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THE GEOMETRY AND TOPOLOGY OF THREE-MANIFOLDS THE GEOMETRY AND TOPOLOGY OF THREE-MANIFOLDS IV William P. Thurston with a Prbeyf ace Steven P. Kerckhoff • •• ...... AMERICAN AMS \;f MATHEMATICAL o:i. SOCIETY ProvidencReh,o deI sland Editorial Board JanGei lman( Chair) JoseSpihl verman Andras Vasy 202M0a thematics SubjePcrti marCyl a5s7sKi3fi2c,a t5i7oRn3.0 , 53C15. ParItV o ft hese otf W illiTahmu rstCoonl'lse cted WTorhkeGs e,o mtehatenr dy TopoloofTg hyr ee-Manbiasfeoodlna d sc,o uorfls eec tufrreosm PriUnncievteorns ity (unpubliissuh seedd) , wtihteph e rmisosfi oJnu Tlhiuanr ston. Chapt7e,"r C omputaotfVi oolnu meb,y" J.M iWl.n oirsp, u blishheerwdei th permisbsyi oJno hnM iWl.n or. Library Coofn gressC ataloging-in-PublicDaattiao n NamesT:h ursWtiolnl,i Pam. , 194a6u-t2h0o1r2., TitlTeh:e geomettorpyo lanoodftg hyr ee-manIiW fiollldiPsam.T hurs;tw ointahP reface by StevKeenr ckPh.o ff DescripPtriodovenin:c Rheo,d eI sland AMmaetrhiectmania cSalo cie[t2y0,2II 2 n]c ludes index. IdentiLfieCrC�N2 02103I7I 7S0B2N9 7814704(6h3ar9d1b5aI cI kS)B N9 781470468361 (eboIoI kS)B 9N7 814704(5�1e6t4)6 SubejctLsC:S HT:h ree-mani(fToolpdoslI oG gye)o metDriyff,e raelnI.tT iopololgAyM.S : Manifoladnsdc elcolm ple-xLeosw -dimentsoipoonliaonlsg pye cdiifimce ns-iHoynpse rbolic 3-manifoIlM dansi.f oanldd sc eclolm ple-xDeisff erale tnotpiol-oFgoyl iaitnido inffse rential topolgoegoym;e ttrhieco rIyD .i fferale gnetoime-tGrly obdali fferential -gGeeonmeertarly geomesttrriucc otnum raneisf o(laldmso csotm plaelxm,op srto duscttr urcetesut,.c ). ClassifiLcCatCQi Ao6n1:3. .T252 80 2ID1 DC5 143j4.-dc23 LCr ecoarvdlai abalthe t pts://lccn.loc.gov/2021037702 © 202b2y tAhmee rcianM athetmiacSalo cietryi.g hrAtelssl e rved. Prinitnet dh Uen itSetda toefAs m ercia. TheA meircaMna thetmiacSalo cireettyai nrsi gsh t all excetphto gsrean tteod the SUtnaitGteoesvd e rnme.n t o Thep apuesr eidnt hibso oiksac id-fanrde feal lwsi thtihnge u idelines establtioes nhseupdre er maneanndc e durability. Vistihte AhMoSm pea gaeth ttps:llwww.ams.org/ 109 8 7 6 5 2742 623 5 2 42 22123 Editorial Board JanGei lman( Chair) JoseSpihl verman Andras Vasy 202M0a thematics SubjePcrti marCyl a5s7sKi3fi2c,a t5i7oRn3.0 , 53C15. ParItV o ft hese otf W illiTahmu rstCoonl'lse cted WTorhkeGs e,o mtehatenr dy TopoloofTg hyr ee-Manbiasfeoodlna d sc,o uorfls eec tufrreosm PriUnncievteorns ity (unpubliissuh seedd) , wtihteph e rmisosfi oJnu Tlhiuanr ston. Chapt7e,"r C omputaotfVi oolnu meb,y" J.M iWl.n oirsp, u blishheerwdei th permisbsyi oJno hnM iWl.n or. Library