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PAGE 1 HYPERBOLICSTRUCTURESONWEAVECOMPLEMENTSbyIndraShottlandAthesissubmittedinpartialfulllmentoftherequirementsforthedegreeofBachelorofArtsMathematicsinNewCollegeofFlorida2010ThesisCommittee:PatrickMcDonald,SponsorDavidMullinsDonColladay PAGE 2 ACKNOWLEDGEMENTSIwishtothankmyfellowmembersoftheHyperbolicGeometrygroupduringSMALL2008:ShawnRafalskiforyourenthusiasticsupportduringandafterSMALL,KarinKnudsonandJeremyLeachforbeingfuncollaborators.ii PAGE 3 HYPERBOLICSTRUCTURESONWEAVECOMPLEMENTSIndraShottlandNewCollegeofFlorida,2010ABSTRACTWestudyaclassofhyperbolic3-manifoldwithnon-nitelygeneratedfundamentalgroup:weavecomplements.Wedescribehowtoendowthesespaceswithahyper-bolicstructurebydecomposingthemintoidealoctahedraandrealizingtheseidealoctahedrainhyperbolicspace.Conditionsaregivenwhichensurethatanal-ternatingweaveishyperbolic.Wethenextendourresultstoaugmentedalternatingweaves.PatrickMcDonaldDivisionofNaturalScienceiii PAGE 4 TABLEOFCONTENTSACKNOWLEDGEMENTS::::::::::::::::::::::::::::::::::iiLISTOFFIGURES::::::::::::::::::::::::::::::::::::::vCHAPTERI.Introduction.......................................1II.AlternatingWeaveComplements..........................32.1Denitions.....................................32.2TopologicalConstruction.............................52.3HyperbolicAlternatingWeaves..........................92.4Bigons.......................................17III.AugmentedAlternatingWeaves...........................213.1Denitions.....................................213.2TopologicalConstruction.............................223.3HyperbolicAugmentedAlternatingWeaves...................273.4Bigons.......................................34IV.Conclusion........................................36APPENDICES::::::::::::::::::::::::::::::::::::::::::38BIBLIOGRAPHY::::::::::::::::::::::::::::::::::::::::74iv PAGE 5 LISTOFFIGURESFigure 2.1APortionoftheSquareWeave.............................42.2PlacingOctahedraatCrossings.............................52.3PrescriptionforIdentifyingFaces............................52.4TruncatedOctahedra...................................72.5FillingtheComplement.................................92.6DecomposinganIdealOctahedron...........................112.7Top-DownViewofDecompositon............................112.8ParametrizingIdealOctahedraAlongaStrand....................122.9OctahedraWithSomeLinks...............................122.10LinksArrangedSideBySide..............................132.11EdgeInvariantsInsideaRegion.............................142.12ArrangementofLinksFromInsideaRegion......................152.13OctahedrawithCorrespondingStrandLink......................162.14APortionofaStrandLink...............................162.15PlacementandParametrizationforaBigonRegion..................182.16FacePairingforaBigonRegion.............................192.17PortionsofLinksfromBigonStrands..........................192.18PortionofNorthernPlanefromBigonRegion.....................202.19PlacementforAdjacentBigonRegions.........................202.20FacePairingforAdjacentBigonRegions........................203.1AnAugmentedSquareWeave..............................22v PAGE 6 3.2AddingaCircleAroundaCrossing...........................223.3DrillingOutanAugmentingCircle...........................233.4CollapsinganAugmentingCircle............................233.5RemovingIdealTriangles................................243.6AMoreSymmetricStar.................................243.7DecomposingtheNeighborsofaStar..........................253.8FacePairingofaStarandIt'sNeighbors........................263.9TheTwoTypesofAugmentingCircleNotAroundaCrossing............273.10TwicePuncturedDiskRedinaStar.........................273.11EdgeInvariantsofaStar.................................283.12EdgeInvariantsofaNeighbortoaStar........................293.13PortionsofLinksfromOverandUnderStrands....................303.14ParametrizationAlongtheNorthern"NeighboringStrand.............313.15PortionofLinkfromtheNorthern"NeighboringStrand...............313.16LinkofanAugmentingCircle..............................323.17ParametrizationoftheInner"RegionsofanAugmentedCrossing.........323.18PortionofNorthernPlanefromAugmentedCrossing.................333.19AugmentingCirclewithBigon..............................343.20AugmentingCircleAroundaBigon...........................343.21AugmentingCirclesnotConsidered...........................35A.1AnIdealTetrahedroninUpperHalfSpace.......................40A.2TheDihedralAnglesofaTetrahedron.........................41A.3TheVertexInvariantzuoftheTriangle4u;v;w.................43A.4TheTriangle4;1;zu................................44A.5TheEdgeInvariantsofanIdealTetrahedron.....................46B.1ArrangingTetrahedraSideBySide...........................50B.2DistanceFromLinkEdgeToNon-incidentVertex...................65vi PAGE 7 CHAPTERIIntroductionKleiniangroups,thediscretegroupsoforientationpreservingisometriesofhyper-bolicspace,havebeenstudiedformanyyears.ThoughfewexamplesofhyperbolicmanifoldsquotientsofhyperbolicspaceMG=H3=GbytheactionofaKleiniangroupGwereknownpriortotheworkofWilliamThurstoninthe1970's,hisGeometrizationConjecturemadecleartheimportanceofthetheoryofhyperbolicmanifoldsintheclassicationof3-manifolds.Inthespecialcaseofknotcomple-ments,itstatesthatforaknotKinR3,exactlyoneofthefollowingholds:Kisatorusknot,Kisasatelliteofanontrivialknot,orthecomplementofKinS3admitsacompletemetricwhichistopologicallyequivalenttotheeuclideanmetricandislocallyisometrictothehyperbolicmetricofhyperbolicspaceH3.SeeBonahon[4]fordenitionsandanelementarydiscussion.Onemayask,whataresucientinvariantstodetermineahyperbolic3-manifolduptoisometry?Theanswertothisquestionisknowninmanycases.Inthenitevolumecase,theMostow-PrasadRigidityTheoremtellsusthatthefundamentalgroupGissuchaninvariant.Inotherwords,thehyperbolicstructureonanitevolumehyperbolic3-manifoldMGisrigidanddeterminedbyG.SeeBenedetti-Petronio[3]foraproof.1 PAGE 8 2Inthecaseofahyperbolic3-manifoldMGwithnitelygeneratedG,wehavetheEndingLaminationTheorem.OriginallyaconjectureofThurston,itstatesthatahyperbolicmanifoldisdetermineduniquelyuptoisometrybyitstopologicalstructureanditsendinvariants"theendsofanitevolumehyperbolicmanifoldareeitheremptyorcusps,theendinvariantsareempty,andtheproblemreducestotheMostow-PrasadRigidityTheorem.TheEndingLaminationTheoremhasbeenestablishedforalltopologicallytamehyperbolic3-manifoldshomeomorphictotheinteriorofacompactmanifoldin[5],[6],and[9],whiletheproofthatallhyperbolic3-manifoldswithnitelygeneratedfundamentalgrouparetopologicallytamewasgivenindependentlybyAgol[2]andCalegari-Gabai[7].Onemayask,cantheseresultsbeextendedinanywaytohyperbolic3-manifoldswithinnitelygeneratednon-nitelygeneratedfundamentalgroup?Kleiniangroupscanbequitecomplicated,andindeedmanytheoremsthataretruefornitelygen-eratedgroupsfailforgroupsingeneral.Inthisthesis,westudyaspecialclassofhyperbolic3-manifoldwithinnitelygeneratedfundamentalgroup:weavecomplements.Tounderstandthesecomple-ments,wefollow[11]anddecomposethemintotopologicalidealoctahedra,solidoctahedrawithverticesremoved.Takingtheidealoctahedraoneforeachcross-ingintheprojectionandidentifyingtheirfacesinaprescribedfashionbuildsthecomplementoftheweavetopologically.Thehyperbolicstructureisthengivenbyrealizingtheseidealoctahedrainhyperbolicspace.Conditionsaregivenwhichen-surethatanalternatingweavecomplementadmitsahyperbolicstructure.Wethendetermineconditionsunderwhichanaugmentedalternatingweaveishyperbolic,apartialanaloguetothetheoremofAdams[1]. PAGE 9 CHAPTERIIAlternatingWeaveComplementsInthischapter1wedescribehowtoconstructthetopologyofanalternatingweavecomplementusingidealoctahedra,solidoctahedrawithverticesremoved.Werstconsideralternatingweaveswithprojectionsfreeofbigonsdenitionbelow.Wedescribesucientconditionstoendowtheircomplementswithahyperbolicstruc-ture,andgiveconditionsunderwhichthehyperbolicstructureofadoublyperiodicweaveiscomplete.Wethenextendourresultstothecaseofweaveswithisolatedbigonsdenitionbelow.2.1DenitionsAweaveisacountablyinnitecollectionofdisjointlinesandcirclesembeddedinR3.Throughoutthisthesis,weonlyconsiderthoseweaveswithregularprojectionsonthehorizontalplaneP=fx;y;z2R3jz=0gsatisfyinganumberofproperties.Tolisttheseproperties,werestrictourattentiontoweaveswhoseprojectionontoPisa4-valentgraphGwhoseverticesarethecrossingsoftheprojection.Unlessotherwisenoted,wefurtherrestrictattentiontoweaveswithGsatisfying:iGisinnite,planar,connected,andhasnitelymanyverticesinanyball.iiAllconnectedregionsofP)]TJ/F33 11.9552 Tf 11.9552 0 Td[(Garebounded. 