarenotautomatic,andGerstenshowedthatsomearenotCAT(0)groups[36].Ontheotherhand,manysuchgroupsdohavetheseproperties,andtheyallsatisfyaquadraticisoperimetricin-equality[18].Question25Classifythose2Aut(Fn)forwhichFnoZisautomaticandthoseforwhichitisCAT(0).Ofcentralimportanceintryingtounderstandmappingtoriis:Question26IsthereanalogrithmtodecideisomorphismamonggroupsoftheformFoZ.Inthepurestformofthisquestiononeisgiventhegroupsasnitepresentations,soonehastoaddressissuesofhowtondthedecompositionFoZandonehastocombatthefactthatthisdecompositionmaynotbeunique.Buttheheartofanysolutionshouldbeananswerto:Question27IstheconjugacyproblemsolvableinOut(Fn)?MartinLustigpostedadetailedoutlineofasolutiontothisproblemonhiswebpagesomeyearsago[65],butneitherthisproofnoranyotherhasbeenacceptedforpublication.Thisproblemisofcentralimportancetotheeldandaclear,compellingsolutionwouldbeofgreatinterest.TheconjugacyproblemformappingclassgroupswasshowntobesolvablebyHemion[52],andaneectivealgorithmfordeterminingconjugacy,atleastforpseudo-Anosovmappingclasses,wasgivenbyMosher[70].TheisomorphismproblemforgroupsoftheformSgoZcanbeviewedasaparticularcaseofthesolutiontotheisomorphismproblemforfundamentalgroupsofgeometrizable3-manifolds[76].ThesolvabilityoftheconjugacyproblemforGL(n;Z)isduetoGrunewald[39] 3thismeansthatGactsproperlyandcocompactlybyisometriesonaCAT(0)space13 5CohomologyIneachoftheseriesofgroupsfngweareconsidering,theithhomologyofnhasbeenshowntobeindependentofnfornsucientlylarge.ForGL(n;Z)thisisduetoCharney[24],formappingclassgroupstoHarer[45],forAut(Fn)andOut(Fn)toHatcherandVogtmann[48,50],thoughforOut(Fn)thisrequiresanerratumbyHatcher,VogtmannandWahl[51].Withtrivialrationalcoecients,thestablecohomologyofGL(n;Z)wascomputedinthe1970'sbyBorel[11],andthestablerationalcohomologyofthemappingclassgroupcomputedbyMadsenandWeissin2002[66].ThestablerationalcohomologyofAut(Fn)(andOut(Fn))wasveryrecentlydeterminedbyS.Galatius[34]tobetrivial.TheexactstablerangefortrivialrationalcoecientsisknownforGL(n;Z)andformappingclassgroupsofpuncturedsurfaces.ForAut(Fn)thebestknownresultisthattheithhomologyisindependentofnforn�5i=4[49],buttheexactrangeisunknown:Question28WherepreciselydoestherationalhomologyofAut(Fn)stabi-lize?AndforOut(Fn)?Thereareonlytwoknownnon-trivialclassesinthe(unstable)rationalhomologyofOut(Fn)[49,26].However,Morita[69]hasdenedaninniteseriesofcycles,usingworkofKontsevichwhichidentiesthehomologyofOut(Fn)withthecohomologyofacertaininnite-dimensionalLiealgebra.TherstofthesecyclesisthegeneratorofH4(Out(F4);Q)=Q,andConantandVogtmannshowedthatthesecondalsogivesanon-trivialclass,inH8(Out(F6);Q)[26].BothMoritaandConant-Vogtmannalsodenedmoregeneralcycles,parametrizedbyodd-valentgraphs.Question29AreMorita'soriginalcyclesnon-trivialinhomology?ArethegeneralizationsduetoMoritaandtoConantandVogtmannnon-trivialinhomology?NootherclasseshavebeenfoundtodateinthehomologyofOut(Fn),leadingnaturallytothequestionofwhetherthesegivealloftherationalhomology.Question30DotheMoritaclassesgeneratealloftherationalhomologyofOut(Fn)?ThemaximumdimensionofaMoritaclassisabout4n=3.Morita'scyclesliftnaturallytoAut(Fn),andagainthersttwoarenon-trivialinhomology.14 