Cet ouvrage fait partie de la bibliothèque YouScribe
Obtenez un accès à la bibliothèque pour le lire en ligne
En savoir plus

CRITICAL EXPONENTS AND RIGIDITY IN NEGATIVE

20 pages
CRITICAL EXPONENTS AND RIGIDITY IN NEGATIVE CURVATURE GILLES COURTOIS 1. Introduction The goal of this lecture is to describe a theorem of M.Bonk and B.Kleiner on the rigidity of discrete groups acting on CAT(-1)-spaces whose limit set's Hausdor and topological dimension coincide. We will give the proof of M.Bonk and B.Kleiner and also an alternative proof in particular cases. Before going into it we rst set up some historical background. A famous theorem of G.D.Mostow states that a compact hyperbolic manifold of dimension n 3 is determined up to isometry by its fundamental group. In other words, if is a cocompact lattice in PO(n; 1), with n 3, there is a unique faithfull and discrete representation : ! PO(n; 1) up to conjugacy. On the other hand, for some lattices of PO(n; 1) there exist many faithfull discrete nonconjugate representations : ! PO(m; 1) 2 n < m as described in the following example. Bendings: Let us assume that a lattice in PO(n; 1) is a free product A C B of its subgroups A and B over the amalgamated subgroup C such that C cocompactly preserves a totally geodesic copy of the hyperbolic space H n1 in H n .

  • any convex cocompact

  • dimensional compact

  • convex cocompact

  • faithfull discrete

  • dimensionel sphere

  • hadamard manifolds

  • discrete nonconjugate

  • hausdor dimension

  • hyperbolic space

  • unique faithfull


Voir plus Voir moins

)CRITICALlargerEXPONENTSBAND:RIGIDITYyINaNEGA1)TIVECCURtoVrAcopTUREmGILLES(COURATOISC1.olicIntrtheoductionusThe(goalinofthethistolecturecoisoftofordescribeobtainedadtheoremrepresenof1M.Bonkcopand+1B.Kleiner(oninthe=rigiditOyforoftdiscreteCgroups)athatcpreservt+1iangucon1)CAset.T(-1)-spacesfucwhosetlimitrepresenset'sfucHausdorPandntopy:ologicalAdimensionOcoincide.hWofeOwillgroupgivaehthe1prooftralizedof;M.BonkaroundandwhicB.KleinerFandSalsotan1alternativaeAprorof2inwithparticularnocases.BeforeCgobing=in()tototallyitnwcannotecf.rstbsetanduptionsomePhistoricalObactokground.ofA()famoustationtheoremofof=:G.D.Mostoucw.states1),thatrepresenaofcompact(hmypberbwolic0manifold!of0dimension2nm;or3fucisdeterminedAupintonisometry1)b0ypreservitsgeofundamenoftalpgroup.HInHotherThew(ords,thenifPis+abcoofcompactnlatticeninisP1Or(itn;,1),newith!nn1)3,tthere=isaaunique)faithfulltandalldiscrete.represencommtation0there:biguit!denitionP(Oc(An;It1)shoupttosmallconjugacy.esOnanthedeotherofhand,Hforthsomeelattices0ofOnePofOw(ucn;non1)representhereofexistlatticeman(ytofaithfullm;discretennonconjugatetheirrepresenthetationslimit=::a!nPisOthe(limitm;G1)02ysian<Omn;asadescribhsianedtationin0theinfolloOwingm;example.withBendings:>Letcanuseassumethisthataalattice:in2PO0(In;1)Pis(a1)freeFprosucductaAhsiantationC0B=ofitsBsubgroupsPA(and+B;othever(the)amalgamatedessubgrouptotallyCdesicsucyhthethatyCerbcospacecompactlynpreservinesna.totgroupally0geoCdesiciscopcenyinofOthenh1yp1)erbyolicsubgroupspacerotationsHHn11Hin+1Hhnisomorphic.SF.ororsucthea2group1theletquotiendetmanifold:MP=(H+n;=bisa(compact)haypallerb2olicandmanifotldbwith=a1totallybrgeofordesicbemBbAseddtedutesandseparating(h)ypisersurfaceamNy=theHofnt1c=Cfor.2One=can\consider.acanFeucwnhsianforrepresen6tation0enough,0group:0!doPnotOe(yngeo+sic1y;H1).inAnrepresenandtationusbofconjugatealattice,of[11].PwOy(distinguishingn;et1)eeninFPhsianOa(Fm;hsian1)ta-with2acompactnof<Omn;isincalledPfuc(hsian1),if>is()comparepreservlimitesBasicallyasizetotallythegeosetdesicGcopyforofnonthehsiahrepresenyperbstriclyothanlicsizespacetheHsenofin0Hm().anLetFbhenatationlattice0of1PF2satisesGILcLEalSbCOUR)TOISwingBeforecgoinginfurther,;letectusEturnMtoifa=moreeacgentheerainlThesettingtheands,inthetroeduceEsomelimnotations.Let(Xb.etheatheCAaT(-1)-space,Xcf.o[4].bExamples(ofgeoCAT(-1)-spaceXaretCartgivan(HadamardsetmanifoldmetricofAnegativ=eumcurvhature's,KA(1,1ie.(i)simplyconnected>manifolds))ofadiusnegativ.earesectionalarecurvtatureGKG1-dimensional1.T(-Fdeneorasaxeddiscretegoroupdenoteof0isometryeenGandofdaeCaAdistanceT(-1)-spacetheXw,basewequivedenotedeneHausdorthedlimitLetsetH((Gfollo)ofGjasgtheonclosuresubsetofAtheor(bHitsaofsome0(and(henceaney)metricp)oinhlforstaoC2BXinrtheMidealcompactnessbGoundaryt@endingsXtofvXcompact,enough,namelyset()Gsuc)is=ologicalGo[15],Xa[)@letXdistance\b@ws.Xe.oinA.subsetYtoinXXisysaid;quasi-conthevtexandifjoiningthere0iswingaconstan)t(C)>on0TsucendshhoicethatpevbuterydierengeoofdesicoinsegmenrisettwithWendpyoin)tswithintheYtheliesGinrecallthe-HausdorCon-neighcb;orhodenedoFdMof0,YH.AThefgdiamroup)Gtheistaksaidsequencesquasi-congvMexvcowhosecompactjifallthereHexist)a!G-inWvthaarianHausdortHquasi-con)vdexHsubset)YdTheXwillwithDe

Un pour Un
Permettre à tous d'accéder à la lecture
Pour chaque accès à la bibliothèque, YouScribe donne un accès à une personne dans le besoin