Arithmetic of Hyperbolic Three-Manifolds

469 Pages·2003·16.19 MB·English

by  Colin Maclachlan

GraduTaetxeit nsM athematics
AKBUTIIZARING.
T Introductitoon 34 SPITZERP.r inciopfRanl deosm Walk.
1
AximoatiScet T heory2.n de d. 2nd
ed.
2 OXTOBYM.e asuaren dC ategory2.n de d. 3 ALE BRIWERMER. SeveraClo mplex
5
3 SCHABFBR.T opologiVceaclt Sopra ces. VarXANDiabanlde Bsa nacAhl gebr3arsd.
ed.
2nd 3 K.BLLEYINAMIOKA eta tL.i near
ed. 6
4 HILTON/STAMMBAAC HC.o ursien TopoloaglSi pcaces.
HomologiAclagle b2rnad. 37 MoNK.M athematLoigciacl.
ed.
5 MACLANB . Categorfioestr h Weo rking 38 GRAUBRTIFRITZSCHB. SeveraClo mplex
Mathematician. 2nd Variables.
ed.
HuoHBSIPIPBRP.r ojectPilvanee s. 39 ARVBSONAn. Invitattoi onA Jgebras.
6 C*
7 . SERRE. Coursie nA rithmetic. 4 K.B /SNBLLIKN APPD.e nu.. merable
J.. p . A 0
8 TAKB UTIIZARING. AxiomaStietcTh eory. MarMBNYkovC hain2sn.d
ed.
HUMPHREYS. IntroductitoonL ieA lgebras 41 APosTOL.M odulFaru nctioannsd
9
andR epresentThaeotiryo.n DirictSh elreiienNs u mbeThre ry.
o
CoHEN. A Coursie nS impHloemo topy 2nde d.
10
Theory. 42 J SERRE.Li near Reprenetsationosf
..p .
,
1 CoNWAY.F unctioofOnn es Complex FiniGrtoeup s.
1
Variable2 nd 43 GILLMANIJERISRiOnNg.s Coofnt inuous
I. ed.
12 BEALS. AdvancMeadt hmeaticaAln alysis. Functions.
13 ANDERSON/FuLLERRi.n gsan d Categories 44 KENDIGE.l ementaryA lgebrGaeiocme try.
ofM odule2sn.de d. 45 ProbbailTihteyo ry 4th
LosvB. I. ed.
14 GoWBITSKYIGUILLBMIN.S tabMlaep pings 4 ProbabThieolryi tyI I4.t ehd .
6 Losvs.
and TheirS ingularities. 47 MOISGeoEme.tri c Topoloigny
15 BBRBBRIAN. Lectureisn F unctional Dimensi2oan nds 3 .
Analysanids Operator Theory. 48 SAcHs!Wu. General Relativity for
WINTERTh.e StructuroefF ields. Mathematicians.
16
17 RosBNBLATI. Random Process2ensde. d . 49 GRUBNBBRGIWBILRi.n eGare ometry.
8 HALMos. MeasuThreoery . 2nde d.
1
HALMOSA. HilbSepratc Per oblBeomo k. EDWARDS.F t'Lsa sThte orem.
19 50 erma
KLINGBNBBRG.
2nde d. A CourisneD ifferential
51
2 HUSBMOLLER.F.i bBreu ndl3esr.d Geometry.
0 ed.
2 HUMPHRBYS. LineAalrg ebrGroauipcs . HARTSHORNEA.l gebrGaeiocme try.
1 52
22 BARNBS!MACK. AnA lgebraInitrco duction5 3 MANIN. A CourisneM athematLoigciacl.
toM athematicLoagli c. 4 VB A S.C ombinatwiotrhi cs
5 ORA RIW T
23 GRBUB. LiAnlegaerb 4rat.h EmphasointKINs h Thee oryo fGra phs.
ed.
24 HoLMES.G eometric FunctAinoanlayls is BROWNIPBARCY. Introductitoo nO perator
55
ItApsp lications. Theory ElemenotfFs un ctionAanla lysis.
and I:
HEWITT/STROMBERG.
