On the Complexity and Volume of Hyperbolic Manifolds

Description

ar X iv :0 81 1. 42 74 v1 [ ma th. GT ] 26 N ov 20 08 On the Complexity and Volume of Hyperbolic 3-Manifolds. Thomas Delzant and Leonid Potyagailo Abstract We compare the volume of a hyperbolic 3-manifold M of finite volume and the complexity of its fundamental group. 1 1 Introduction. Complexity of 3-manifolds and groups. One of the most striking corollaries of the recent solution of the geometrization conjecture for 3-manifolds is the fact that every aspherical 3- manifold is uniquely determined by its fundamental group. It seems to be natural to think that a topological/geometrical description of a 3-manifold M produces the simplest way to describe its fundamental group π1(M); on the other hand, the simplest way to define the group π1(M) gives rise to the most efficient way to describe M. More precisely, we want to compare the complexity of 3-manifolds and their fundamental groups. The study of the complexity of 3-manifolds goes back to the classical work of H. Kneser [K]. Recall that the Kneser complexity invariant k(M) is defined to be the minimal number of simplices of a triangulation of the manifold M . The main result of Kneser is that this complexity serves as a bound of the number of embedded incompressible 2-spheres in M , and bounds the numbers of factors in a decomposition ofM as a connected sum.

• torsion any


• group


• since every


• every finitely generated


• finite discrete


• all parabolic


• manifold


• kneser complexity


• hyperbolic

Sujets

• Cooper
• Institute
• Manifold

Informations

• Publié par: pefav
• Nombre de lectures: 20
• Langue: English

Extrait

OntheComplexityandVolumeofHyperbolic3-Manifolds. ThomasDelzantandLeonidPotyagailo Abstract Wecomparethevolumeofahyperbolic3-manifold M offinitevolumeandthecomplexity ofitsfundamentalgroup. 1 1Introduction. Complexityof3-manifoldsandgroups. Oneofthemoststrikingcorollariesoftherecent solutionofthegeometrizationconjecturefor3-manifoldsisthefactthateveryaspherical3- manifoldisuniquelydeterminedbyitsfundamentalgroup.Itseemstobenaturaltothinkthata topological/geometricaldescriptionofa3-manifoldMproducesthesimplestwaytodescribeits fundamentalgroupπ 1 (M);ontheotherhand,thesimplestwaytodefinethegroupπ 1 (M)gives risetothemostefficientwaytodescribeM.Moreprecisely,wewanttocomparethecomplexity of3-manifoldsandtheirfundamentalgroups. Thestudyofthecomplexityof3-manifoldsgoesbacktotheclassicalworkofH.Kneser [K].RecallthattheKnesercomplexityinvariantk(M)isdefinedtobetheminimalnumberof simplicesofatriangulationofthemanifoldM.ThemainresultofKneseristhatthiscomplexity servesasaboundofthenumberofembeddedincompressible2-spheresinM,andboundsthe numbersoffactorsinadecompositionofMasaconnectedsum.Aversionofthiscomplexitywas usedbyW.Hakentoprovetheexistenceofhierarchiesforalargeclassofcompact3-manifolds (calledsincethenHakenmanifolds).Anothermeasureofthecomplexityc(M)forthe3-manifold MisduetoS.Matveev.ItistheminimalnumberofverticesofaspecialspineofM[Ma].Itis shownthatinmanyimportantcases(e.g.ifMisanon-compacthyperbolic3-manifoldoffinite volume)onehask(M)=c(M)[Ma]. 1 2000MathematicsSubjectClassification. 20F55,51F15,57M07,20F65,57M50 Keywords:hyperbolicmanifolds,volume,invariant T . 1

Therank(minimalnumberofgenerators)isalsoameasureofcomplexityofafinitelygener- atedgroup.AccordingtotheclassicaltheoremofI.Grushko[Gr],therankofafreeproductof groupsisthesumoftheirranks.Thisimmediatelyimpliesthateveryfinitelygeneratedgroupis afreeproductoffinitelymanyfreelyindecomposiblefactors,whichisanalgebraicanalogueof Knesertheorem. ForafinitelypresentedgroupGameasureofcomplexityofGwasdefinedin[De].Hereis itsdefinition: Definition1.1. LetGbeafinitelypresentedgroup.WesaythatT(G)≤tifthereexistsa simply-connected2-dimensionalcomplexPsuchthatGactsfreelyandsimpliciallyonPandthe thenumberof2-facesofthequotientΠ=P/Gislessthant. IfthegroupGisdefinedbyapresentation<a 1 ,...a r ;R 1 ,...R n >thesumΣ( | R i |− 2)serves asanaturalboundforT(G). NotethataninequalitybetweenKnesercomplexityandthisinvariantisobvious.Indeed,by contractingamaximalsubtreeofthe2-dimensionalskeletonofatriangulationofMoneobtains atriangularpresentationofthegroupπ 1 (M).Sinceevery3-simplexhasfour2-facesitfollows T(π 1 (M)) ≤ 4k(M). Inordertocomparethecomplexityofamanifoldandthatofitsfundamentalgroup,itis enoughtofindafunctionθsuchthatθ(π 1 (M))≤T(π 1 (M)).Notethattheexistenceofsucha functionfollowsfromG.Perelman’ssolutionofthegeometrizationconjecture[Pe1-3].Indeed therecouldexistatmostfinitelymanydifferent3-manifoldshavingthefundamentalgroups isomorphictothesamegroupG(forirreducible3-manifoldswithboundarythiswasshown muchearlierin[Swa]).Thequestionwhichstillremainsopenistodescribetheasymptotic behaviorofthefunctionθ. Notethatforcertainlensspacesthefollowinginequalityisprovenin[PP]: c(L n,1 )≤lnn≈const
T( Z /n Z ). However,theaboveproblemremainswidelyopenforirreducible3-manifoldswithinfinite fundamentalgroup.IfMisacompacthyperbolic3-manifold,D.Coopershowed[C]: VolM≤π
T(π 1 (M))(C). whereVolMisthehyperbolicvolumeofM.Notethattheconverseinequalityindimension3 isnottrue:thereexistsinfinitesequencesofdifferenthyperbolic3-manifoldsM n obtainedby DehnfillingonafixedfinitevolumehyperbolicmanifoldMwithcuspssuchthatVolM n <VolM [Th].Theranksofthegroupsπ 1 (M n )areallboundedbyrank(π 1 (M))andsinceπ 1 (M n )are notisomorphic,wemusthaveT(π 1 (M n ))→∞.SotheinvariantT(π 1 (M))isnotcomparable 2