Coxeter covers of the symmetric groups

arXiv:math/0405185v1[math.GR]11May2004COXETERCOVERSOFTHESYMMETRICGROUPSLOUISROWEN,MINATEICHERANDUZIVISHNEAbstract.WestudyCoxetergroupsfromwhichthereisanat-uralmapontoasymmetricgroup.Suchgroupshavenaturalquo-tientgroupsrelatedtopresentationsofthesymmetricgrouponanarbitrarysetToftranspositions.Thesequotients,denotedherebyCY(T),areaspecialtypeofthegeneralizedCoxetergroupsde-finedin[4],andalsoariseinthecomputationofcertaininvariantsofsurfaces.WeuseasurprisingactionofSnonthekernelofthesurjectionCY(T)→Sntoshowthatthiskernelembedsinthedirectproductofncopiesofthefreegroupπ1(T)(withtheexceptionofTbeingthefullsetoftranspositionsinS4).Asaresult,weshowthatthegroupsCY(T)areeithervirtuallyAbelianorcontainanon-Abelianfreesubgroup.1.IntroductionThesymmetricgrouponnlettersisgeneratedbythetranspositionssi=(ii+1),i=1,...,n−1.Thesegeneratorssatisfythewellknownrelationss2i=1,sisj=sjsi(|i−j|≥2)andsisi+1si=si+1sisi+1.Moreover,theabstractgroupdefinedbythesiwiththegivenrelationsisaCoxetergroup,isomorphictoSn.Thisset{si}canbepresentedbyagraphonthevertices1,...,n,wheresiistheedgeconnectingiandi+1.Moregenerally,onecanuseanyconnectedgraphTonnverticestodefineaCoxetergroupC(T),fromwhichthereisanaturalprojectionontothecorrespondingsymmetricgroup.Thekernelofthisprojectionisgeneratedbyelementscomingfromtwofamilies;onecorrespondingtotriplesofverticesinTwhichmeetinacommonvertex,andonetothecyclesofT.LetCY(T)denotethequotientofC(T)obtainedbyassumingthefirstfamilyofrelationstohold.OnemotivationtostudythegroupsCY(T)comesfromalgebraicgeometry,wherethesegroupsareakeyingredientinstudyingcertainDate:15Nov.2003.ThethirdnamedauthorwaspartiallysupportedbytheFulbrightVisitingScholarProgram,UnitedStatesDepartmentofState.12LOUISROWEN,MINATEICHERANDUZIVISHNEinvariantsofsurfaces(see[10]foradiscussiononthecomputationofthoseinvariants).Fromanotherdirection,signedgraphsareusedin[4]todefinegeneralizedCoxetergroups,whicharequotientsofordi-naryCoxetergroups.OurgroupsCY(T)belongtothisclass.ThegeneralizedCoxetergroupswhichourresultsenableustocomputearediscussedinSubsection7.4.TheseincludethegroupD2whosecompu-tationoccupiesalargeportionof[5],andacertainfamilyofTsaranovgroups.Forexample,theTsaranovgroupofahexagon(whichat-tractedmuchattention,see[4,Example8.6]),isidentifiedinCorollary7.16.ThereareotherindicationsthatCY(T)isanaturalquotientoftheCoxetergroupC(T).Forexample,theirparabolicsubgroupsarewellbehaved:ifT′⊆Tisasubgraph,thenthesubgroupofCY(T)gener-atedbytheelementsofT′isisomorphictotheabstractgroupdefinedonT′.ThisisshowninSubsection7.2.Weprovethat(withtheexceptionofTequalsK4,thecompletegraphon4vertices),CY(T)iscontainedinthesemidirectproductSn⋉π1(T)n,thussolvingthewordproblemforthesegroups.OntheotherhandCY(T)containscopiesofπ1(T),showingthatitisvirtuallysolvable(thatis,hasasolvablesubgroupoffiniteindex)iffThasatmostonecycle.Moreover,ifThasonecyclethenCY(T)isvirtuallyAbelian.ThissupportsTeicher’sconjecturethattheinvariantsmen-tionedaboveareeithervirtuallysolvable,orcontainafreesubgroup[9].Recently,MargulisandVinberg[7,Cor.2]provedthatinfinitenon-affineCoxetergroupsarelarge(i.e.virtuallyhaveafreequotient).InparticularthegroupC(T)islargeforeverygraphTotherthanaline,aY-shapedgraph(onfourvertices),oracycle.OurresultsprovidemoreinformationonC(T),provingthatalreadyCY(T)islargeifThasatleasttwocycles,whilethekernelofC(T)→CY(T)islargeotherwise.InSection2wegivethebasicdefinitionsandproperties,andbrieflydescribeanapplicationforourresultstoalgebraicgeometry.SpanningsubtreesofTareanimportanttoolthroughout,andinSection3weprovethatthesubgroupgeneratedbyaspanningsubtreeisthesym-metricgroup.WethendescribeanactionofSnonthekerneloftheprojectionCY(T)→Sn,whichusestwodifferentembeddingsofSntoCY(T).InSection6weprovethemainresult,thatthiskernelisiso-morphictoagivenabstractgroup,givenbygeneratorsandrelations.ThisgroupisstudiedinSection5,whereweshowitembedsinadirectproductoffreegroups.TheapplicationstoTheorem6.1aregiveninSection7:inCorollary7.1wegivethecriterionforCY(T)tobevirtuallysolvable.AnotherCOXETERCOVERSOFTHESYMMETRICGROUPS3immediateresultisthatCY(T)dependsonlyonthenumberofverticesandcyclesofT.InSubsection7.3wediscusstheCoxetergraphofC(T)andsomespecialcases.2.PresentationsofSnontranspositionsLetTbeagraphonnvertices.Considerthegroupgeneratedbythetranspositions(ab)∈Snfortheedges(a,b)inT;obviouslythisisthefullsymmetricgroupSniffTisconnected.Throughoutthepaper,allourgraphsaresimple(i.e.norepeatededgesorloops).RecallthataCoxetergroupisagroupwithgeneratorss1,...,sk,anddefiningrelationss2i=1and(sisj)mij=1,wheremij∈{2,3,...,∞}.ThefiniteCoxetergroupsarecompletelyclassified(see[3]),andtheyarethefinite(real)reflectiongroups.WeusethegraphTtodefineaCoxetergroupC(T),asfollows.Definition2.1.ThegroupC(T)isgeneratedbytheedgesu∈T,subjecttothefollowingrelations:(1)u2=1forallu∈T,(2)uv=vuifu,varedisjoint,and(3)uvu=vuvifu,vintersect.Notethatthelastrelationisequivalentto(uv)3=1,soC=C(T)isindeedaCoxetergroup.Definition2.2.LetTbeagraph.Themapφ:C(T)→Snisdefinedbysendingu=(a,b)tothetransposition(ab).Thismapiseasilyseentobewelldefined.