25 Reaaln dA bstract MASSEYA.l gebrTaoipco loAgny :
56
Analysis. Introduction.
CRoWBulFox.
MANEsA.l gebrThaeoirice s. IntroductiotnoKn ot
26 51
27 KBLLBY. GenerTaolp ology. Theory.
28 ZAiusiCJ!SAMUEL. CommutatiAvleg ebra. 8 KOBLITZ. adiNcu mbe,rs adic
5 p p ..
..
Vol.I. Analysiansd, Z eta-Func2tnideo dn.s .
LANG.
29 ZARisiCJ!S L.C ommutatAilvgee bra. CycoltomicF ields.
AMUB 59
ARNoLD.
VolI.I . MathematMiecthaold isn
60
3 JACOBSOLeNc.t ureisn A bstractA lgebIr.a ClassMicecahla nic2sn.d
0 ed.
WmTBHBAD.
BasiCco ncepts. 1 ElementosfH omotopy
6
JACOBSLeOcNt.u reisn A bstracAtl gebra Theory.
31 II.
LineaArl gebra. I<ARGAPOLOVIMBRLZJAFKuOVn.d amentals
62
JACOBSLeOcNt.u riensA bstraAcltg ebra oft hThee oroyf Gr oups.
32
Theoryo fF ieladnsdG aloThieso ry. Bot..LOBGArSa.p h
III. 63 Theory.
l y
33 IIJRDSiCfIfIe.r Teonptooigal
.
ajttl<(�'xl)' ill
(t�OIIIiiUU'd
ColiMna clachlan
Alan W. Reid
TheA rithmeotfi c
Hyperbo3l-iMca n
Wit5h7 I llustrations
Springer
ColiMna clachlan Alan W. Reid
DepartmoefnM ta thematSicciale nces DepartmoefnM ta thematics
UniversoifAty b erdeen UniversoifTt eyx aast A ustin
KingCso llege AustiTnX, 7 8712
AberdeAeBn2 4 USA
3UE
UK arei@dm ath.utex.aesdu
[email protected]
Editorial Board:
S.A xler F.W.G ehring K.A.R ibet
MathematiDcesp artmenMta thematiDcesp artment MathematiDcesp artment
San FrancisSctoa te EasHta ll UniversoifCt ayl ifornia,
University UniversoifMty i chigan Berkeley
SanF ranciscCoA, 9 4132 AnnA rboMr,I 48109 BerkelCeAy 9,4 720-3840
USA USA USA
axlesrf@s u.edu [email protected]@tcmha.the.dbue rke.leedyu
MathematSiucbsj eCclta ssific(a2t00i0o)n5: 7 -0517,N 105,7 Mxx5,1 H20l,l Rxx
LibraorfCy o ngresCsa taloging-in-PDuabtliac ation
MaclachCl.a n,
Thea rithmeotfih cy perbo3li-cm anifoClodlsiM na clachAllaann , Reid.
I W.
p.e m.- (Gradutaetxeit nsm athematic2s1 9)
Inclubdiebsli ographriecfaelre nancde si nd;e x.
ISBN0 -387-968-34(8 acid-pfraepee r)
Three-manifolds( TopologyR)e idA,l an Title. Series.
W. II. ill.
QA16.13 .2 .M292 002 I.
514'.3-dc21 2002070472
ISBN0 -387-8968-34 Printed ona cid-frpeaepe r.
2003S prinVgeerrl-aNegw orlc, Inc.