1Muchofthecontentofthischapterhasbeenobtainedfrom[8].3 PAGE 10 4iiiEachvertexofGmeetstheclosureoffourdistinctregionsofP)]TJ/F33 11.9552 Tf 11.9552 0 Td[(G.ivTherearenoregionssurroundedbytwoedgeswithonepassingbelowatbothvertices.Thenextpropertyistheonlyonethatisn'trequiredforthemethodsweuse.Werequireittosimplifyourarguments,thoughwewillremarkonhowitcanberelaxed.AbigonisaconnectedregionintheplaneofprojectionPsurroundedbyexactlytwoedges.TwobigonsareisolatediftheyhavedisjointclosuresinP.Wefurtherrequire:vAllbigonsinPareisolated.Aweavesatisfyingtheabovepropertiesiscalledalternatingifallthecrossingsoftheprojectionalternate.Perhapsthesimplestalternatingweave,thesquareweave,isshowninFigure2.1.AweavecomplementisthecomplementofaweaveinR3. Figure2.1:APortionoftheSquareWeave PAGE 11 52.2TopologicalConstructionToconstructthetopologyofanalternatingweavecomplement,rstapplyanambientisotopywhichplacestheweaveclosetotheprojectionplaneP.ThenplaceanidealoctahedronateachcrossingasinFigure2.2.Finally,gluethefacesofeachoctahedrontothefacesofit'sneighboringoctahedraaccordingtotheprescriptioninFigure2.3.Notethateachoctahedronhasfourneighbors,buttheprescriptionforthegluinginFigure2.3showsonlytwoneighbors.ThegluingforthetwoneighborsnotshownisgivenbyrotatingthepictureinFigure2.3upsidedown". Figure2.2:PlacingOctahedraatCrossings Figure2.3:PrescriptionforIdentifyingFaces PAGE 12 6Theorem2.2.1.LetWbeanalternatingweavewithanassociatedcollectionoftopologicalidealoctahedra.IfweidentifytheoctahedraaccordingtoFigure2.3insuchawaythatnotwopointsonthesameedgearegluedtogether,oneobtainsaspacehomeomorphictotheweavecomplementR3)]TJ/F33 11.9552 Tf 11.9551 0 Td[(W.Proof.Withoutlossofgenerality,assumethatWliesentirelyintheslabdenedbyS=fx;y;z2R3:jzj PAGE 13 7 Figure2.4:TruncatedOctahedra PAGE 14 8Itisclearfromthepicturethatwhenthesetruncatedoctahedraaregluedtogether,theyllaportionoftheclosureoftheslabS.Asthislocalpictureisthesamealongeverystrand,thetruncatedoctahedragluetogethertoformaspacehomeomorphictoS)]TJ/F33 11.9552 Tf 10.9904 0 Td[(NW,whereNWSiscomposedofdisjointopentubularneighborhoodsofeverystandinW.Byinspectionofthefacepairing,thecuspscorrespondingtoagivenstrandgluetoformacylinderortorus,dependingonthetypeofstrandcrossanopeninterval.ThisllseachcorrespondingcomponentofNW)]TJ/F33 11.9552 Tf 11.1826 0 Td[(W.Furthermore,theremainingtwocuspscorrespondingtothefacespulledupanddowntotheboundaryofSgluetoformtwoopenballs,llingtheremainderofthecomplementR3)]TJ/F16 11.9552 Tf 13.9518 3.022 Td[(S.Thisshowsthatthegluedidealoctahedracanbedeformed,withoutaectingthetopology,tollR3)]TJ/F33 11.9552 Tf 11.9551 0 Td[(W.Thisprovesthatthesetwospacesarehomeomorphic. TheedgeconditioninthestatementofTheorem2.2.1doesnotlimittheclassofweavestowhichwecanapplythisconstruction.Thatis,allweavecomplementscanbebuiltfromidealoctahedrawiththisconditionsatised.Onewaytodothis,forexample,istousepairwiseisometricregularidealeuclidianoctahedra.Withtheseoctahedraandeuclideanisometriesforgluingmaps,theedgeconditionwillalwaysbesatised.Thencuttingoeuclideanballscenteredattheidealverticeswithaxedbutsucientlysmallsothattheballsaredisjointradiusgivestruncatedoctahedrasatisfying-intheproofofTheorem2.2.1.Onecanthendeformtheoctahedratocompletetheproof.Thisconstructiongivesamethodforbuildingthetopologyofanalternatingweavecomplementwhereeachstrandoftheweaveisidentiedwiththeequivalenceclassofit'sincidentidealvertices.Conceptually,wearellingallofR3abovetheweavebyidentifyingthepairofrededgesinFigure2.5andpullingtheminthepositive PAGE 15 9z-directionwithoutbound.Similarly,identifyingtheindicatedpairofblueedgesandpullingtheminthenegativez-directionllsoutspacebelowtheweave.Finally,identifyingedgesandfacesasinFigure2.3llstheremainderofthecomplement. Figure2.5:FillingtheComplement2.3HyperbolicAlternatingWeavesNowthatwehaveamethodforconstructingthetopologyofanyalternatingweavecomplementusingidealoctahedra,wecanturntohowweendowthesespaceswithahyperbolicstructure,thatisametriconaweavecomplementwhichistopologicallyequivalenttotheeuclideanmetricinthesensethattheidentitymapisahomeomorphism,locallyisometrictothemetricofH3,andcomplete.Thisisjustamatterofrealizingeachidealoctahedronasanidealhyperbolicoctahedron.AhyperbolicpolyhedronisaclosedregionXinH3delimitedbynitelymanyidealpolygons,calleditsfaces.ApolygoninH3isaclosedsubsetFofahyperbolicplaneH3whichisdelimitedinbynitelymanygeodesicscalleditsedges.Werequirethatedgescanonlymeetattheirendpoints,calledvertices,andthateachvertexisadjacenttoexactlytwoedges.Wealsorequirethatfacesonlymeetalongedgesandverticesandthatanedgeisadjacenttoexactlytwofaces.Nowanidealhyperbolicpolyhedronisahyperbolicpolyhedronwithonlyidealvertices,verticesinthesphereatinnity^C=C[f1gforupperhalf-space.Inorderto PAGE 16 10avoidunnecessarycomplicationse.g.twofacestouchingatvinthesphereatinnitywithoutvtheendpointofanedge,wealsorequirethatforeachidealvertexv,thereisasmalleuclideanballBcenteredatvsothatBXisconnected.Toconstructametricsatisfying-above,breakeachidealhyperbolicocta-hedronintofouridealhyperbolictetrahedraalongtherepole"asinFigures2.6and2.7andlabeleachidealtetrahedronwiththeedgeinvariantasindicatedinFigure2.8.SeeAppendixAfordetailsonedgeinvariants.Thefollowingtheoremgivesconditionsunderwhichwehaveametriconanalternatingweavecomplementsatisfyingconditionsandabove.Theorem2.3.1.LetWbeanalternatingweavesatisfyingtheconditionsofSection2.1withassociatedidealhyperbolictetrahedrafTigandfacepairingfromFigure2.3.IfforeachequivalenceclassofedgesinX=[iTi,theassociatededgeinvariantshaveproductoneandargumentsum2,thenthequotientX;dunderthegluingismetric,locallyisometrictoH3,andhomeomorphictoR3)]TJ/F33 11.9552 Tf 12.0542 0 Td[(W.Furthermore,theassociatedhomeomorphisminducesametricdWonR3)]TJ/F33 11.9552 Tf 11.9551 0 Td[(WsuchthatidWistopologicallyequivalenttotheeuclideanmetricofR3)]TJ/F33 11.9552 Tf 11.9551 0 Td[(W,andiiR3)]TJ/F33 11.9552 Tf 11.9552 0 Td[(WwiththemetricdWislocallyisometrictoH3.Proof.ByTheoremB.1.4andandTheoremB.1.7,XismetricandlocallyisometrictoH3.Sincetheedgeinvariantshaveproductofmodulusone,weareguaranteedbyLemmaB.1.1thatnotwoedgepointsaregluedtoeachother.BythesameargumentasinTheorem2.2.1,thequotientXmustbehomeomorphictothecomplementR3)]TJ/F33 11.9552 Tf 11.9551 0 Td[(W.Leth:R3)]TJ/F33 11.9552 Tf 10.92 0 Td[(W!XbetheassociatedhomeomorphismnotethathereR3)]TJ/F33 11.9552 Tf 10.92 0 Td[(Whasthetopologyfromtheeuclideanmetric.ThenhinducesthedesiredmetricdWon PAGE 17 11R3)]TJ/F33 11.9552 Tf 9.3832 0 Td[(WbydeningdWP;P0=dhP;hP0.ThemaphisthenahomeomorphismfromR3)]TJ/F33 11.9552 Tf 12.7714 0 Td[(WwiththeeuclideanmetrictoXandalsoahomeomorphismfromR3)]TJ/F33 11.9552 Tf 13.1891 0 Td[(WwiththemetricdWtoX.Thus,theidentitymapi=h)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1hisahomeomorphismfromR3)]TJ/F33 11.9552 Tf 12.933 0 Td[(WwiththeeuclideanmetrictoR3)]TJ/F33 11.9552 Tf 12.9331 0 Td[(WwiththemetricdW.ByconstructionofdW,thehomeomorphismhisalsoanisometryfromR3)]TJ/F33 11.9552 Tf 12.2289 0 Td[(WwiththemetricdWtoX.SinceXislocalyisometrictoH3,R3)]TJ/F33 11.9552 Tf 12.4894 0 Td[(WwiththemetricdWislocallyisometrictoH3aswell. Figure2.6:DecomposinganIdealOctahedron Figure2.7:Top-DownViewofDecompositonThistheoremshowsthatforthemetricdWonanalternatingweavecomplementR3)]TJ/F33 11.9552 Tf 12.208 0 Td[(Wtosatisfyand2,weneedonlycheckthattheedgeinvariantsaroundeachedgehaveproductoneandargumentsum2. PAGE 18 12 Figure2.8:ParametrizingIdealOctahedraAlongaStrandTocheckthesetwoconditionsalonganequivalenceclassofedges,wecanin-spectthelinksdenitioninSectionA.2ofthecorrespondingidealverticesthataretheendpointsoftherelevantedges.Forexample,Figure2.9showsthelinkscorrespondingtocertainverticesthataregluedtogetherthosepulledupinthepositivez-directionwithoutbound.Theselinksaredetermineduptosimilaritybytheedgeinvariantofthetetrahedrafromwhichthelinkisobtained.Figure2.10showstheselinksarrangedsidebyside.ThetwoconditionsontheparametersfromTheorem2.3.1areequivalenttotheabilitytoarrangetheselinkswhilepreserv-ingsimilarityclassnexttooneanotherwithedgesgluedtogetherconsistentlyandanglesaroundeachvertexaddingto2. Figure2.9:OctahedraWithSomeLinksAsanotherexample,supposeFigure2.11isaportionofthecrossing-indicated PAGE 19 13 Figure2.10:LinksArrangedSideBySideprojectionofanalternatingweavewithindicatedparametersinside"theregion.ThenthearrangementoftherelevantlinksisshowninFigure2.12andtheconditionaroundthisvertexwillread4Yi=1zi=1and4Xi=1argzi=2:Thisappliestoeveryequivalenceclassofedges,andsincetheedgeinvariantsassociatedwitheachsuchequivalenceclassappeartogetherinsomelinkitsucestojustcheckthelinksofequivalenceclassesofverticesgluingoflinksofidealverticesaccordingtofacepairingoftetrahedraforconsistentgluingandanglesaddingto2.Therearethreetypesoftheselinksthatwemustconsider,dependingonthetypeofthecorrespondingequivalenceclassofidealvertices.Thethreetypesofverticesarethosecorrespondingtoastrandoftheweaveallidealverticesincidenttoastrandareidentied,thosepulledinthepositivez-directionwithoutboundandthosepulledinthenegativez-directionwithoutbound.Werefertothesetypes PAGE 20 14 Figure2.11:EdgeInvariantsInsideaRegionoflinksasstrandlinks,thenorthernplane,andthesouthernplane,respectivelytopologically,thestrandlinksarecylinders,andthenorthern/southernplanesareplanes.Wealsorefertotheidealverticesofthecorrespondingequivalenceclassesasstrandvertices,northernvertices,andsouthernvertices,respectively.Figures2.9and2.10correspondtothenorthernplane,andFigures2.13and2.14correspondtoastrandlinkwithparametrizationfromFigure2.8notethatthehyperbolicconditionsassociatedwiththerepoleedgesareincludedintheconditionsfromthestrandlinkinFigure2.14.CompletenessofthequotientXismorediculttocheck,soweproceedwiththespecialcasethatWisadoublyperiodicweave.Oneessentialingredientforcompletenessofthequotientiscompletenessofeverylinkofequivalenceclassofidealvertices,asmadeprecisebyconditionsiandii