20 Pages
English
Gain access to the library to view online
Learn more

On the Complexity and Volume of Hyperbolic Manifolds

Gain access to the library to view online
Learn more
20 Pages
English

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


Subjects

Informations

Published by
Reads 20
Language English

Exrait

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