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Preface
'hGee omaettPirroionzg orTfah mu rhsabtseo tenhnd e r ifvoifrnrocrg ee ­
'I
s�(racihn a3no-iltmfdol pooigtnyh e2 5yl eas.at r Tshhasii nissr apes du rge
acitviinetvsyt gih gyeaprtbi3onm-laioncli adfnsKd l eignrio,au npas s
{ )
of
tshmeea onlifdfrosmt hlea ragneldse wtae sltl s-tuocnoldodaecf sro smc pta
:�-o1ln.Fda asinmlriai fnande two frooldmsi vaerroesmfaea s t hematics have
IHe �nu tiiltnih seiesndevi egsat-tfioromtn osop gogyle,o rmyae,nt ai,ls ys
p,rteohouarpny, d frtohpmeio notvf i oefw btohoaikls,g ebnruaem bra nd
tl.aeTohireym .pa onortbt serivatnth ciiotosennx i ts t M tothsoaRwi gidity
irutphltaihtmee as t ernitxor fti heeesl emoefSnL (tC2s),,r eptrniegs en
licnoivtoeKl-eluimngeir ao,cnu a pbnet k aetnol iieanfi ewlhdi icash
:t
l1ieae ixtteonQfs. Tih ohinasl s et dto h ueso etf ol froso aml genburmeabri c
I.IH�iotnrh syet uoKdfyl i eaigrnno uopfifsn ciotveo alntudhm ouehfs py re­
3-moalnodififsfn ivtoel. u pmAraei tcrus luabcofilfnas icsto levuo-me
IH)lic
nigarno ufrpow sh itchnheu em-rbthecoornenteaicrcste ti roonnsg est
l!
1\<i
t.chleas asir tohfmK elteiigrcnop iusaT.nh egsreo aurppesa rltaircluy
is
;ueanb•alteoe xihbitthiienn gtl eabrytep wetehgnee ormyoe,ntt he one
antdhn eu mtbeheoror ny ,ot th.he e r
l1aud
bokio dse sitgionn etdrt ohrdeedu ae,cwr ehh oasb egtuhsnet uodfy
'rhis
l.YtJ>e3r-abmnooillofidKrcsl ieaigrnnop ust,to h eisnet ecroensintoeincnstg
nurtnebhoearrny td h teo tohlwasit bl erl e quiretdh .et mo pursue
'vit.h
nuronftbe exrwt hhsic covtehtreoo plo,gg iecoamalentdr ic
'l'lu�re
area.s a peoehfty sep rb3om-laionlcisd Tf.h bioskio cso nsttrou cted
allalyt.ical
aspwehcithcvsahen obte deins ciuonst shteeetdrxsA .
··ova(r�ilt'. htll<!Lie
thien ivaannrutm fibeelrd
d
is s
c'(•utth.(rHatlt tei rteu y au<l <loterotfn iuatiou
invariaquula.f r<'ruiou .. a
�Hocifa.fo.K (l�cc�li lliau
;nutl. h<' al�!;<�ha ra groupo f
Praecfe
vi
finictolevu o,mt eheaisrteh moebtjsbie ecci tinnagvr iwainrtteh s tpoe ct
thceo mmensculrasatsbh gieorl foiW utepsy .h opuoliodnu tt tthhiast
bookd oneosit ns vteigactlea sasisrotimhcemao elbt jsaie sccs totcoi ated
Kleignriosvau intpa hS ee rlTrgba ecFero mau.l I,wn ewd ouelesddu ggest
tah,ti pfr osepr eedcaetarirsuve n swuhreett hheweriy ts ohfl o ltoehw
rodaodw wnh itcihbhso lodke,s at hsehyo duilipdn C tahop t4ean rd5s
tose weh airtse veaabloeeuxdtal meapsnp dr obwliewtmhhhsi t cehya re
alrfemaaidl.Tyi h aturshb iosio wskr iftoatrnea und iaeldnrycfe aeaim lri a
withb astiashcpe e ochfty sep rb3om-laionclia dfnsKd el ingirpao,snu to
exptahnredei pre traotr oiitrahepml peitciaicttn ih fioiens.ls ds Buyi table