PAGE 21 15 Figure2.12:ArrangementofLinksFromInsideaRegionofTheoremB.2.2.Forthestrandlinks,thiscompletenessconditionisjustthatthelinkisacompleteEuclideancylinderortorus.TheconditionfortheportionofthelinkofthestrandinFigure2.13tobepartofacompleteEuclideancylinderortorusisequivalenttotheparameterssatisfying)]TJ/F16 11.9552 Tf 15.9216 8.0878 Td[(1 u11 1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(u2=1 1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(v3)]TJ/F16 11.9552 Tf 15.4199 8.0877 Td[(1 v4and1 1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(v1)]TJ/F16 11.9552 Tf 15.42 8.0877 Td[(1 v2=)]TJ/F16 11.9552 Tf 16.7743 8.0877 Td[(1 w31 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(w4:SeethediscussionafterLemmaA.3.2inAppendixAfordetailsontheformulasforcalculatingtherelevantedgeinvariants.Therstconditionensuresthatthequadrilateralinthisportionofthelinkobtainedbyconnectingtheheadsofthe PAGE 22 16 Figure2.13:OctahedrawithCorrespondingStrandLink Figure2.14:APortionofaStrandLinksinglearrowededgesandthenthetailsofthesinglearrowededgesisaparallelo-gram.Similarly,thesecondconditionensuresthatthequadrilateralfromthedoublearrowededgesformsaparallelogram.Ifthesestrandequationsaresatisedalongtheentirestrand,thenthelinkofthatstrandiscomplete.ByCorollaryB.2.4,weneedonlycheckthestrandequationsandthecomplete-nessofthenorthern/southernplanesforadoublyperiodicalternatingweavewithparametersrespectingthisperiodicity.Incheckingthenorthernandsouthernplane,wenotethattherearetwotypesofverticesintheselinks.Thersttypeliesattheintersectionoftwoquadrilateralscomingfromasingleidealoctahedron.Ifthestrandequationsaresatisedalong PAGE 23 17thestrandsoftheweave,thenthehyperbolicconditionsassociatedwiththesetypesofverticesaresatisedforfree.ThesecondtypeiscomposedofthoseverticesthatlieatthecornersofthetriangleswheretheparametersarelabelledasinFigure2.12.Thestrandequationsdonottakecareofthesevertices,however,andwestillmusthavetheproductofallparametersinside"thecorrespondingregionoftheprojectiontomultiplytooneandhaveargumentssumto2.2.4BigonsThetopologicalconstructionabovealsoappliestoalternatingweaveswithbigonregionsintheprojection,buttherewillbeproblemswhentryingtoputacompletehyperbolicstructureonthecomplement.Inparticular,therewillbeanedgeinthequotientwithexactlytwotetrahedrathoseinside"thebigonregiongluedaroundit.Toavoidanydicultiesfromanedgeinvarianthavingargument,weproceedwithaminorchangeintheconstruction.Weplaceanidealoctahedronateachcrossingexceptforthosecrossingsincidentwithabigon.Atcrossingsincidentwithabigon,placeanidealoctahedronwithoneidealtetrahedronremoved.Thisplacement,alongwiththeassociatededgeinvariants,isshowninFigure2.15.Recallthatallbigonsareisolated,sothatweareremovingatmostonetetrahedronfromeachoctahedron.ThefacepairingsofthesenewshapesarethengivenbyFigure2.16,whileallotherfacepairingsremainthesameasintheconstructionfromtheprevioussections.TherelevantportionofthelinksfromthetwotwistedstrandsaredepictedinFigure2.17.ByinspectionofFigure2.17,thestrandequationsbecomez11z21z31z12z22z32=1;1 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(z11=1 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(z12)]TJ/F16 11.9552 Tf 9.2985 0 Td[(1 z11z21z31; PAGE 24 18and1 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(z21=1 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(z22)]TJ/F16 11.9552 Tf 9.2985 0 Td[(1 z11z21z31:Theargumentsofthefactorsintherstequationalsomustsumto2.Allotherstrandequationsremainthesame.TheportionofthenorthernplanecorrespondingtothebigonregionisdepictedinFigure2.18.Wecanseethattheusualconditionsoftheinside"parametersmultiplyingto1applyinallregionsexcept,ofcourse,thebigonregion.Justasinthebigon-freecase,thetworemainingconditionsfromFigure2.18aresatisedifthestrandequationsaresatised.ThecasewithbigonsthatarenotisolatedcanbehandledsimilarlybyplacingidealoctahedrawithcertaintetrahedraremovedasshowninFigure2.19.Thecor-respondingpairingoffacesisthengiveninFigure2.20. Figure2.15:PlacementandParametrizationforaBigonRegion PAGE 25 19 Figure2.16:FacePairingforaBigonRegion Figure2.17:PortionsofLinksfromBigonStrands PAGE 26 20 Figure2.18:PortionofNorthernPlanefromBigonRegion Figure2.19:PlacementforAdjacentBigonRegions Figure2.20:FacePairingforAdjacentBigonRegions PAGE 27 CHAPTERIIIAugmentedAlternatingWeavesInthischapterweexpandtheclassofexamplesatourdisposalbytakinganalternatingweaveandaddingcircleswhoseinteriorintersectsexactlytwostrandsoftheweave.Firstwedescribethelocalpictureofaugmentinganalternatingweavewithasinglecirclearoundacrossing.Thenwederiveconditionsunderwhichanon-alternatingdoublyperiodicaugmentedalternatingweaveadmitsacompletehyperbolicstructure.Thenweremarkonhowtogeneralizetothecasewithbigons.3.1DenitionsSupposeWisanalternatingweavesatisfyingthepropertiesfromthepreviouschapterbutwithnobigonsintheprojectionplaneP.LetC1;C2;:::benonisotopicembedded1-spheresinR3)]TJ/F33 11.9552 Tf 11.7169 0 Td[(WsuchthateachCiintersectsPinexactlytwopointswhereeachofthetwointersectionsoccurinadierentregionoftheprojectionplaneandsuchthateachCiboundsaverticaldiskDiinR3whereiDiDj=;fori6=j,iiEachDiintersectsWinexactlytwopoints.ThenwecallW[SiCianaugmentedalternatingweave.Figure3.1showsanaugmentedsquareweave.21 PAGE 28 22 Figure3.1:AnAugmentedSquareWeave3.2TopologicalConstructionLetWbeanaugmentedalternatingweaveobtainedbyaddingasinglecircleCaroundacrossingofWfreeofbigonsasinFigure3.2.Asdonepreviously,weconstructthetopologyofR3)]TJ/F33 11.9552 Tf 12.6024 0 Td[(Wusingidealpolyhedra.ProceedbyplacinganidealoctahedronateachcrossingofWasinFigure2.2.AfacepairingisthengivenfromFigure2.3,butbeforeidentifyingfaceswemustrsttakeintoaccounttheaugmentingcircle. Figure3.2:AddingaCircleAroundaCrossing.ConsidertheoctahedronOatthecrossingofWwhereCisadded.Sincetheredandblueedgesgetidentiedaboveandbelowthecorrespondingstrands,drilling PAGE 29 23outthesegmentsofOinFigure3.3istopologicallyequivalenttodrillingoutCinthecomplementofW. Figure3.3:DrillingOutanAugmentingCircleWedesireadecompositionofOandit'sneighborsinFigure2.2intoidealtetrahedraandacorrespondingfacepairingwithneighboringoctahedrawhereCisidentiedtoanequivalenceclassofidealvertices.Toobtainsuchadecomposition,collapseCtotwoidealverticesasinFigure3.4. Figure3.4:CollapsinganAugmentingCircleTheresultingshapecontainstwothricepuncturedspheresthatcanbeattenedintoidealtriangleswithoutchangingthetopologyoftheweavecomplement.TheseidealtrianglescanthenberemovedasinFigure3.5becausethefacesoftheneigh-boringoctahedrathatarepairedtoit'soppositesidescaninsteadbepairedtoeachother.Wecalltheresultingidealpolyhedronastar. PAGE 30 24 Figure3.5:RemovingIdealTrianglesConnectingtheidealverticescorrespondingtoCandtriangulatingthesurfaceofthestarwiththecolorededgesofFigure3.6,weobtainthedesireddecompositionofthestarintosixidealtetrahedra. Figure3.6:AMoreSymmetricStarSincewedrilledpointsoutoftheredandblueedgesinFigure3.3,wemustremovethecorrespondingpointsfromtheneighboringoctahedraaswell.Toaccountforthesenewidealvertices,theneighboringoctahedracanbedecomposedasinFigure3.7.Finally,thestarandit'sneighboringoctahedrainheritafacepairingfromthefacepairingoftheoriginalidealoctahedraofW.ThisisshowninFigure3.8. PAGE 31 25 Figure3.7:DecomposingtheNeighborsofaStarToextendthisconstructiontoanaugmentingcirclenotaroundacrossing,therearetwocasestobeconsidered.IfthecircleisofthetypeinFigure3.9left,werstnotethatthistypecanbeobtainedbycuttingalongthediskboundedbyanaugmentingcircleofFigure3.2,performingahalf-twist,andreidentifying.Inourconstruction,thiscorrespondstocuttingalongthetwicepunctureddiskinFigure3.10,performingahalftwist,andreidentifying.Thus,toextendtheconstructiontothiscaseonesimplyhastoaddacrossingintheoriginalweave,followtheaboveconstructionwiththenewcrossingaugmented,andperformahalf-twistofthetwicepunctureddiskinthestarpriortoidentication.IfthecircleisofthetypeinFigure3.9right,thenaseparatedecompositionisrequiredwhichwewillnotdiscuss.Thisdecompositionwillalsoworkforanynumberofaugmentingcirclesthataresucientlyfarapart"intheweave.Sucientlyfarapartmeansthatthedecom-positionoftheneighborstoastarwon'tinterferewiththedecompositionoftheneighborsofanotherstar.Alternatingweaveswithaugmentingcirclesthataretooclosetogetherrequireamoreinvolveddecompositionwhichwewillnotdiscuss. PAGE 32 26 Figure3.8:FacePairingofaStarandIt'sNeighbors PAGE 33 27 Figure3.9:TheTwoTypesofAugmentingCircleNotAroundaCrossing Figure3.10:TwicePuncturedDiskRedinaStar3.3HyperbolicAugmentedAlternatingWeavesTheproofofTheoremB.1.7carriesdirectlyovertoaugmentedalternatingweavessothatwehavethesameconditionsonthetetrahedrathatensurewhenthecomple-mentofanaugmentedalternatingweaveWadmitsametricdWsatisfyingiandiiofTheorem2.3.1.TheproofofTheoremsB.2.2andB.2.4alsocarryover,sothatwegetthesameconditionsensuringcompletenessaswell.Todescribetheseconditionsmoreexplicitly,weproceedbyrepeatingtheanalysisfromthepreviouschapter.Inordertolistalltheconditionstogether,werestricttothecasethattheweaveisadoublyperiodicaugmentedalternatingweavewith