ByGalatius'result,allofthesecyclesmusteventuallydisappearunderthestabilizationmapAut(Fn)!Aut(Fn+1).ConantandVogtmannshowthatinfacttheydisappearimmediatelyaftertheyappear,i.e.oneapplicationofthestabilizationmapkillsthem[25].IfitistruethattheMoritaclassesgeneratealloftherationalhomologyofOut(Fn)thenthisimpliesthatthestablerangeissignicantlylowerthanthecurrentbound.WenotethatMoritahasidentiedseveralconjecturalrelationshipsbe-tweenhiscyclesandvariousotherinterestingobjects,includingtheimageoftheJohnsonhomomorphism,thegroupofhomologycobordismclassesofhomologycylinders,andthemotivicLiealgebraassociatedtothealgebraicmappingclassgroup(seeMorita'sarticleinthisvolume).SincethestablerationalhomologyofOut(Fn)istrivial,thenaturalmapsfrommappingclassgroupstoOut(Fn)andfromOut(Fn)toGL(n;Z)areofcoursezero.However,theunstablehomologyofallthreeclassesofgroupsremainslargelyunkownandintheunstablerangethesemapsmightwellbenontrivial.Inparticular,wenotethatH8(GL(6;Z);Q)=Q[30];thisleadsnaturallytothequestionQuestion31IstheimageofthesecondMoritaclassinH8(GL(6;Z);Q))non-trivial?ForfurtherdiscussionofthecohomologyofAut(Fn)andOut(Fn)wereferto[81].6GeneratorsandRelationsThegroupsweareconsideringareallnitelygenerated.Ineachcase,themostnaturalsetofgeneratorsconsistsofasingleorientation-reversinggen-eratorofordertwo,togetherwithacollectionofsimpleinnite-orderspecialautomorphisms.ForOut(Fn),thesespecialautomorphismsaretheNielsenautomorphisms,whichmultiplyonegeneratorofFnbyanotherandleavetherestofthegeneratorsxed;forGL(n;Z)thesearetheelementaryma-trices;andformappingclassgroupstheyareDehntwistsaroundasmallsetofnon-separatingsimpleclosedcurves.Thesegeneratingsetshaveanumberofimportantfeaturesincommon.First,implicitinthedescriptionofeachisachoiceofgeneratingsetforthegroupBonwhichisacting.InthecaseofMod(Sg)this\basis"canbetakentoconsistof2g+1simpleclosedcurvesrepresentingthestandardgeneratorsa1;b1;a2;b2;:::;ag;bg;of1(Sg)togetherwithz=a12b3a3b13.15 InthecaseofOut(Fn)andGL(n;Z),thegeneratingsetisabasisforFnandZnrespectively.Notethatinthecases=Out(Fn)orGL(n;Z),theuniversalpropertyoftheunderlyingfreeobjectsB=FnorZnensuresthatactstransitivelyonthesetofpreferredgeneratingsets(bases).InthecaseB=1Sg,thecorrespondingresultisthatanytwocollectionsofsimpleclosedcurveswiththesamepatternofintersectionnumbersandcomplementaryregionsarerelatedbyahomeomorphismofthesurface,hence(atthelevelof1)bytheactionof.IfweidentifyZnwiththeabelianizationofFnandchoosebasesaccord-ingly,thentheactionofOut(Fn)ontheabelianizationinducesahomo-morphismOut(Fn)!GL(n;Z)thatsendseachNielsenmovetothecorre-spondingelementarymatrix(andhenceissurjective).Correspondingly,theactionMod(Sg)ontheabelianizationof1SgyieldsahomomorphismontothesymplecticgroupSp(2g;Z)sendingthegeneratorsofMod(Sg)givenbyDehntwistsaroundtheaiandbitotransvections.Anothercommonfeatureofthesegeneratingsetsisthattheyallhavelineargrowth(seesection4).Smaller(butlesstransparent)generatingsetsexistineachcase.IndeedB.H