seclote,ni ciatn alsaso a nibt ner oduustceoatd ri iotnKh lmeenit niica
grposeu,v, ei nnd,e aetrhdoim teFutcihcsg iraosnu. p
Wen opwr oavg iuditedto eh c eo natneidnn tto etfnh ctehp eatrs and
tehiirn teercct,foi ronto enhnr edae,tr ecahoesrrt nutwd heoy w miatsoh
bes eilveieccno ht oisnrgo uatt hoeru tghhbi osoA kste. hn umebriiisnn g­
tednetdio n dtieCc,hpa atre iasr eefneccrhpeat ceorn ttareimniinnogl ogy
0
anbdca kgroiuonnrfdm aotnai logne bratihrceyM o.ann yur medbaecerarsn
byaspst hise orcn fih rasptt ,er seepcaidialtiflhny ayeg r mei lfwaiiattrhh e
basciocn coaeflp gtesnb urmatbiherceryC o .h apitenes rcs ede,en fitnhees
1,
tragaeutd iaetsonh scweehh ova ,ea lte ,asp taas fsaiimnlrgiiw atiyt h some
of ttohpieitn ch csih sea .rpI tCna hpt2te 5ort, se h s truur,cce otnstruction
anadp pliocfa itvtniarhioeann nustem fibre alndid na vriqaunatt earl­nion
gebarsas otcoi afiatnneicydtoo evl-uKmelei ngrioauanprd ee vee.dTl hoep
algesbtrrauoicqfctu euarrtnneal i goebigrsia viseCn nh ap2an tdet irhis s
frutehxepra innd peCtdh6e aa rns7,dw he,r peianr ticaurliatrh,m etthiec
struocqftu euarrtneai lognei sbser otau.C st h patre g3i tvehtseo aonlds
fromutldoase te,frr omamig nievKelinen girao,niu taspss otceiivadnar iant
numfibeealrnd qd u eartnaileogbnTr haii.tss h peuntt eo eff citCn ah prt e
4i mnan ye xampuletsii lCnaih nsadep5 dt t ioevn rse titghgaeet oem etric
rmaificaodtfei etornmsit nhieinsngivae atr ns.
FromC hpatr6e o nrw,dat heem phiasosin ds e veltohtpeehi onorgfy
aritKhelmienngt iripaocscu ,o ntcreaontnti hnoasgspe ecsw t hhih cvagee o­
metarpipcl itcoha yetpribo3on-alsmnif oicla dns3d - oodrl.sbO iurdf enfii­
toin aiortfh mKeliteniigrcaop nu,sa nadrh imteFutc�hcsg irao,ngu ipvsen
iCnh ap8,pt reorc veiead sq uatearnsndnoi a otnu praralolggrleeybs rsaess
formt he cehpaatrr.elsT ihgeeer o maeptprilioifcnclo sali toCnwh ap9t,e rs
11a n1d2I .pn a rtrii,mc puaolnasratpt escutacstsh h dee veltoo tfph mee n
volfurommeua lntadh dee mtietniraoonmf a ximgarloi uanpc so memn­
suracblifalrosimtst h fyeco uosCf h ap11tb euri lodnti hgnerg o wuonrdk
iCnh ap6ta enrd s
7.
Usiqnuga taelergbntridooase n fi anrei tiKhcelm nieingta rofuaipcaltsie ts
tehfl oowif d ebtasew eteehnn u mtbeehorro ynte, ho ne hatenh d and
geoymo,en tt hroet .h er teh ieof
Thiisn teripsol naeoy f b
Hpeeiae]au t s
Teher
which ha.ve every (�Jupha.sise.
Hnhjc�et. opportutu.iof .y
the w<� ta.k<�ll
<'qua.lly approa('tI.Uao'l H' it.ln1n\<l'<t'.ig;inrcio annp H,
an'o l.lu'r, llH'rit.orious
..
Praecfe
Vll
utrltaviricqyauu aidfcrro asmtT.h easrdeei sciuCnsh psateedwr h ere
10,
1
alsshwoho otwh eear sithmetic fiKtl etiihwnneiti dorae enra lgmr oups
\V<�
,gendeirsaaclrrh iemttesetu ibcg orLfoi guerpp os.su
• .