Coofn gressC ataloging-in-PublicDaattiao n NamesT:h ursWtiolnl,i Pam. , 194a6u-t2h0o1r2., TitlTeh:e geomettorpyo lanoodftg hyr ee-manIiW fiollldiPsam.T hurs;tw ointahP reface by StevKeenr ckPh.o ff DescripPtriodovenin:c Rheo,d eI sland AMmaetrhiectmania cSalo cie[t2y0,2II 2 n]c ludes index. IdentiLfieCrC�N2 02103I7I 7S0B2N9 7814704(6h3ar9d1b5aI cI kS)B N9 781470468361 (eboIoI kS)B 9N7 814704(5�1e6t4)6 SubejctLsC:S HT:h ree-mani(fToolpdoslI oG gye)o metDriyff,e raelnI.tT iopololgAyM.S : Manifoladnsdc elcolm ple-xLeosw -dimentsoipoonliaonlsg pye cdiifimce ns-iHoynpse rbolic 3-manifoIlM dansi.f oanldd sc eclolm ple-xDeisff erale tnotpiol-oFgoyl iaitnido inffse rential topolgoegoym;e ttrhieco rIyD .i fferale gnetoime-tGrly obdali fferential -gGeeonmeertarly geomesttrriucc otnum raneisf o(laldmso csotm plaelxm,op srto duscttr urcetesut,.c ). ClassifiLcCatCQi Ao6n1:3. .T252 80 2ID1 DC5 143j4.-dc23 LCr ecoarvdlai abalthe t pts://lccn.loc.gov/2021037702 © 202b2y tAhmee rcianM athetmiacSalo cietryi.g hrAtelssl e rved. Prinitnet dh Uen itSetda toefAs m ercia. TheA meircaMna thetmiacSalo cireettyai nrsi gsh t all excetphto gsrean tteod the SUtnaitGteoesvd e rnme.n t o Thep apuesr eidnt hibso oiksac id-fanrde feal lwsi thtihnge u idelines establtioes nhseupdre er maneanndc e durability. Vistihte AhMoSm pea gaeth ttps:llwww.ams.org/ 109 8 7 6 5 2742 623 5 2 42 22123 Contents PubliNsohteer 's ix EditPorre'fsac e Xl Introduction XVll Chap1t.eG re omeantdrt yh ree-manifolds 1 ChapteErl l2ia.pn thdiy cp ergbeoolmiect ry 7 2.1T.hP eo indciamsroked el 9 2.2T.h seo uthheemrins phere 10 2.3T.h uep phear-l sfpmaocdee l 11 2.4T.h per ojemcotdievle 11 2.5T.h sep hoefirm ea girnaadriyu s 14 2.6T.r igornyo met 15 Chap3t.eG re ometrico nsm tarnouilcfdtsu res 23 3.1A. h ypersbtorluiocctnt u hfirege u rek-neoictgo hmtp lement 25 3.2A. h ypermbaonliiwfcio tlgdhe odbeosuincd ary 26 3.3T.h e Whiltiecnhokem apdl ement 27 3.4T.h Beo rrormienacgnos m plement 28 3.5T.h dee velmoappi ng 29 3.8H.o rospheres 32 3.9H.y persbuorlfioacbc teaisfn reoidmd eal triangles 34 3.1H0y.p ermbaonliifocob ltdasbi yng eldui idnegpa oll yhedra 36 Chap4t.eH ry perDbeohlsniu cr gery 37 4.1I.d etale traihnHe3 d ra 37 4.2G.l uicnogn sicsotnednictyi ons 40 4.3H.y perbolic sfitgruurcetk-uneroicetgo hmotpn l etmheen t 41 4.4T.h ceo mploefht yipoenr 3b-omlaincio fbotlaifdnrseo idmd epaoll yhed4r4a 4.5T.h gee nerDaelhisnzu erdgi enrvyar iant 45 4.6D.e hsnu rgoentr hyfie g urek-neoitgh t 46 4.8D.e geneorfha ytpieornsb torluicct ures 49 4.1I0n.c omprseusrsfiiacntbe hlsfiee g urek-neoictgo hmtp lement 57 Chap5t.eF