PAGE 34 28augmentingcirclessucientlyfarapart.Beginbyassigningedgeinvariantstoeachtetrahedroninthestar,andtheNorth,South,EastandWestneighborsaccordingtothenumberinginFigures3.11and3.12replacingzwithnfortheNorthneighbor,etc.. Figure3.11:EdgeInvariantsofaStarTheconditionsaroundtheedgesinsideouridealpolyhedrathegreenedgeinFigure3.6,forexampleareY1i6ci=1;Y1i3ni=1;Y3i6ni=1;Y1i3si=1;Y3i6si=1;Y1i3ei=1;Y3i6ei=1;Y1i3wi=1;Y3i6wi=1:Figure3.13showstherelevantportionofthelinksofthetwocenter"strandsincidentwiththestar.Fromthesediagramswecanobtaintherelevanthyperbolic PAGE 35 29 Figure3.12:EdgeInvariantsofaNeighbortoaStarconditionsfortheselinks.Theyaren4=c2;s4=c1;e4=c5;w4=c4:IfthetetrahedraalongtheNorthern"strandarelabelledasinFigure3.14,thenFigure3.15showstherelevantportionofthelinkfromthisstrand.Thecorrespondingconditionsarethesameasthosefromthepreviouschapterwithoneparametermodied.ByinspectionofFigure3.15,therelevantconditionsfromthisstrandare)]TJ/F16 11.9552 Tf 15.9216 8.0878 Td[(1 u11 1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(u2=1 1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(n2[)]TJ/F16 11.9552 Tf 16.0841 8.0877 Td[(1 n31 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(n6];[)]TJ/F16 11.9552 Tf 16.0841 8.0877 Td[(1 n61 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(n3])]TJ/F16 11.9552 Tf 16.0841 8.0878 Td[(1 n1=)]TJ/F16 11.9552 Tf 15.42 8.0878 Td[(1 v21 1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(v1:Figure3.16showsthelinkassociatedwiththeaugmentingcircle.Byinspection,thecorrespondingconditionsare PAGE 36 30 Figure3.13:PortionsofLinksfromOverandUnderStrands)]TJ/F16 11.9552 Tf 16.0842 8.0878 Td[(1 n5)]TJ/F16 11.9552 Tf 15.1093 8.0878 Td[(1 c31 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(c22=1;)]TJ/F16 11.9552 Tf 15.3473 8.0878 Td[(1 s5)]TJ/F16 11.9552 Tf 15.1093 8.0878 Td[(1 c61 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(c12=1;)]TJ/F16 11.9552 Tf 15.303 8.0878 Td[(1 e5)]TJ/F16 11.9552 Tf 15.1094 8.0878 Td[(1 c61 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(c52=1;)]TJ/F16 11.9552 Tf 16.7743 8.0878 Td[(1 w5)]TJ/F16 11.9552 Tf 15.1094 8.0877 Td[(1 c31 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(c42=1;andn5s5c1c2=e5w5c4c5:Ifthetetrahedrainside"thefourregionsadjacenttotheaugmentedcrossingarelabelledasinFigure3.17,thenFigure3.18showstherelevantportionofnorthernplane.Byinspection,thecorrespondingconditionsare PAGE 37 31 Figure3.14:ParametrizationAlongtheNorthern"NeighboringStrand Figure3.15:PortionofLinkfromtheNorthern"NeighboringStrand)]TJ/F16 11.9552 Tf 16.0842 8.0878 Td[(1 n51 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(n6)]TJ/F16 11.9552 Tf 15.3031 8.0878 Td[(1 e51 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(e6kYi=1ri=1;1 1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(w5)]TJ/F16 11.9552 Tf 27.0067 8.0878 Td[(1 1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(w6c31 1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(n5)]TJ/F16 11.9552 Tf 16.0841 8.0878 Td[(1 n6`Yi=1ti=1;)]TJ/F16 11.9552 Tf 15.3473 8.0878 Td[(1 s51 1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(s6)]TJ/F16 11.9552 Tf 16.7743 8.0878 Td[(1 w51 1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(w6mYi=1ui=1;and1 1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(e5)]TJ/F16 11.9552 Tf 25.5354 8.0878 Td[(1 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(e6c61 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(s5)]TJ/F16 11.9552 Tf 15.3473 8.0878 Td[(1 s6nYi=1vi=1: PAGE 38 32Theconditionsfromthesouthernplaneareredundant,andallotherconditionsontheparametersremainthesameasinthepreviouschapter.Furthermore,thefactorsinallproductsabovemultiplyingtoonemusthaveargumentssumto2. Figure3.16:LinkofanAugmentingCircle Figure3.17:ParametrizationoftheInner"RegionsofanAugmentedCrossing PAGE 39 33 Figure3.18:PortionofNorthernPlanefromAugmentedCrossing PAGE 40 343.4BigonsToextendtheconstructiontothecaseofanaugmentedalternatingweavewithisolatedbigons,weconsidertwocases.ForanaugmentingcircleofthetypeinFigure3.19,themethodsdescribedintheprevioussectionsapplycrossthepiercingstrands,applyconstruction,performhalf-twistpriortoidentication.Foranaugmenting Figure3.19:AugmentingCirclewithBigoncirclearoundabigonasinFigure3.20,notethatintheconstructionfromtheprevioussections,performingafulltwistofthetwicepunctureddiskinastarresultsinamanifoldhomeomorphictotheoriginal.Therefore,wecanuntwistthestrandspiercingtheaugmentingcirclewithoutchangingthetopologyofthecomplementsothattheaugmentingcircleisofatypealreadyconsidered.Augmentingcirclesofthe Figure3.20:AugmentingCircleAroundaBigontypesshowninFigure3.21requireaseparatedecompositionandarenotconsideredhere. PAGE 41 35 Figure3.21:AugmentingCirclesnotConsideredWenotethatthesameargumentsusedforisolatedbigonsapplytothecaseofanaugmentingcirclearoundasinglestrandfromabigoninseries. PAGE 42 CHAPTERIVConclusionInthisthesiswehavedescribedaconstructionthatgivesametricdWonanalternatingweavecomplementR3)]TJ/F33 11.9552 Tf 11.9552 0 Td[(WsuchthatidWistopologicallyequivalenttotheeuclideanmetricofR3)]TJ/F33 11.9552 Tf 11.9551 0 Td[(W,iiR3)]TJ/F33 11.9552 Tf 11.9551 0 Td[(WwiththemetricdWislocallyisometrictoH3,andiiidWiscomplete.Thenweextendedourresultstocertainnon-alternatingaugmentedalternatingweaves.Thereareanumberofpossibleavenuesforfurtherrenementanddevelopmentofthesetopics.WeconjecturethatconditioniiiinTheoremB.2.2canbeweakenedbutcannotbeeliminatedwithoutsomeconditiontoreplaceit.Thoughitiseasyenoughtocheckfordoublyperiodicweaves,itwouldcertainlybedesirabletohaveamorenaturalreplacementconditionthatiseasiertocheckforensuringcompletenessofdWonanarbitraryweavecomplement.Nowhereinthisthesishavewementionedamethodforconstructinganexplicithyperbolicstructureintermsofacollectionofedgeinvariantsonagivenweavecomplement.Furtherworkinthisdirectionwouldshedlightonthequestionofwhichweavecomplementsadmitahyperbolicstructure.36 PAGE 43 37Itisstillanopenquestionwhethertheweavecomplementsadmittingahyperbolicstructurearerigid,thatiswhethertherearemultipleisometricallydistincthyperbolicstructuresonagivenweavecomplement. PAGE 44 APPENDICES38 PAGE 45 39APPENDIXAParameterizaitonofIdealHyperbolicTetrahedraThisappendix1introducestheparameterizationofidealhyperbolictetrahedraviacomplexnumbers.Weusethefactsthatorientation-preservingsimilaritiesofCarefunctionsoftheformfz=az+bwherea;b2Canda6=0.Unlessotherwisenoted,weworkintheupperhalf-spacemodelofH3'CR+andwrite^C=C[f1g.A.1Linear/AntilinearFractionalMapsandHyperbolicIsometriesWebeginbyrecordingaresultwhichrelatestheisometriesofhyperbolicspaceH3tothelinearfractionalmaps:^C!^C,z=az+b cz+d,andantilinearfractionalmaps:^C!^C,z=az+b cz+donitsboundary^C.TheoremA.1.1.Everylinearorantilinearfractionalmap:^C!^Chasauniquecontinuousextension^:H3[^C!H3[^CwhoserestrictiontoH3isanisometryofH3.Furthermore,everyisometryofH3isobtainedinthisway.Proof.SeeBonahon[4]. TheproofofTheoremA.1.1reliesonthefactthateverylinearorantilinearfractionalmapcanbewrittenasthecompositionofreectionsinnitelymanyoflinesandcirclesin^C,andeachofthesefactorsextendstoareectioninasphereor 1MuchofthecontentofthisappendixhasbeenobtainedfromRatclie[10]. PAGE 46 40planeorthogonaltoC.ThenthereectionsinthesespheresrestricttoisometriesofH3.A.2IdealHyperbolicTetrahedraSupposeTisanidealtetrahedroninH3andletbeahorospherebasedatanidealvertexofTthatdoesnotintersecttheoppositesideofTahorospherebasedatanidealvertexv2CisaeuclideansphereinH3tangenttoCatvorifv=1aplaneinH3paralleltoC.ThenLv=TisaEuclideantriangle,calledthelinkofvinT.AscanbeseenfromFigureA.1,theorientationpreservingsimilarityclassofLvdoesnotdependonthechoiceof. FigureA.1:AnIdealTetrahedroninUpperHalfSpaceTheoremA.2.1.TheorientationpreservingsimilarityclassofthelinkLvofavertexvofanidealtetrahedronTinH3determinesTuptoorientationpreserving PAGE 47 41isometry.Proof.Weassume,withoutlossofgenerality,thatv=1.ThentheotherthreeverticesofTformatriangleinCthatisintheorientationpreservingsimilarityclassofLv.SeeFigureA.1.SupposeT0isanotheridealtetrahedroninH3,withavertexv0suchthatLvissimilartoLv0.Afterarrangingforv0=1,thereisanorientation-preservingsimilarityfofCsendingthetrianglefromTtothetrianglefromT0.Thenfisoftheformfz=az+b,andextendstocontinuouslytoamap^f:H3[^C!H3[^Cwhichrestrictstoanorientation-preservingisometryofH3.Bycontinuityof^fandthefactthatitsrestrictiontoH3isanisometry,wehavethat^fsendsTtoT0,andthatthesetwotetrahedraareisometric. FigureA.2:TheDihedralAnglesofaTetrahedronTheoremA.2.2.LetTbeanidealtetrahedroninH3.ThenTisdetermined,uptoisometry,bythethreedihedralangles;;oftheedgesincidenttoavertexofT.Moreover,++=andthedihedralanglesofoppositeedgesareequal.Furthermore,if;;arepositive,realnumberssuchthat++=,thenthere PAGE 48 