,
Somree amdaewyri ss h tithso k b uoass oeta rno ditunaocr tiitohnm etic
grposAu.s hocrotuc rorsvieetn hgge e ntehreaoolqfr u yea rtnion
leinian
1\
lp;eobvrrne ausme brfi ed,lss uaibtflroes uacnhi ntrotdoeuh icettri on
a
aitrhmKeeltiinngc ir aoouarpir st hcmFu ecthnisg irpao,siu eses nstsie­allfly
C'onitCnah ianp2et6,dae nr7ds.T hceo nstorafur hcimtteiKtoliniec ai nn
p,nfnor1mqp aust ieorannl geigbsir viasten hn fie rp sator Cftah pet8ran d
ntcaoiennqs ueontfch ceiossn starppuercia Ctnhi paotne Hro weerv,
11.
tIu. •
readert oin wtviiesgshtaehrtseoe p l lyeae bdya i rthmKelteiicn ian
t.he
i r
Hisip tnh gee nfearrmaelwo oafrlK kll eignrpio,stau hnhe eonsr h meu st
J'.n assitmhimela itaeitrCneih aalp3 st,ueh ecr x almeipsCn h ap4 ter
further
itnreetsht,e mthe rCehmaap8i,tCn eahdrpe t9ra e nroasd fm ucohf
ns
�1lap1t2aset rh weihys.
<
thoitsnhe e ofife elhrdyb3p o-lmiacna in3fd-o ololrdd,bwss ie hfv ae
For
laveoutrmoea dtk eehe xopsihteiarossene - lctfoiannaepsdo s,sg iibvleen
4· u•
enostroansi onfmtmaesi r liiwtaiybt ahas psicecto saf l genburmabiecr
l.lu'
'oryas,m enteiaoern.rTle hidea rr,hee o wrec,ve ertaienxe cpstpieocnisfi c
llu
whi,wc ebh lei,ew veuerneoa ivdiatnbh lieenr teseos kfte peing the
this,
In otfh itsi wstiernt e harieasboondua.sTnb wolo etf h aersieen voilnv ed
:z;(i'
whiacrcher itttoih gceea nllde ervameeltono ifpd .eF ai,srw set state
:d.('PH
proionCf aht pre thHeas sek-woMsiTknhie oornqe uma dratic
\vit.ltout 0,
and tuhstiae Cn h ap2tt poer vroep arottf h cel fiacsasttihieoonr em
H'lrtH
lc
qutaeranligoeonbv aren rau sem fibre .Sl cedo,nw ded on ogtit vhfeleu l
•r
r.
l•nuC>hpfatr 7e ti hnta htTe ma aganwuam obtfeeh rq ouitent is
A�/Al
wed od evaellolotfe p hs urrotuehnodTrihynTi.mag sa wgaa
;t.lt.hough
I.
11111eh1ri1u ss ienCd h paetr1 1t oo bativno lfurommeu flroaa irst hmetic
agnr oaunpadsrh imteFutcihcsg irpaosnWu. es hoaullmsdeo n tion
1l\
(•rtii
thiem paontrteht oroefMm a rsgw,uh leiyrt ehaberh imtiectaintdy
hat.
1I1 ,11 1r-iaetthimicKnil teyig nriocauanpbn sed etebcytt hdeeed n soern ess
iscno�tfth ceeon memsestn o,isr dsui rsac,bu usntsop etrvd oe,d Cihna pter
•I
llvoe,wtr eihrse siunslo utts cerdi ltliiyntc hsaee q.Au le,sol ona
I 0.
numobofec rsc iaiolnnta sec rh ear,pss tpecriealsiousnale ltdgs e braic
all
:an
111111>tetlrh eaoreremyl p oyteoodb tsapienac pipfitolcini. sc a
Mauotyfe h a rhimtiemcte thdoidssci utnsh bsioesaod rkn e oa wav ilable
comrpp urtoegSrnpaa.Omn creed aehrascv oe mteot erwmist h
t.lu�
i11
these, w meser ttohnogdlasyg teeh nteecomxo epurritwm iettnhhi s
:;••JIH'
of
progtrdoae mva ef leefolrop itnhtee brtaewcetheiyenpor nb olic
\\'l(<H�Ir<f ul
anndu mtbeheorr y.