rl exianbdir liigtioydfg i etoym esttrriucc tures 71 5.1D.ef oiromonafgst e ometric structure 71 v vi CONTENTS 5.2A. c rude dciomuennts ion 72 5.3T.eh imciislplaecre 74 5.4S.p ecailagle pbrraoipceo rfgt rioeuospfi s s omeotfHr 3i es 79 5.5T.h dei menosfti hodene foiromsnap tacoef h yap ertbhorleiec- mani8f2o ld 5.7M.o stoTwhe'osr em 86 5.8G.e neraDleihsznue rdga enrhdyy perbolic structure8s7 5.9A. P rooofMf o stoThwe'osr em 90 5.1A0 d.e compoosfci otmipohlnye pteer mbaonolilifdcs 97 5.1C1o.m plheytpee rmbaonolilifwdcis t bho unvdoeldu me 99 5.1J2o.r ge'Tnshs eeonrem 102 Chap6t.eG rr om'oisvn varitanhtve o alnuodmf eh yap ermbaonliifco ld 105 6.1G.ro m'oisvn variant 105 6.3G.r om'opsvr ooofMf o stoTwh'eosr em 111 6.4S.t rviecrtso ifGor no moTvh'eso rem 111 6.5M.a nifwoiltbdhos u ndary 115 6.6O.r dinals 119 6.7C.o mmensurability 120 6.8S.o meex amples 123 Chap7t.eC ro mputoafvt oilounm e 135 7.1T.h e Lobafcuhnecvtsikoyn 135 JI(B) 7.2V.ol umoefss o mpeo lyhedra 138 7.3S.o mmea noilfds 143 7.4A.r ithemxeamtpilce s 144 References 146 Chap8t.eK rl eignrioaunp s 149 8.1T.he limit set 149 8.2T.h deo maoifdni scontinuity 151 8.3C.o nvheyxp ermbaonolilifdcs 154 8.4G.e ometrfiincigatrleol uyp s 157 8.5T.hg ee omeotftr hbyeo undoaftrh cyeo nvheuxl l 161 8.6M.e asulraimnign ations 165 8.7Q.u asi-Fgurcohuspisa n 167 8.8U.n crumspulrefda ces 174 8.9T.hs et ruocftgue roedle asmaiitcni otnrsat:ir na cks 178 8.1R0e.a lizing ilntah mrienea-tmiaonnisf olds 182 8.1T1h.se t ruocftcuu rsep s 188 8.1H2ar.m onfiucn ctaniedor ngso dicity 191 Chap9t.eA rl gebcroanivce rgence 195 9.1L.im oifdt iss cgrreotuep s 195 9.2G.e omettarmiecn ess 198 9.3T.h een doinfang e nd 202 9.4T.am intgh e toopfano elnodg y 208 CONTENTS vii 9.5I.n terpnoelgaattciiunvrgevs leuydr faces 210 9.6S.t rcoonngv erfgreoalnmgc eeb croanivce rgence 225 9.7R.e alizaotfgie oondlsea smiicn aftosiruo rnfgsar coewu ipste hx tcruas wpist,h ad igreosnss tiegorrnea opc hoiocr dinates 228 9.9E.r godoiftch igete yo dfleoswi c 242 NOTE 249 Chap1t1eD.re forKmlienigmn ainainf boyhl odmse omoropfthh isesp mhsae tr e infinity 251 11.E1x.t enosfvie ocnfitseo lrd s 251 Chap1t3e.Orr bifolds 261 13.S1o.me ex almeposfq u otsipeancte s 261 13.B2a.s diecfi nitions 264 13.T3w.o -dimeonrsbiiofnoall ds 271 13.F4i.b rations 279 13.T5et.r ahedral orbifolds 284 13.A6n.d r'etseh veoarnegdme neralizations 290 13.C7o.n strpuactttioenfcrg in rsc les 297 13.A8 g.e omectormipca ctiffiotcrha Tete hiimocini slplaeocrefp s o lygonal orbifolds 305 13.A9 g.e omectormipcac tiffioctrah tdeie ofno rmsaptaiocofcenes r tKalieni nian groups. 309 Index 313