42isanidealtetrahedroninH3whosedihedralanglesare;;.Proof.LetvbeanidealvertexofT.Bytheprevioustheorem,Tisdetermined,uptoisometry,bythesimilarityclassofLv,whichinturnisdeterminedbythedihedralangles;;oftheedgesofTincidenttov.ToseethatthedihedralanglesoftheoppositesidesofTareequal,labelthedihedralanglesofTasinFigureA.2.Thenwehave++=+0+0=0++0=0+0+=:Byaddingthersttwoandthelasttwoequations,weobtain2++0++0=20++0++0=:Therefore,=0.Thesameargumentshowsthat=0and=0.Let;;bepositiverealnumberssuchthat++=.Thenthereisatriangle4inCwithangles;;.LetTbetheidealtetrahedroninH3whoseverticesaretheverticesof4and1.Thenthelinkof1inTissimilarto4.Hence,TisanidealtetrahedroninH3whosedihedralanglesare;;. ItfollowsthattheorientationpreservingsimilarityclassofthelinkLvofavertexvofTdoesnotdependonthechoiceofv.AgeometricexplanationofthisfactisthatthegroupoforientationpreservingsymmetriesofTactstransitivelyonthesetofverticesofT. PAGE 49 43A.3ParameterizationofEuclideanTrianglesinCWebegintheparameterizationofidealhyperbolictetrahedrabyrstparameter-izingEuclideantrianglesinC.Let4u;v;wbeaEuclideantriangleinthecomplexplaneCwithverticesu;v;wlabelledcounterclockwisearound4.Toeachvertexof4weassociatetheratioofthesidesadjacenttothevertexinthefollowingmanner.zu=w)]TJ/F33 11.9552 Tf 11.9552 0 Td[(u v)]TJ/F33 11.9552 Tf 11.9551 0 Td[(u;zv=u)]TJ/F33 11.9552 Tf 11.9552 0 Td[(v w)]TJ/F33 11.9552 Tf 11.9552 0 Td[(v;zw=v)]TJ/F33 11.9552 Tf 11.9551 0 Td[(w u)]TJ/F33 11.9552 Tf 11.9551 0 Td[(w:Thecomplexnumberszu,zvandzwarecalledthevertexinvariantsofthetriangle4u;v;w.SeeFigureA.3. FigureA.3:TheVertexInvariantzuoftheTriangle4u;v;wLemmaA.3.1.Thevertexinvariantszu,zvandzwdependonlyontheori-entationpreservingsimilarityclassofthetriangle4u;v;w.Proof.AnarbitraryorientationpreservingsimilarityofCisoftheformx7!ax+bwitha6=0.Observethatzau+b=aw+b)]TJ/F16 11.9552 Tf 11.9552 0 Td[(au+b av+b)]TJ/F16 11.9552 Tf 11.9552 0 Td[(au+b=aw)]TJ/F33 11.9552 Tf 11.9552 0 Td[(u av)]TJ/F33 11.9552 Tf 11.9551 0 Td[(u=zu: LemmaA.3.2.Letzubeavertexinvariantofatriangle4u;v;w.ThenImzu>0andargzuistheangleof4u;v;watu. PAGE 50 44Proof.DeneasimilarityofCbyx=x v)]TJ/F33 11.9552 Tf 11.9552 0 Td[(u)]TJ/F33 11.9552 Tf 23.5007 8.0878 Td[(u v)]TJ/F33 11.9552 Tf 11.9552 0 Td[(u:Thenu=0,v=1,andw=zu.Aspreservesorientation,thetriangle4;1;zuislabelledcounterclockwise.SeeFigureA.4.HenceImzu>0,andargzuistheangleof4;1;zuatu. FigureA.4:TheTriangle4;1;zuItisevidentfromFigureA.4thatzudeterminestheorientationpreservingsimilarityclassof4u;v;w.Consequentlyzudetermineszvandzw.ByLemmaA.3.1,wecancalculatezvandzwfromthetriangle4;1;zu.Thisgivestherelationshipszv=1 1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(zu;zw=1)]TJ/F16 11.9552 Tf 21.093 8.0877 Td[(1 zu:TheoremA.3.3.Let4u;v;wbeaEuclideantriangleinC,withverticeslabelledcounterclockwiseandletz1=zu,z2=zv,andz3=zwbeitsvertexinvariants.Thenz1;z2;z3satisfytheequationsiz1z2z3=)]TJ/F16 11.9552 Tf 9.2985 0 Td[(1,andii1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(z2+z1z2=0. PAGE 51 45Conversely,ifz1;z2;z3areinCwithImzi>0andsatisfyiandii,thenthereisaEuclideantriangle4inCthatisuniqueuptoorientationpreservingsimilaritywhosevertexinvariantsareincounterclockwiseorderz1;z2;z3.Proof.Bythetwopreviousformulas,wehavez1z2z3=z11 1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(z1)]TJ/F16 11.9552 Tf 15.3093 8.0877 Td[(1 z1=)]TJ/F16 11.9552 Tf 9.2985 0 Td[(1:Asz2=1 1)]TJ/F34 7.9701 Tf 6.5865 0 Td[(z1,wehavez2)]TJ/F33 11.9552 Tf 12.5745 0 Td[(z1z2=1.Conversely,supposethatz1;z2;z3areinCwithImzi>0andsatisfyequationsiandii.Thenthevertexinvariantsof4;1;z1arez1;z2;z3. A.4ParameterizationofIdealHyperbolicTetrahedraWecannowparametrizetheidealtetrahedrainH3.LetvbeavertexofanidealtetrahedroninH3,withthecorrespondingvertexinvariantsz1;z2;z3ofthelinkofv.ThenoppositeedgesofThavethesamelabel.Thethreeparametersz1;z2;z3areindexedaccordingtotheright-handrulewithyourthumbpointingtowardsavertexofT.SeeFigureA.5.Thecomplexparametersz1;z2;z3arecalledtheedgeinvariantsofT.TheoremA.4.1.Letz1;z2;z3becomplexnumberswithImzi>0satisfyingz1z2z3=)]TJ/F16 11.9552 Tf 9.2985 0 Td[(1;and1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(z2+z1z2=0:ThenthereisanidealtetrahedroninH3,uniqueuptoorientationpreservingisom-etry,whoseedgeinvariants,inright-handorder,arez1;z2;z3.Proof.ThisfollowsimmediatelyfromTheoremsA.2.1andA.3.3. PAGE 52 46 FigureA.5:TheEdgeInvariantsofanIdealTetrahedronAsnotedabove,onecancalculatethevertexinvariantsofatriangleinCpro-videdthatoneisalreadygiven.Itthenfollowsthatoneneedonlyspecifyasingleedgeinvariantofanidealtetrahedroninordertodeterminethetetrahedronuptoorientationpreservingisometry.Thisjustiesouruseofasingleedgeinvariantforeachtetrahedroninthedecompositionofaweavecomplement. PAGE 53 47APPENDIXBHyperbolicGluingandCompletenessWehaveseenaconstructionwhichbuildsthetopologyofaweavecomplementfromtopologicalidealoctahedra.Wewishtoendowtheweavecomplementwithamet-ricwhichistopologicallyequivalenttotheeuclideanmetricinthesensethattheidentitymapisahomeomorphismandlocallyisometrictothemetricofhyperbolicspace.Thisisdonebybuildingtheweavecomplementfromgeometric,i.e.hyper-bolic,idealoctahedra.InSectionB.11,wediscusstheconditionsontheoctahedrawhichensurethatourconstructiongivesametricwiththesetwoproperties.InSec-tionB.2wegiveconditionswhichensurethemetricarisingfromourconstructioniscomplete.Weworkintheupperhalf-spacemodelofH3throughoutanddenotethehyperbolicmetricbydh.Furthermore,weusethefactthatifz1;z2;z3andw1;w2;w3aretriplesofdistinctpointsin^C,thenthereisauniquelinearantilin-earfractionalmapoftheformz=az+b cz+dz=az+b cz+dsuchthatzi=wi,for1i3.B.1HyperbolicGluingBeginwithanalternating2weavesatisfyingthepropertiesofsection2.1,andassociateahyperbolicidealoctahedronwitheachcrossingoftheprojection.De1ManyargumentsweretakenfromBonahon[4],whichdealswithgluingtheedgesofasinglepolygon.2Ourargumentsworkforanaugmentedalternatingweavealso,butforthesakeofconcretenessweassumeourweavealternates. PAGE 54 48composingeachoctahedronintofourtetrahedragivesacollectionfTigofdisjointhyperbolicidealtetrahedraafterseparatingthetetrahedrawithacorrespondingfacepairingfromthetopologicalconstructionandtheoctahedraldecompositions.Inordertodeneametricontheweavecomplement,rstdeneanextendedmetricdametricwhichtakesvaluesintheextendedrealnumbersonX=STi.ForpointsP;QinthesametetrahedronT,denedP;QasthehyperbolicdistancebetweenPandQinH3.ForP;Qindierenttetrahedra,denedP;Q=1.Thefactthatdisanextendedmetricisimmediate.NextindexthefacesofthetetrahedrasotheycanbegroupedtogetherintopairsfF1;F2g;fF3;F4g;:::consistentwiththefacepairingabove.Sinceeachfaceisanidealtriangleahyperbolictrianglewithverticesatinnityandidealtrianglesareisometricinhyperbolicspace,thereisanorientation-preservingisometry2k)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1:F2k)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1!F2kassociatedwitheachpairfF2k)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1;F2kg.Bydening2kby2k=)]TJ/F31 7.9701 Tf 6.5865 0 Td[(12k)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1,wehavethateveryfaceFiisgluedtoafaceFi1byanorientationpreservingisometryi:Fi!Fi1wherethe1dependsontheparityofi.DeneXtobethequotientunderthisgluing.Moreprecisely,denetheelementsP2Mbythefollowing:IfPisintheinteriorofatetrahedronT,thenPisnotgluedtoanyotherpointsandP=fPg.IfPisinafaceFiandnotanedge,thenPconsistsofthepointsP2FiandiP2Fi1.IfPisinanedge,thenPconsistsofallpointsoftheformikik)]TJ/F32 5.9776 Tf 5.7561 0 Td[(1i1Pwheretheindicesi1;:::;ikaresuchthatij)]TJ/F32 5.9776 Tf 5.7562 0 Td[(1i1P2Fijforeveryj.Whileweknowfromtheconstructionthateachedgeisgluedtoonlynitelymanyotheredges,weneedmoreinformationtoensurethatPisniteinthislastcaseaswell.LemmaB.1.1.SupposePisapointontheedgeEinX.Ifz1;:::;znaretheedgeinvariantsoftheedgesgluedtoE,thenPisniteifandonlyifjz1znj=1. PAGE 55 49Proof.FirstlabeltheedgesgluedtoEbyE1;E2;:::Enwherethetetrahedracon-tainingtheedgesEiandEi+1havepairedfacestakingindicesmodn.ThenletzibetheedgeinvariantcorrespondingtoEi.Arrangethersttetrahedroncon-tainingE1inupperhalf-spacesothatit'sidealverticesare0;1;1;z1.ThenplacethetetrahedroncontainingE2besidetherst,asinFigureB.1,sothattheidealverticesofthesecondtetrahedronare0;1;z1;z1z2andthegluingmap1betweenthesetwotetrahedraistheidentity.Arrangingtherestofthetetrahedrainthisfashion,wegetthatthenthtetrahedronhasidealvertices0;1;Qn)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1j=1zj;Qnj=1zjandtheisometryifor1i PAGE 56 50 FigureB.1:ArrangingTetrahedraSideBySidethesum`dw=nXi=1dPi;Qi:WewouldliketodeneadistancefunctiondbydP;Q=inff`dw:wadiscretewalkfromPtoQgforanytwopointsP;Q2X.LemmaB.1.2.ThenumberdP;QisindependentofthechoiceofPandQusedtorepresentPandQ,anddiswelldened.Proof.ConsidertwopointsP0andQ0suchthatP0=PandQ0=Q.WeneedtoshowthatdP0;Q0=dP;Q.IfwisadiscretewalkP=P1;Q1P2;Q2P3;:::;Qn)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Pn;Qn=Q,fromPtoQ,wecanconsideranotherdiscretewalkw0oftheformP0=P0;Q0P1;Q1P2;Q2P3;:::;Qn)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Pn;QnPn+1;Qn+1=Q0bytakingP0=Q0=P0andPn+1=Qn+1=Q0.Thisnewdiscretewalkw0startsatP0,endsatQ0,andhasthesamed-length`dw0=`dwasw.Takingtheinmumoverallsuchdiscretewalksw,weconcludethatdP0;Q0,denedusingP0andQ0islessthanorequalto PAGE 57 