:1'n auifolds
shocuolmdm oenon utmr e thoofred e nrfecing.
Finally,
We
have
wet hsep ortee"rfc eeansnh dva ep laaclreleed rf ences
give11
"ou
:1\'oid"d in
the a.
c-lnaptt..<lF·u'r�n· r tR..leua�dri snege t.iou of
at eaeh
in appearing end chapt.er.
\\'s1h•o luda lslo' e.trutka. lt1.ah.etFH.'e n t·ltl.<I�Ha�drHi P1<1�gt. ious
ar<i�u t.<'t.loH I<'d
ustt.. ltaantd.a ,n 'b,y nlol H'adu<s•,s i�tIoHp; 'idv a< 'l tist.oarciccoauln t.
j
IH' of
eha.pt'tl�'llh·".ut s� �n·lH.otunlp.e.nh'pl ty�,cr r··si t­
theev olutiiodne a.'i
of iut lu�
ictaotl h dee veloofp rnent
uta.hya .Vt! oh1en(i�ll.lt .(�d
certationp ieH whil<�,
prehpasl,ar rt eefinteasmn eedxn poastictory inlcu.d ed
il orb ookH,
r e�
Noo effnocprerj deuiicisen tebnyd seaudnoc ymh i nssws,hair ciao crsheu rely
the orfe ssuhlotor nte ca touhmt'ihp noagprrsost s sdiutbeol yst ohmee ­
whuants yswtyaeb mywa htitichchte hye vmebsseea clmaceq ueadiw nitth
thmea tecroinathlae irnee.d
Weo wae gdreeaamlta nctyoo gl ulaeensafrd i ewnhdhosva ec ontrib­
utteoodu urn dnedrisonttfgah s eue bcmjta tter conptaa.gi ensed in these
Thecsotenr tiibohuvnaersn a gterhdo uignhs plierci,ten ungrlt neieisggn h
covenrtsia,oh neuslclp oflltai,bo onorsnaegnocaiognuegrma encndrt i lt ica
feedcbkta on uame brl eocfct ouurwrehs itechsahe uh tohrvases erpalatye
givoenpn a rotfts hm iast leW.re ei saplelcyi tawoti hnsakTh de C ihn­
bu,rE gduaFrrideod KmrearnJy,o ,nD erasrLeonn Mgu,ry Mr aacbteha,
GvaeMna ir,nWt atlreN eumanadnG nre haRrodse eren.grWb ea lwsios h
ttoah nk GFreehdrw ihanodg d,i teinocanogauuelrstdlw o yr itthteixe st ,
anOdil vGeormo andf ros uppSlnpyaD itanawgh hii cisn clutdehe d in
appnediFnxila.l wyeo, w aep raitlcaur odfger ibatttut dote wp ooe lp:e
DoryoM tachclhalaanEndd maCrvaaacnl aitR eiDdro.o thhasby ee na n
esnstemiealm botefehr b acokmsr tffow,ai etnhd pltaeiseasnn scdue p port
ovtehrye er asM.o rrceene tEldym,'sa p raatiaenndc esh uaspb epeno rt
impaonirtnt e c tohmplotefeth b ioookn.
Iand idoitntco o lltai,abn nogwdro ariknidngil v,lai yotdu huroa mien sti­
tutoiAfob rnedesenU nivaenrtdshU ient iyv oeTfrxe sasai Attu ys ,t iwnork
ont htxeeth asb neeefidftor mp reisoepdtnsa tt h Uenie vrsoiAftu yca knld
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• • • • • • • • • • • • •
(0) A dleesI daen.0ld e s0 0 0 0 0 0 35
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lo2S ubgorPfoS uLp(so2 ,o o o o o 50
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14o.lB inacGhrio ou ps o o o 58
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14..C2xo etGoerur.p soo o o o • 59
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o o
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5. 13 - • • • • • • • • • • • • •