Quotient isomorphism invariants of a finitely gene

arXiv:math/0607428v1[math.GR]18Jul2006QuotientIsomorphismInvariantsofaFinitelyGeneratedCoxeterGroupMichaelMihalik,JohnRatcliffe,andStevenTschantzMathematicsDepartment,VanderbiltUniversity,NashvilleTN37240,USA1IntroductionTheisomorphismproblemforfinitelygeneratedCoxetergroupsistheprob-lemofdecidingiftwofiniteCoxetermatricesdefineisomorphicCoxetergroups.Coxeter[4]solvedthisproblemforfiniteirreducibleCoxetergroups.Recentlytherehasbeenconsiderableinterestandactivityontheisomor-phismproblemforarbitraryfinitelygeneratedCoxetergroups.InthispaperwedescribeafamilyofisomorphisminvariantsofafinitelygeneratedCoxetergroupW.EachoftheseinvariantsistheisomorphismtypeofaquotientgroupW/NofWbyacharacteristicsubgroupN.ThevirtueoftheseinvariantsisthatW/NisalsoaCoxetergroup.Forsomeoftheseinvariants,theisomorphismproblemofW/NissolvedandsoweobtainisomorphisminvariantsthatcanbeeffectivelyusedtodistinguishisomorphismtypesoffinitelygeneratedCoxetergroups.WeemphasizethateveniftheisomorphismproblemforfinitelygeneratedCoxetergroupsiseventuallysolved,severalofthealgorithmsdescribedinourpaperwillstillbeusefulbecausetheyarecomputationalfastandwouldmostlikelybeincorporatedintoanefficientcomputerprogramthatdeterminesiftwofiniterankCoxetersystemshaveisomorphicgroups.In§2,weestablishnotation.In§3,wedescribetwoelementaryquoti-entingoperationsonaCoxetersystemthatyieldsanotherCoxetersystem.In§4,wedescribethebinaryisomorphisminvariantofafinitelygeneratedCoxetergroup.In§5,wereviewsomematchingtheorems.In§6,wedescribetheevenisomorphisminvariantofafinitelygeneratedCoxetergroup.In§7,wedefinebasiccharacteristicsubgroupsofafinitelygeneratedCoxeter1group.In§8,wedescribethesphericalranktwoisomorphisminvariantofafinitelygeneratedCoxetergroup.In§9,wemakesomeconcludingremarks.2PreliminariesACoxetermatrixisasymmetricmatrixM=(m(s,t))s,t∈Swithm(s,t)eitherapositiveintegerorinfinityandm(s,t)=1ifandonlyifs=t.ACoxetersystemwithCoxetermatrixM=(m(s,t))s,t∈Sisapair(W,S)consistingofagroupWandasetofgeneratorsSforWsuchthatWhasthepresentationW=hS|(st)m(s,t):s,t∈Sandm(s,t)∞i.WecalltheabovepresentationofW,theCoxeterpresentationof(W,S).If(W,S)isaCoxetersystemwithCoxetermatrixM=(m(s,t))s,t∈S,thentheorderofstism(s,t)foreachs,tinSbyProp.4,p.92ofBourbaki[3],andsoaCoxetersystem(W,S)determinesitsCoxetermatrix;moreover,anyCoxetermatrixM=(m(s,t))s,t∈SdeterminesaCoxetersystem(W,S)whereWisdefinedbythecorrespondingCoxeterpresentation.If(W,S)isaCoxetersystem,thenWiscalledaCoxetergroupandSiscalledasetofCoxetergeneratorsforW,andthecardinalityofSiscalledtherankof(W,S).ACoxetersystem(W,S)hasfiniterankifandonlyifWisfinitelygeneratedbyTheorem2(iii),p.20ofBourbaki[3].Let(W,S)beaCoxetersystem.Avisualsubgroupof(W,S)isasubgroupofWoftheformhAiforsomeA⊂S.IfhAiisavisualsubgroupof(W,S),then(hAi,A)isalsoaCoxetersystembyTheorem2(i),p.20ofBourbaki[3].WhenstudyingaCoxetersystem(W,S)withCoxetermatrixMitishelpfultohaveavisualrepresentationof(W,S).Therearetwographicalwaysofrepresenting(W,S)andweshallusebothdependingonourneeds.TheCoxeterdiagram(C-diagram)of(W,S)isthelabeledundirectedgraphΔ=Δ(W,S)withverticesSandedges{(s,t):s,t∈Sandm(s,t)2}suchthatanedge(s,t)islabeledbym(s,t).CoxeterdiagramsareusefulforvisuallyrepresentingfiniteCoxetergroups.IfA⊂S,thenΔ(hAi,A)isthesubdiagramofΔ(W,S)inducedbyA.2ACoxetersystem(W,S)issaidtobeirreducibleifitsC-diagramΔisconnected.AvisualsubgrouphAiof(W,S)issaidtobeirreducibleif(hAi,A)isirreducible.AsubsetAofSissaidtobeirreducibleifhAiisirreducible.AsubsetAofSissaidtobeacomponentofSifAisamaximalirreduciblesubsetofSorequivalentlyifΔ(hAi,A)isaconnectedcomponentofΔ(W,S).TheconnectedcomponentsoftheΔ(W,S)representthefactorsofadirectproductdecompositionofW.Thepresentationdiagram(P-diagram)of(W,S)isthelabeledundirectedgraphΓ=Γ(W,S)withverticesSandedges{(s,t):s,t∈Sandm(s,t)∞}suchthatanedge(s,t)islabeledbym(s,t).PresentationdiagramsareusefulforvisuallyrepresentinginfiniteCoxetergroups.IfA⊂S,thenΓ(hAi,A)isthesubdiagramofΓ(W,S)inducedbyA.TheconnectedcomponentsofΓ(W,S)representthefactorsofafreeproductdecompositionofW.Forexample,considertheCoxetergroupWgeneratedbythefourreflec-tionsinthesidesofarectangleinE2.TheC-diagramof(W,S)isthedisjointunionoftwoedgeslabeledby∞whiletheP-diagramofWisasquarewithedgelabels2.Let(W,S)and(W′,S′)beCoxetersystemswithP-diagramsΓandΓ′,respectively.Anisomorphismφ:(W,S)→(W′,S′)ofCoxetersystemsisanisomorphismφ:W→W′suchthatφ(S)=S′.Anisomorphismψ:Γ→Γ′ofP-diagramsisabijectionfromStoS′thatpreservesedgesandtheirlabels.Notethat(W,S)∼=(W′,S′)ifandonlyifΓ∼=Γ′.WeshalluseCoxeter’snotationonp.297of[5]fortheirreduciblespher-icalCoxetersimplexreflectiongroupsexceptthatwedenotethedihedralgroupDk2byD2(k).SubscriptsdenotetherankofaCoxetersysteminCoxeter’snotation.Coxeter’snotationpartlyagreeswithbutdiffersfromBourbaki’snotationonp.193of[3].Coxeter[4]provedthateveryfiniteirreducibleCoxetersystemisisomor-phictoexactlyoneoftheCoxetersystemsAn,n≥1,Bn,n≥4,Cn,n≥2,D2(k),k≥5,E6,E7,E8,F4,G3,G4.Fornotationalconvenience,wedefineB3=A3,D2(3)=A2,andD2(4)=C2ThetypeofafiniteirreducibleCoxetersystem(W,S)istheisomorphismtypeof(W,