51dP;QusingPandQ.ExchangingtherolesofP;QandP0;Q0,wesimilarlyobtainthatdP;QdP0;Q0,andsodP;Q=dP0;Q0. NotethattheabovenumberdP;Qisnite,becauseforeverypairofpointsP;Q2TthereisadiscretewalkwdenedbyP=P1;Q1P2;Q2P3;:::;Qn)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Pn;Qn=QwithPiandQiinthesametetrahedronforeachi.Thed-length`dwisthenanitesumofniterealnumbers,andhenceisnite.Inordertoshowthatd:XX!Risinfactametric,werstprovealemmawhichstatesthatforsucientlysmall,theballBdP;isexactlytheunionoftheimagesunderoftheballsBdP0;asP0rangesoverallpointsofP.Here:X!Xisdened,asusual,byP=P.LemmaB.1.3.SupposetheequivalentconditionsinLemmaB.1.1holdforeveryequivalenceclassofedgesinX.ThenforeveryP2X,thereisan0>0suchthatforevery<0andeveryQ2X,thepointQ2XisintheballBdP;ifandonlyifthereisaP02PsuchthatdP0;Q<.Proof.Toprovetheif"partofthestatement,observethatdP;QdP;QforallP;Q2X.Toseethis,letwbethediscretewalkfromPtoQdenedbyP=P1;Q1=Q.Thenbythedenitionofd,dP;Q`dw=dP;Q.Thus,theif"partholdswithoutrestrictionon.Asfortheonlyif"partofthestatement,supposeP2X.Foranumber0whichwewillspecifydependingonthetypeofP,thatiswhethertheelementsofPareinterior,face,oredgepoints,weconsideranypointQ2XwithdP;Q<0.WewanttondapointP02PsuchthatdP0;Q<.SincedP;Q<,thereisadiscretewalkwfromPtoQoftheformP=P1;Q1P2;Q2P3;:::;Qn)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Pn;Qn=Qandwhosed-lengthissuchthat PAGE 58 52`dw=Pni=1dPi;Qi<.Wewanttoprovebyinductionthatforeveryjn,B.1thereexistsP02PsuchthatdP0;QjPji=1dPi;Qi<:Ifwedothis,thecasej=nwillprovethelemma,sinceQn=Q.Wecanbegintheinductionwithj=1,inwhichcaseB.1istrivialbytakingP0=P.SupposeasaninductionhypothesisthatB.1holdsforj.Wewanttoshowthatitholdsforj+1.Forthis,wewilldistinguishcasesaccordingtothetypeofthepointP2X.Wewillalsospecify0ineachcase.Case1:PisintheinteriorofatetrahedroninX.Werstspecifythenumber0neededinthiscase.Wechooseitsothatthecloseddiskofradius0centeredatPiscompletelycontainedintheinteriorofthetetrahedron.Inthiscase,PistheonlypointofP.BytheinductionhypothesisB.1andbychoiceof0>,thepointQjisintheinteriorofthetetrahedron.Inparticular,itisgluedtonootherpointssothatPj+1=Qj.CombiningtheTriangleInequalitywiththeinductionhypothesisis,weconcludethatdP;Qj+1dP;Qj+dPj+1;Qj+1Pj+1i=1dPi;Qi<.ThisprovesB.1forj+1.Case2:PisinafaceFiandnotanedgeofatetrahedroninX.Inthiscase,PconsistsofPandexactlyoneotherpointiPinthefaceFi1thatisgluedtoFi.Choose1sothatPisatdistance>1fromanyfaceotherthanthefaceFithatcontainsit.Similarly,let2besuchthatiPisatdistance>2fromanyfaceotherthanthefaceFi1thatcontainsit.Choose0asthesmallerof1and2. PAGE 59 53IfQj=Pj+1,combiningtheinductionhypothesisB.1withtheTriangleInequal-itygives,asinthecaseofinteriorpoints,dP0;Qj+1dP0;Qj+dPj+1;Qj+1j+1Xi=1dPi;Qi<;whichprovesB.1forj+1inthiscase.OtherwiseQjandPj+1aredistinctbutgluedtogether.BecausedP0;Qj<0andbychoiceof0,thesetwopointscannotbeedgepoints,sothatoneofthemisinthefaceFiandtheotheroneisinthefaceFi1gluedtoFibythemapi.Inparticular,Pj+1=1iQj.SetP00=1iP0.NotethatP00isjustequaltoPoriP;inparticularitisinP.Sinceiisanisometryandrespectsdistances,dP00;Pj+1=dP0;Qj.WethenhavethatdP00;Qj+1dP00;Pj+1+dPj+1;Qj+1dP0;Qj+dPj+1;Qj+1Pj+1i=1dPi;Qi PAGE 60 54compositionoffacepairingisometriesik:::i1sendingtheedgeEtotheedgecontainingPj+1,andsendingQjtoPj+1.ThenP00=ik:::i1P02Pisdened.Sincethismappreservesdistances,wehavethatdP00;Pj+1=dP0;QjanddP00;Qj+1dP00;Pj+1+dPj+1;Qj+1dP0;Qj+dPj+1;Qj+1Pj+1i=1dPi;Qi PAGE 61 55Let0beassociatedtoPbyLemmaB.1.3.SincePandQareniteanddisjoint,thereisan1>0suchthateverypointofPisatadistance>fromeverypointofQ.Set=min0;1.ThendP;Q>0.Indeed,LemmaB.1.3wouldotherwiseprovideapointP02PsuchthatdP0;Q<1,therebycontradictingthedenitionof1.Thisprovesi.Asforii,notethateverydiscretewalkfromPtoQoftheformP=P1;Q1P2;Q2P3;:::;Qn)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Pn;Qn=QprovidesadiscretewalkQ=Qn;PnQn)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1;Pn)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Qn)]TJ/F31 7.9701 Tf 6.5866 0 Td[(2;:::;P2Q1;P1=P.Sincethesetwodiscretewalkshavethesamed-length,wehavedP;Q=dQ;P.Finallyforiii,consideradiscretewalkwoftheformP=P1;Q1P2;Q2P3;:::;Qn)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Pn;Qn=QgoingfromPtoQ,andadiscretewalkw0oftheformQ=Q01;R1Q02;R2Q03;:::;Rm)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Q0m;Rm=RgoingfromQtoR.Thesetwodiscretewalkscanbeconcatenatedtogivethediscerewalkw00oftheformP=P1;Q1P2;:::;Qn)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Pn;QnQ01;R1Q02;:::;Rm)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Q0mRm=RgoingfromPtoQ.Since`dw00=`dw+`dw0,takingtheinmumoverallsuchdiscretewalkswandw0,weconcludethatdP;RdP;Q+dQ;R. InordertoaddressthequestionofwhenXislocallyisometrictoH3,wenowprovetwolemmas.LemmaB.1.5.Let:T!T0beanisometrybetweentwoidealtrianglesTandT0.Thenextendstoanisometry:H3!H3ofH3.Inaddition,ifwechooseonesideofTandanothersideofT0,wecanarrangethatsendstheselectedsideofTtotheoneselectedforT0.Theisometryisthenuniquelydeterminedbytheseproperties.Proof.Bytheassumptionintheintroductionofthisappendix,thereisaunique PAGE 62 56linearfactionalmap:^C!^Candauniqueantilinearfractionalmap:^C!^CsendingtheidealverticesofTtotheverticesofT0consistentwith.ThenbyTheoremA.1.1,thesetwomapsextendcontinuouslytotwomaps^and^ofH3[^CwhichrestricttoisometriesofH3.Bycontinuityoftheextensions^and^,andthefactthattheyrestricttoisometriesofH3,^and^bothsendTtoT0.Thisprovestheclaimofexistence.IfwechooseasideofTandT0,wecanthenarrangefortheextensiontosendthesideofTtothesideofT0bychoosingeither^or^whichwouldgiveanorientation-preservingororientation-reversingextension,respectively.ByTheoremA.1.1,anyisometrywhichextendsarisesfromtheextensionafalinearorantilinearfractionalmap,butandaretheonlytwosendingtheverticesofTtotheverticesofT0consistentwith.Thisprovesuniqueness. Asforthesecondlemma,letP2XandsupposetheequivalentconditionsinLemmaB.1.1holdforeveryedgeinX.FixansatisfyingtheconclusionsofLemmaB.1.3.Inaddition,choosesmallenoughthateachP02Pisatahyperbolicdistance>3fromanyfacethatdoesnotcontainit.Inparticular,theballsBdP0;arepairwisedisjointandareballs,half-balls,orwedgesinH3,accordingtothetypeofP02P.Here,ahyperbolicwedgeisoneofthetwopiecesofahyperbolicballBdhP0;rinH3delimitedbytwoclosedhyperbolichalf-planeswhoseboundariesintersectinacompletehyperbolicgeodesiccontainingP0.Inaddition,theTriangleInequalityshowsthattheballsBdP0;areatadistance>apart,inthesensethatdhQ0;Q00>ifQ02BdP0;andQ002BdP00;withP06=P002P.LemmaB.1.3saysthattheballBdP;isobtainedbygluingtogethertheballsBdP0;inXcenteredatthepointsP0thataregluedtoP.Let PAGE 63 57B=SP02PBdP0;denotetheunionoftheseballs.ThissubsetBXcomeswithanaturalmetric,namelytherestrictionofthemetricd.However,wewanttodeneanewextendedmetricdBonBbysettingdBQ;Q0=dQ;Q0whenQandQ0areinthesameballBdP0;,anddBQ;Q0=1whenQandQ0areindistinctballsBdP0;andBdP0;ofB.TheadvantageofdeningthisnewextendedmetriconBisthatitisdenedwithoutreferencetopointsofXthatlieoutsideofB.ThisletsusdeneanothermetricdBonB=BdP;inadditiontod.WedothisbyconsideringdiscretewalksinBinthesamewaywedeneddforX.Briey,ifwisadiscretewalkinBdenedbyP=P1;Q1P2;Q2P3;:::;Qn)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Pn;Qn=QwhereallPi;QilieinB,thenthedB-lengthofwisthesum`Bw=nXi=1dBPi;Qi:ThenwesetdBP;Q=inff`Bw:wadiscretewalkinBfromPtoQgforanytwopointsP;Q2B.ThesameproofinLemmaB.1.2usedfordshowsthatdBiswelldened.TheproofthatdBisametricisthesameasthatfromLemmaB.1.4,withtheexceptionofthesecondhalfofpropertyi.ThisisreplacedwiththeobservationthatdR;R0dBR;R0foreveryR;R02B,sothatdQ;Q0dBQ;Q0everydiscretewalkinBisadiscretewalkinX.ThendBP;Q=0impliesP=Q,anddBisindeedametric.ThemetricsdanddBmaynotcoincideontheentireballB,becausetheremay PAGE 64 58beashortcut"throughXbetweentwopointsmakingthemcloserinXthaninB.However,ifwerestrictattentiontoasmallenoughball,wehavethefollowingresult.LemmaB.1.6.ThemetricsdanddBcoincideontheballBdP;1 3.Proof.LetQ;Q02BdP;1 3.WealreadyobservedabovethatdQ;Q0dBQ;Q0,soitremainstoprovethereverseinequality.SinceQ;Q02BdP;1 3,wehavethatdQ;Q0<2 3bytheTriangleInequality.LetwbeadiscretewalkfromQtoQ0inXoftheformQ=Q1;Q01Q2;Q02Q3;:::;Q0n)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Qn;Q0n=Q0,andwhosed-length`dwissucientlyclosetodQ;Q0that`dw<2 3.ThenQ0i=Qi+1inXand,usingthefactthatthequotientmapisdistancenonincreasingobservedinproofofLemmaB.1.3,nXi=1dQi;Qi+1nXi=1dQi;Q0i<2 3:ArepeateduseoftheTriangleInequalitythenshowsthatdP;QidP;Q1+i)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Xj=1dQj;Qj+1<1 3+2 3=;sothatallQiareinBdP;.SincesatisestheconclusionofLemmaB.1.3,weconcludethatallQiandQ0iareinthesubsetB.IfP06=P002P,thendP0;P00>3bychoiceof,andtheTriangleInequalityshowsthatanypointoftheballBdP0;isatadistance>fromanypointofBdP00;.SincedQi;Q0i<1 3,weconcludethatQiandQ0iareinthesameballBdP0;anddBQi;Q0i=dQi;Q0i.WhatthisshowsisthatwisalsoadiscretewalkfromQtoQ0inBwhosedB-lengthisequaltoitsd-length.Asaconsequence,dBQ;Q0`dw.Sincethisholdsforeverydiscretewalkwwhosed-length`dwissucientlyclosetodQ;Q0,weconcludethatdBQ;Q0dQ;Q0. PAGE 65 59WehaveshownthatdBQ;Q0=dQ;Q0forallQ;Q02BdP;1 