CRITICAL EXPONENTS AND RIGIDITY IN NEGATIVE

Publié par

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


Publié le : lundi 18 juin 2012
Lecture(s) : 21
Source : www-fourier.ujf-grenoble.fr
Nombre de pages : 20
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

Soyez le premier à déposer un commentaire !

17/1000 caractères maximum.