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On the Complexity and Volume of Hyperbolic Manifolds

20 pages
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


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OntheComplexityandVolumeofHyperbolic3-Manifolds.ThomasDelzantandLeonidPotyagailoAbstractWecomparethevolumeofahyperbolic3-manifoldMoffinitevolumeandthecomplexityofitsfundamentalgroup.11Introduction.Complexityof3-manifoldsandgroups.Oneofthemoststrikingcorollariesoftherecentsolutionofthegeometrizationconjecturefor3-manifoldsisthefactthateveryaspherical3-manifoldisuniquelydeterminedbyitsfundamentalgroup.Itseemstobenaturaltothinkthatatopological/geometricaldescriptionofa3-manifoldMproducesthesimplestwaytodescribeitsfundamentalgroupπ1(M);ontheotherhand,thesimplestwaytodefinethegroupπ1(M)givesrisetothemostefficientwaytodescribeM.Moreprecisely,wewanttocomparethecomplexityof3-manifoldsandtheirfundamentalgroups.Thestudyofthecomplexityof3-manifoldsgoesbacktotheclassicalworkofH.Kneser[K].RecallthattheKnesercomplexityinvariantk(M)isdefinedtobetheminimalnumberofsimplicesofatriangulationofthemanifoldM.ThemainresultofKneseristhatthiscomplexityservesasaboundofthenumberofembeddedincompressible2-spheresinM,andboundsthenumbersoffactorsinadecompositionofMasaconnectedsum.AversionofthiscomplexitywasusedbyW.Hakentoprovetheexistenceofhierarchiesforalargeclassofcompact3-manifolds(calledsincethenHakenmanifolds).Anothermeasureofthecomplexityc(M)forthe3-manifoldMisduetoS.Matveev.ItistheminimalnumberofverticesofaspecialspineofM[Ma].Itisshownthatinmanyimportantcases(e.g.ifMisanon-compacthyperbolic3-manifoldoffinitevolume)onehask(M)=c(M)[Ma].12000MathematicsSubjectClassification.20F55,51F15,57M07,20F65,57M50Keywords:hyperbolicmanifolds,volume,invariantT.1
Therank(minimalnumberofgenerators)isalsoameasureofcomplexityofafinitelygener-atedgroup.AccordingtotheclassicaltheoremofI.Grushko[Gr],therankofafreeproductofgroupsisthesumoftheirranks.Thisimmediatelyimpliesthateveryfinitelygeneratedgroupisafreeproductoffinitelymanyfreelyindecomposiblefactors,whichisanalgebraicanalogueofKnesertheorem.ForafinitelypresentedgroupGameasureofcomplexityofGwasdefinedin[De].Hereisitsdefinition:Definition1.1.LetGbeafinitelypresentedgroup.WesaythatT(G)tifthereexistsasimply-connected2-dimensionalcomplexPsuchthatGactsfreelyandsimpliciallyonPandthethenumberof2-facesofthequotientΠ=P/Gislessthant.IfthegroupGisdefinedbyapresentation<a1,...ar;R1,...Rn>thesumΣ(|Ri|−2)servesasanaturalboundforT(G).NotethataninequalitybetweenKnesercomplexityandthisinvariantisobvious.Indeed,bycontractingamaximalsubtreeofthe2-dimensionalskeletonofatriangulationofMoneobtainsatriangularpresentationofthegroupπ1(M).Sinceevery3-simplexhasfour2-facesitfollowsT(π1(M))4k(M).Inordertocomparethecomplexityofamanifoldandthatofitsfundamentalgroup,itisenoughtofindafunctionθsuchthatθ(π1(M))T(π1(M)).NotethattheexistenceofsuchafunctionfollowsfromG.Perelman’ssolutionofthegeometrizationconjecture[Pe1-3].Indeedtherecouldexistatmostfinitelymanydifferent3-manifoldshavingthefundamentalgroupsisomorphictothesamegroupG(forirreducible3-manifoldswithboundarythiswasshownmuchearlierin[Swa]).Thequestionwhichstillremainsopenistodescribetheasymptoticbehaviorofthefunctionθ.Notethatforcertainlensspacesthefollowinginequalityisprovenin[PP]:c(Ln,1)lnnconstT(Z/nZ).However,theaboveproblemremainswidelyopenforirreducible3-manifoldswithinfinitefundamentalgroup.IfMisacompacthyperbolic3-manifold,D.Coopershowed[C]:VolMπT(π1(M))(C).whereVolMisthehyperbolicvolumeofM.Notethattheconverseinequalityindimension3isnottrue:thereexistsinfinitesequencesofdifferenthyperbolic3-manifoldsMnobtainedbyDehnfillingonafixedfinitevolumehyperbolicmanifoldMwithcuspssuchthatVolMn<VolM[Th].Theranksofthegroupsπ1(Mn)areallboundedbyrank(π1(M))andsinceπ1(Mn)arenotisomorphic,wemusthaveT(π1(Mn))→∞.SotheinvariantT(π1(M))isnotcomparable2