3,asrequired. Wearenowreadytoprovethemainresultofthissection.TheoremB.1.7.ThemetricspaceXislocallyisometrictoH3ifandonlyifforeachedgeEinXtheedgeinvariantscorrespondingtoalledgesgluedtoEhaveproductofmodulusoneandargumentssumto2.Proof.FirstassumethatconditionsandholdforeachedgeinX,andletPbeapointinX.FixansatisfyingtheconclusionsofLemmaB.1.3.Inaddition,choosesmallenoughthateachP02Pisatadistance>fromanyfacethatdoesnotcontainit.WewillndanisometrybetweenaballBdP;1 3andahyperbolicballBdhP0;1 3forapointP0inhyperbolicspaceH3.Case1:PisintheinteriorofatetrahedroninX.Inthiscase,PisgluedtonootherpointsothatPconsistsonlyofP.ThenB=BdP;and,byourchoiceof,theballBdP;iscompletelycontainedintheinteriorofX.Inparticular,theballBdP;isthesameasthehyperbolicballBdhP;H3,anopenballinhyperbolicspaceH3.Also,therearenogluingsbetweendistinctpointsofB=BdP;,sothateveryQ2BdP;correspondstoexactlyonepointQ2BdP;.Dene:BdP;!BdhP;bythepropertythatQ=QforeveryQ2BdP;.Themapmaynotbeanisometryoverthewholeball,butweclaimthatdhQ;Q0=dQ;Q0foreveryQ;Q02BdP;1 3.Indeed,dQ;Q0=dBQ;Q0byLemmaB.1.6.SincetherearenogluingsinB,oneeasilyseesthatdBQ;Q0=dBQ;Q0.Finally, PAGE 66 60dBQ;Q0=dQ;Q0=dhQ;Q0=dhQ;Q0.ThisprovesthattherestrictionoftotheballBdP;1 3isanisometryfromBdP;1 3;dtothehyperbolicballBdhP;1 3;dh,asrequired.Case2:PisinanedgeofatetrahedroninX.WriteP=fP1;P2;:::;PkgwithP=P1.Namely,P1;P2;:::;PkaretheedgepointsofXthataregluedtoP.LemmaB.1.3saysthattheballBdP;inXistheimageinderthequotientmap:X!XoftheunionBoftheballsBdP1;;BdP2;;:::;BdPk;inX.Becauseofourchoiceof,eachoftheballsBdPj;inthemetricspaceX;disawedgeofradiusinH3,andthesewedgesarepairwisedisjoint.Wenowneedtorearrangethesewedgesintoafullball,usingaproceduresimilartotheoneusedintheproofofLemmaB.1.1.EachPjbelongstoexactlytwofacesFijandFi0j.Wecanchoosetheindexingssothatforeveryjwith1jk,thegluingmapijsendsthevertexPjtoPj+1andthefaceFijtoFi0j+1withtheconventionthatPk+1=P1andi0k+1=i01.Wewillconstructourisometry:BdP;!BdhP;piecewisefromsuitableisometriesjofH3;dh.ByLemmaB.1.5,foreveryj,wecanextendthegluingmapij:Fij!Fi0j+1toanisometryij:H3!H3ofH3;dhthatsendsthetetrahedroncontainingthefaceFijtothesideofFi0j+1thatisoppositethetetrahedroncontainingFi0j+1.Todenej,webeginwithanyisometry1ofH3;dhandinductivelydenej+1=j)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1ij=1)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1i1)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1i2)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1ij:ByinductiononjandbecausePj+1=ijPj,themapjsendsthevertexPjtothesamepointP0=1Pforeveryj.Inparticular,theisometryjsendsthewedgeBdPj;toawedgeoftheballBdhP0;.Similary,theimageofthe PAGE 67 61faceFi0j+1=ijFijunderj+1isequaltotheimagesofFij,underj.Bydeni-tionoftheextensionofijtoanisometryofH3,thetwowedgesjBdPj;andj+1BdPj+1;sitonoppositesidesofjFij=j+1Fi0j+1.ItfollowsthatthewedgesjBdPj;alltsidebysideandinorderofincreasingjaroundtheircommonedgethroughP0.SincetheinternalanglesofthewedgesBdP1;;BdP2;;:::;BdPk;addupto2,thewedgek+1BdPk+1;=k+1BdP1;isequalto1BdP1;.Inparticular,thetwoisometriesk+1and1ofH3sendP1=Pk+1tothesamepointP0,sendthefaceFi0k+1=Fi01tothesameidealtriangle,andsendasideofFi0k+1=Fi01tothesamesideofk+1Fi0k+1=1Fi01.BytheuniquenesspartofLemmaB.1.5,itfollowsthatk+1=1.Finally,notethatwhenQ2FijisgluedtoQ0=ijQ2Fi0j+1,thenjQ=j+1Q0.Wecanthereforedeneamap:BdP;!BdhP0;bythepropertythatQisequaltojQwheneverQ2BdPj;.Theaboveconsiderationsshowthatiswelldened.Wewillshowthatinducesanisometrybetweenthecorrespondingballsofradius1 3.Forthis,considertwopointsQ;Q02BdP;1 3.ByLemmaB.1.6andbytheTriangleInequality,dBQ;Q0=dQ;Q0<2 3.LetwbeadiscretewalkfromQtoQ0inBoftheformQ=Q1;Q01Q2;Q02Q3;:::;Q0n)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Qn;Q0n=Q0,andwhosedB-length`dBwissucentlyclosetodBQ;Q0that`dBw<2 3.Inparticular,eachdBQi;Q0iisnite,sothatQiandQ0ibelongtothesameballBdPj;.Asa PAGE 68 62consequence,dhQi;Q0i=dhjiQi;jiQ0i=dhQi;Q0idBQi;Q0isinceeachjiisahyperbolicisometry.Then,byiteratingtheTriangleInequalityandusingthefactthatQ0i=Qi+1,dhQ;Q0nXi=1dhQi;Q0inXi=1dBQi;Q0i=`dBw:SincethisholdsforeverydiscretewalkwfromQtoQ0inBwhoselengthissucentlyclosetodBQ;Q0,weconcludethatB.2dhQ;Q0dBQ;Q0:Conversely,letbetheorientedgeodesicfromQtoQ0intheballBdhP0;1 3.RecallthatBdhP0;1 3isdecomposedintothewedgesjBdPj;1 3.Therefore,wecansplitintogeodesics1;2;:::;n,inthisorder,suchthateachiiscontainedinawedgejiBdPji;1 3.InthewedgeBdPji;1 3X,considertheorientedgeodesic0i=)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1jiicor-respondingtoi.Iftheendpointsof0iarelabelledsothat0igoesfromQitoQ0i,wenowhaveadiscretewalkwfromQtoQ0oftheformQ=Q1;Q01Q2;Q02Q3;:::;Q0n)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1Qn;Q0n=Q0,withdB-length`dBw=nXi=1dBQi;Q0i=nXi=1`h0i=nXi=1`hi=`h=dhQ;Q0:ItfollowsthatB.3dBQ;Q0dhQ;Q0.CombiningtheinequalitiesB.2andB.3,weconcludethatdQ;Q0=dBQ;Q0=dhQ;Q0 PAGE 69 63foreveryQ;Q02BdP;1 3.Inotherwords,inducesanisometryfromtheballBdP;1 3;dtotheballBdhP0;1 3;dh.Case3:PisinafacebutnotanedgeofatetrahedroninX.TheproofinthiscaseisidenticaltotheprooffromCase2.Thisprovesthatifconditions1andholdforeachedgeinX,thenXislocallyisometrictoH3.Toprovetheconverse,rstassumethatdoesnotholdforsomeedgeEinX.Inparticular,theedgeinvariantscorrespondingtoalledgesgluedtoEhaveproductwithmodulusdierentfromone.BytheproofofLemmaB.1.1,weconcludethateachpointPinXcorrespondingtoanedgepointPconsistofinnitelymanypointsalongE,asshowninFigureX.SinceeveryneighborhoodofapointPofthistypehasnon-compactclosure,wecanconcludethatXisnotlocallyisometrictoH3.Next,assumethatcondition1holdsforeachequivalenceclassofedgesinX,butdoesnot.SupposethattheanglescorrespondingtotheedgesgluedtoEdonotaddupto2,andxanyP2XcorrespondingtoapointP2E.ByLemmaB.1.3,thereisan>0forwhichtheballBdP;isexactlytheimageoftheballsBdP0;,whereP0rangesoverallofP.WecanfurtherarrangefortobesmallenoughsothateachP02Pisatadistance>fromthefacesthatdonotcontainit,sothatweareensuredthateachBdP0;isawedge.ForeachP02P,wecanconsiderthehyperbolicplaneP0orthogonaltoEandpassingthroughP0.TakingtheintersectonDP0=P0BdP0;givesahyperbolicdisksectorDP0withangleP0.ThesedisksectorstheninheritagluingfromthegluingofX,andtheirimageinXunderthequotientmapisahyperbolicconeCPwithradiusandconeanglePP02PP0.Everycircle"fQ2CP:dP;Q=rgfor0 PAGE 70 64ballmodelshowsthateacharchaslengthP0sinhr,givingusthetotallengthofthecircleasPP02PP0sinhr6=2sinhr.IftherewereaballaroundPthatwasisometrictoahyperbolicball,everysucientlysmallplanarcirclewouldhavelength2sinhr. B.2CompletenessWecannowgiveconditionswhichensurethatthemetricsconstructedinSectionB.1.7arecomplete,butrstweprovealemma.LemmaB.2.1.LetM;d0bemetricspace.SupposethereisasequencefCig1i=1ofcompactsubsetsinMsuchthatM=[1i=1Ciandforeachi1,Ci+1containsNCi;1,theneighborhoodofCiofradius1.ThenMiscomplete.Proof.LetfPjg1j=1beaCauchysequenceinM.Thenthereisanintegerksuchthatforeverym;nk,wehaved0Pm;Pn<1.SinceM=[1i=1Ci,thereisaC`withfP1;P2;:::PkgC`.SinceNC`;1C`+1,itfollowsthateveryPjiscontainedinthecompactsetC`+1.HencethesequencefPjg1j=1convergestosomeP12C`+1M. WerefertosuchasequencefCig1i=1asanexhaustionofMbycompactsubsets.TheconverseofLemmaB.2.1isalsotrue,thatisifMiscompletethenthereisanexhaustionofMbycompactsubsets.Wewillnotusethisconverse,however.Wearenowreadytoprovethemaintheoremofthissection.Itreducestheques-tionofcompletenessofthethree-dimensionalspaceXtoaproblemonedimensionallowerinvolvingthelinksofequivalenceclassesofidealvertices.AsdenedinAppendixA,thelinkofanidealvertexvinatetrahedronTvincidentwithvistheintersectionofasuitablyhighhorospherebasedatvwithTv.Alinkofanequiv-alenceclassofidealverticesvisthenthesurfaceobtainedbygluingtogetherthe PAGE 71 65 FigureB.2:DistanceFromLinkEdgeToNon-incidentVertexlinksoftheidealverticesinvinthetetrahedraincidentwiththeidealverticesofv,asmadeprecisebyconditionsiandiibelow.TheoremB.2.2.SupposethatconditionsandfromTheoremB.1.7holdforeachedgeinXsothatthemetricspaceX;dislocallyisometrictoH3.Inaddition,assumethatiforeachidealvertexvinX,thereisahorosphereSvbasedatvwhichsatises:aSvandSv0aredisjointforv6=v0andSvonlyintersectsthesidesadjacenttov,bifvandvarepairedbythefacepairingisometry,thenfmapsSvtoSv,iiforeachequivalenceclassofidealverticesv,thecorrespondinglinkfSuTu:u2v;Tuthetetrahedroncontainingugiscomplete.iiithereisaconstantc>0sothatforeveryidealvertexv,thedistanceinXfromanyedgeinSvTvtothecorrespondingnon-incidentvertexinSvTvisgreaterthanc,asinFigureB.2.ThenX;discomplete. PAGE 72 66Proof".ConsideraCauchysequencefPng1n=1inX.LetBvbetheopenhoroballbasedatvwithboundarySv.Thatis,BvconsistsofallthepointsthatareinsidethehorosphereSv.ThendeneS=[vSvandS=XS.SimilarlydeneB=[vBvandB=XB.Wesplittheproofupintotwocases.Case1:ThereareinnitelymanyPninX)]TJ/F16 11.9552 Tf 14.7523 3.022 Td[(B.WeprovethatX)]TJ/F16 11.9552 Tf 14.7586 3.022 Td[(BiscompletewiththerestrictionofthemetricfromXbyndinganexhaustionbycompactsubsets.BeginwithanysingletetrahedronT1XanddeneC1=T1)]TJ/F33 11.9552 Tf 12.6954 0 Td[(B.SinceT1)]TJ/F33 11.9552 Tf 11.9551 0 Td[(Biscompactandiscontinuous,C1iscompact.TodeneC2,weneedtondacompactsubsetofX)]TJ/F16 11.9552 Tf 14.7605 3.022 Td[(BthatcontainsNC1;1inX)]TJ/F16 11.9552 Tf 14.7523 3.022 Td[(B.ConsidertheclosedneigborhoodN1ofT1SinSofradius1.ThereareonlynitelymanytetrahedrafT1;T2;:::TkgthatintersectN1,andtheunion[kj=1Tj)]TJ/F33 