On the Complexity and Volume of Hyperbolic Manifolds

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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.

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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)≤lnn≈constT(Z/nZ).However,theaboveproblemremainswidelyopenforirreducible3-manifoldswithinﬁnitefundamentalgroup.IfMisacompacthyperbolic3-manifold,D.Coopershowed[C]:VolM≤πT(π1(M))(C).whereVolMisthehyperbolicvolumeofM.Notethattheconverseinequalityindimension3isnottrue:thereexistsinﬁnitesequencesofdiﬀerenthyperbolic3-manifoldsMnobtainedbyDehnﬁllingonaﬁxedﬁnitevolumehyperbolicmanifoldMwithcuspssuchthatVolMn<VolM[Th].Theranksofthegroupsπ1(Mn)areallboundedbyrank(π1(M))andsinceπ1(Mn)arenotisomorphic,wemusthaveT(π1(Mn))→∞.SotheinvariantT(π1(M))isnotcomparable2

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