.Neumann[72]provedthatAut(Fn)(henceitsquotientsOut(Fn)andGL(n;Z))isgeneratedbyjust2elementswhenn4.Wajnryb[83]provedthatthisisalsotrueofmappingclassgroups.Ineachcaseonecanalsondgeneratingsetsconsistingofniteorderelements,involutionsinfact.ZuccashowedthatAut(Fn)canbegeneratedby3involutionstwoofwhichcommute[84],andKassabov,buildingonworkofFarbandBrendle,showedthatmappingclassgroupsoflargeenoughgenuscanbegeneratedby4involutions[58].Ourgroupsarealsoallnitelypresented.ForGL(n;Z),ormorepre-ciselyforSL(n;Z),therearetheclassicalSteinbergrelations,whichinvolvecommutatorsoftheelementarymatrices.ForthespecialautomorphismsSAut(Fn),Gerstengaveapresentationintermsofcorrespondingcommu-tatorrelationsoftheNielsengenerators[35].Finitepresentationsofthemappingclassgroupsaremorecomplicated.TherstwasgivenbyHatcherandThurston,andworkedoutexplicitlybyWajnryb[82].Question32IsthereasetofsimpleSteinberg-typerelationsforthemap-pingclassgroup?ThereisalsoapresentationofAut(Fn)comingfromtheactionofAut(Fn)onthesubcomplexofAuterspacespannedbygraphsofdegreeatmost2.Thisissimply-connectedby[48],soBrown'smethod[22]canbeusedtowrite16 downapresentation.Thevertexgroupsarestabilizersofmarkedgraphs,andtheedgegroupsarethestabilizersofpairsconsistingofamarkedgraphandaforestinthegraph.ThequotientofthesubcomplexmoduloAut(Fn)canbecomputedexplicitly,andonendsthatAut(Fn)isgeneratedbythe(nite)stabilizersofsevenspecicmarkedgraphs.Inaddition,alloftherelationsexcepttwocomefromthenaturalinclusionsofedgestabilizersintovertexstabilizers,i.e.eitherincludingthestabilizerofapair(graph,forest)intothestabilizerofthegraph,orintothestabilizerofthequotientofthegraphmodulotheforest.Thusthewholegroupisalmost(butnotquite)apushoutofthesenitesubgroups.IntheterminologyofHae\riger(see[19],II.12),thecomplexofgroupsisnotsimple.Question33CanOut(Fn)andMod(Sg)beobtainedasapushoutofanitesubsystemoftheirnitesubgroups,i.e.iseitherthefundamentalgroupofadevelopablesimplecomplexofnitegroupsona1-connectedbase?6.1IAautomorphismsWeconcludewithawell-knownproblemaboutthekernelIA(n)ofthemapfromOut(Fn)toGL(n;Z).Thenotation\IA"standsforidentityontheabelianization;theseare(outer)automorphismsofFnwhicharetheidentityontheabelianizationZnofFn.Magnusshowedthatthiskernelisnitelygenerated,andforn=3KrsticandMcCoolshowedthatitisnotnitelypresentable[60].Itisalsoknownthatinsomedimensionthehomologyisnotnitelygenerated[77].Butthatistheextentofourknowledgeofbasicnitenessproperties.Question34EstablishnitenesspropertiesofthekernelIA(n)ofthemapfromOut(Fn)toGL(n;Z).Inparticular,determinewhetherIA(n)isnitelypresentableforn�3.ThesubgroupIA(n)isanalogoustotheTorellisubgroupofthemappingclassgroupofasurface,whichalsoremainsquitemysteriousinspiteofhavingbeenextensivelystudied.7AutomaticityandIsoperimetricInequalitiesInthefoundationaltextonautomaticgroups[31],EpsteingivesadetailedaccountofThurston'sproofthatifn3thenGL(n;Z)isnotautomatic.Theargumentusesthegeometryofthesymmetricspacetoobtainanex-ponentiallowerboundonthe(n1)-dimensionalisoperimetricfunctionof17 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