11.9552 Tf 9.8318 0 Td[(BcontainstheneighborhoodNC1;cinX)]TJ/F16 11.9552 Tf 12.6289 3.022 Td[(B.WethushaveacompactsubsetofX)]TJ/F16 11.9552 Tf 14.7523 3.022 Td[(BcontainingNC1;c.NowtherearenitelymanyhorospheresassociatedwiththeidealverticesoftheseTj.BytakinganeighborhoodN2inSofradius1)]TJ/F33 11.9552 Tf 11.7701 0 Td[(cofeachoftheTjSexceptforC1S,andtakingallthetetrahedrathatintersectN1andN2,wegetnitelymanytetrahedrafT1;T2;:::;Tk;Tk+1;:::;T`g.Thentheunion[`j=1Tj)]TJ/F33 11.9552 Tf 11.0351 0 Td[(BiscompactandcontainstheneighborhoodNC1;2cinX)]TJ/F16 11.9552 Tf 13.8322 3.022 Td[(B.ByrepeatingthisprocedureonthehorospheresofTk+1;:::;T`,weobtaincompactsubsetswhichcontainNC1;3c;NC1;4c;:::inX)]TJ/F16 11.9552 Tf 13.4978 3.022 Td[(B.WecaniteratethisprocessuntilweobtainanitefamilyoftruncatedtetrahedracontainingNC1;mcwheremc<1andm+1c1.Thenrepeatingonelasttimewillgiveanitefamilyof PAGE 73 67truncatedtetrahedrainX)]TJ/F16 11.9552 Tf 14.9508 3.022 Td[(Bwhoseunion,C2,iscompactandcontainsNC1;1inX)]TJ/F16 11.9552 Tf 14.7523 3.022 Td[(B.NowthatwehaveacompactC2thatconsistsofnitelymanytruncatedtetrahedrainX)]TJ/F16 11.9552 Tf 14.9287 3.022 Td[(S,wecanrepeattheprocedureoutlinedaboveoneachofthetruncatedtetrahedraofC2.TakingallthetruncatedtetrahedrasoobtainedwillgivenitelymanytetrahedratruncatedatSwhoseunion,C3,containsaunitneighborhoodofC2inX)]TJ/F16 11.9552 Tf 14.7523 3.022 Td[(B.RepeatingthisprocedureinductivelygivesanexhaustionofX)]TJ/F16 11.9552 Tf 15.0738 3.022 Td[(BbycompactsubsetsfCig1i=1.HenceX)]TJ/F16 11.9552 Tf 14.985 3.022 Td[(BiscompleteintherestrictionofthemetricfromX.SincethereareinnitelymanyPninX)]TJ/F16 11.9552 Tf 14.3994 3.022 Td[(B,thereisasubsequenceoffPng1n=1thatconvergestoapointP12X)]TJ/F16 11.9552 Tf 14.8817 3.022 Td[(B.ThesequencefPng1n=1isCauchy,henceitmustconvergetoP1aswell.Wenotethatforeachv,ifwechoosethehorosphereS0vthatisadistanceMclosertovthanSv,thenthesameproofofCase1willhold.Thisnewcollectionofhorosh-eresfS0vgclearlysatisespropertyi-a.IfanisometrysendsthehorosphereSutothehorosphereSv,thenitalsosendstheS0utoS0v,hencepropertyi-bissatisedaswell.Furthermore,foreachequivalenceclassofidealverticesv,thereisasimilaritybetweenthelinkLv=fSuTu:u2v;TuthetetrahedroncontainingugandthelinkL0v=fS0uTu:u2v;TuthetetrahedroncontainingugdenedbytakingapointPinLandprojectingdowntoL0viathegeodesicconnectingPtov.Thisensuresthatconditionsiiandiiiaresatisedaswell.Case2:ThereareinnitelymanyPninB.WereduceCase2toCase1byshowingthatthepointsPnareatmostadistanceMclosertotheidealverticesfromS.Wedeneafunctionh:B!Rasfollows.ForapointPinsomeBv,letQ PAGE 74 68bethepointofSvthatisclosesttoP;thenhP=dhP;Q.ThepointQcanbeeasilyconstructedfromthepropertythatitistheintersectionofSvwiththecompletegeodesicgpassingthroughvandP.Thispropertycanbeeasilycheckedwhenv=1,andforthegeneralcasewecanconjugatebyanisometrysendingvto1.Thisfunctionhhasthefollowingtwoimportantproperties:ifP;P02Baregluedtogether,thenhP=hP0;foranytwoP;P0inthesamehoroballBv,dhP;P0jhP)]TJ/F33 11.9552 Tf 11.9552 0 Td[(hP0j.TherstpropertyisanimmediateconsequenceofthefactthatthegluingmapsareisometriesandsendeachBvtosomeBv0.ThesecondpropertyisaconsequenceoftheestimatedhP;P0jlnz0 zjforP=x;y;zandP0=x;y;z0inthecasewhenv=1.Thegeneralcasefollows,onceagain,byconjugatingbyahyperbolicisometrysendingvto1.Nowxanarbitrary,whoseprecisevaluewillnotbeimportant.Thereisanumbern1suchthatdPn;Pn+1 PAGE 75 69issucientlyclosetodPn;Pn+1that`dw PAGE 76 70ThiscompletesourproofbyinductionthatallQiandRi)]TJ/F31 7.9701 Tf 6.5865 0 Td[(1with1ikbelongtoBandsatisfyB.4.OnemorestepinthesameproofgivesthathPn+1=hRkhQk+dQk;RkhPn+kXj=1dQj;Rj;andhPn+1=hRkhQk)]TJ/F33 11.9552 Tf 11.9552 0 Td[(dQk;RkhPn)]TJ/F34 7.9701 Tf 18.2787 14.944 Td[(kXj=1dQj;Rj;sothatjhPn+1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(hPnjPkj=1dQj;Rj=`dw.SincethisholdsforeverydiscretewalkwfromPntoPn+1whoselengthissu-cientlyclosetotheinmumdPn;Pn+1,weconcludethatjhPn+1)]TJ/F33 11.9552 Tf 11.9552 0 Td[(hPnjdPn;Pn+1:AconsequenceofthisinequalityisthatthesequencefhPng1n=1inRisCauchy,andhenceconvergesinR.Thus,itisboundedbysomenumberM.ThisprovesthatthepointsPnstayatahyperbolicdistanceMfromS.Hence,theproofofCase1appliesandwearedone. CorollaryB.2.3.LetWbeanalternatingweavesatisfyingtheconditionsofSection2.1withassociatedidealhyperbolictetrahedrafTig.AssumethatthetwoconditionsofTheoremB.1.7aresatisedforeachedgeofX=STiandtheconditionsofTheoremB.2.2aresatised.ThenthemetricdWonR3)]TJ/F33 11.9552 Tf 12.1778 0 Td[(WfromCorolarry2.3.1iscomplete.Proof.ByTheoremB.2.2,themetricdonXiscomplete.SincehisanisometryfromR3)]TJ/F33 11.9552 Tf 12.083 0 Td[(WwiththemetricdWtoX,themetricdWonR3)]TJ/F33 11.9552 Tf 12.083 0 Td[(Wiscompleteaswell. PAGE 77 71WhileitmaybediculttochecktheconditionsinTheoremB.2.2foranarbitraryalternatingweave,theyarerelativelyeasytoverifyifWisadoublyperiodicweave,thatisiftheprojectionofWcanbearrangedtohavetwolinearlyindependenteuclideantranslationalsymmetries.Forexample,thesquareweaveinFigure2.1isdoublyperiodic.CorollaryB.2.4.LetWbeadoublyperiodicalternatingweavewithparameterssatisfyingtheconditionsofTheoremB.1.7.ThemetricdWonR3)]TJ/F33 11.9552 Tf 9.7903 0 Td[(WfromCorollary2.3.1iscompleteifitheparametersrespecttheperiodicityofW,iithestandequationsofSection2.3holdforeachstrandofW,andiiithetwotilingsobtainedbyarrangingthelinksofthenorthernandsouthernverticessidebysidearecompletethatis,theybothexhibitdoublyperiodiceuclideansymmetry.Proof.Wemustcheckthatthereisachoiceofhorospheresassociatedwiththeidealverticesofthetetrahedrawhichsatisesi-iiiofTheoremB.2.2.SincetheparametersrespectthedoublyperiodicsymmetryofW,thereareonlynitelymanyisometricallydistincttetrahedra.Hence,thereisachoiceofhoropsheresrespectingthesymmetryofWthatsatisesconditioni-aofTheoremB.2.2.Nowwemustshowthatwecanchoosethehorospheressothattheyarepreservedbythefacepairingisometries,thatisthefacepairingsofthetetrahedrarestricttoedgepairingsofthelinks.WedothisbychoosinghorospheresstillrespectingthesymmetryofWclosertothecorrespondingidealvertices.Thisneitheraects-aofTheoremB.2.2norchangesthesimilarityclassesofthecorrespondinglinks. PAGE 78 72Wehavebyassumptioniiithatwecanarrangebyscalingandpreservingsim-ilarityclassthelinksofthenorthernandsouthernverticessothattheyaregluededgetoedgeandsothattheresultingtilingsaredoublyperiodicandcomplete.Byassumptionii,wecandothiswiththelinksofthestrandverticesaswellstrandequationsandniterepeatingpatternoflinksensurescompletetiling.However,wewishtobeabletoarrangetheselinksinthisfasionwithoutscaling.SupposeTandT0aretwotetrahedrawithfacesFandF0pairedbytheisometry:F!F0andincidentwiththeidealverticesvandv0,respectively.ThenmapsthehorosphereSvtothehorosphereSv0ifandonlyifthehyperboliclengthofFSvisequaltothehyperboliclengthofF0Sv0.Thus,itsucestoshowthattheirisachoiceofhorospheressuchthateachedgeineachlinkhasthesamehyperboliclengthastheedgeinthelinktowhichitispaired.Wecanshowthisasfollows.Fixanequivalenceclassofverticesv,andletGbetheabstractgraphwithverticestheelementofvandedgesthepairsofverticeswhicharegluedbyisometries.ThenGisconnected.ConsiderthesubgraphHofGwiththesameverticesasGwithedgesthepairsofverticesthathavehorospherespreservedbythecorespondinggluingisometry.SincewehavechosenhorospheresrespectingthesymmetryofW,wecanarrangeforHtohaveasmanyedgesaspossiblebychoosinghorospheresclosertothecorrespondingidealvertices.ThesubgraphHmustbeconnectedaswell,becauseifitisnottherewillbeanedgeofGconnectingtwocomponentsofH.Byadjustingtheappropriatehoro-spherescomponentwithhorospheresfartherawayfromtheidealverticesajustedtomatchthosefromothercomponentwhicharecloseruniformlyclosertotheidealverticesandstillrespectingthesymmetryofW,wegetanotheredgeinH.ThiscontradictstheassumptionthatHhasasmanyedgesaspossible. PAGE 79 73SinceHisconnected,thereisasequenceofidealverticesfvjgnj=1ofvwithvjgluedtovj+1andthehorosphereSvjmappedtoSvj+1bytheappropriategluingisometry.SincehorospheresofthevjareisometricallypairedandthelinksofthevjcanbearrangedtogiveaconsistentandcompletetilinginparticularequalityofthelengthsoftheedgesinthisarrangementcorrespondingtoFSvandF0Sv0,weknowthatthelengthoftheedgeFSvisequaltothelengthoftheedgeF0Sv0.Thus,thefacepairingisometrysendsthehorosphereSvtoSv0.SinceWisdoublyperiodic,weonlyhavetodothisfornitelymanyequivalenceclassesofverticestogetachoiceofhorospheessatisfyingi-aandi-bofTheoremB.2.2.Wemustalsocheckthecompletenessofthelinksofeachequivalenceclassofidealvertices.Wehaveassumedthatthearrangingofthelinkscorrespondingtoeachequivalenceclassofidealverticesgivesacompletetiling,andhavejustshowedthatthisarrangementcanbegivenbyrestrictingthefacepairingisometries.Itisclearthenthatthecorrespondingsurfacesarecomplete.ToverifyconditioniiifromTheoremB.2.2,wenoteagainthatthenorthernplaneandsouthernplaneexhibitdoublyperiodicsymmetry.Sincethismeansthereisaniterepeatingpattern,itisclearthereisaconstantcsatisfyingiiiofTheoremB.2.2.Furthermore,thereisaniterepeatingpatternforeachstrandlink,sothatthesameargumentapplies.Sincethereareonlynitelymanypossibledistinctlinks,wecanchooseaconstantsatisfyingconditioniiiofTheoremB.2.2andwearedone. 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