; TeX output 1995.01.12:1652 - hGS !K`y UU cmr10AnnalesAcademiScienĽtiarumFN8ennic SeriesA.I.MathematicaVN8olumen18,1993,45{64"z"V cmbx10ACELLULARPٚARAMETRIZATIONFORCLOSED'^qSURFwxACESWITHADISTINGUISHEDPOINT3 .|"V cmbx10MarjattaTNTuaatanenaUnivĽersityofHelsinki,DepartmenĽtofMathematics >PN8.O.BoĽx4(Hallitusk{atu15),SF-00014UniversityofHelsinki,Finland3,"V UU cmbx10Abstract.8TheconĽvexhullconstructioninMinkowskispaceisusedheretoparametrize DiricĽhlet]fundamentalp;Bolygonsby]$b> UU cmmi10L -lengthsoftheedgesofthecorresp;Bondingconvexhull.>ThebasicKconditionshaĽveKsimplegeometricinĽterpretationsbyprovjectiontothePoincaremo;Bdel,1andthepresenĽtation_isdonealsointermsofconceptsinthePoincaremo;Bdel.nTheparametrizationleadstoacell-decomp;BositionQAofconformalstructuresofclosedsurfacesofgenĽusQAgwithadistinguishedpoinĽt.A connectionb;BetĽweentheentriesofthematrixofaMUobiustransformationandthecorresp;BondingL -length,withGdistinguishedp;BoinĽtattheorigin,isobtained.AFnecessaryandsucienĽtconditionfordiscretenessisobtainedintermsofthematricesofthegeneratorsofthegroup.w 1.pIn9troQductionލ K`y cmr10The+convexhullconstructioninMinkowskispaceforclosedsurfaceswaspre- sented- byNaatanenandPenner[3].dUHereweusethisconstructiontoparametrizeDirichletGfundamentalpGolygonsbyGb> cmmi10L -lengthsoftheedgesofthecorrespGondingconvexhull.Sincethebasicconditionshavesimplegeometricinterpretationsbypro 8jectiontothePoincarGemoGdel,thepresentationisdonealsointermsofcon-cepts=inthePoincarGe=moGdel.?Theparametrizationleadstoacell-decompositionofconformalstructuresofclosedsurfacesofgenusg+withadistinguishedpGoint;eachtop-dimensional(i.e.(6g<!", cmsy10 &c4) -dimensional)cellcorrespGondingtoaxedcom-binatorialtypGeofside-pairingsoftheDirichletpGolygon.pTheboundariesofthetop-dimensionalcellscorrespGondtothe\degenerate"Dirichletpolygons,andarecharacterizedbyPtolemyequationsinMinkowskispace[3].hcExamplesaredoneforgA=?h2 .JWAsconnectionbGetweentheentriesofthematrixofaMobiustransfor-mation0andthecorrespGonding0L -length,83withdistinguishedpointattheorigin,83isderivedAinChapter6,|andinChapter7,anecessaryandsucientconditionfordiscreteness,}writtenuexplicitlyinthecaseofgenusu2 ,isobtainedintermsofthematricesBofthegeneratorsofthegroup.QlW*ewishtothankR.Penner,T.Kuusalo,T.Nakqanishiforhelpfuldiscussions,tT.SorvaliandP*.Tukiaforcomments.fThepicturesUUhavebGeendrawnbyJ.Haata 8jaandM.Nikunen.AsmentionedabGove,theparametrizationintroGducedinthisworkwasinitiallyderivedNbyusingthehypGerboloidmodelforhyperbolicplaneandconstructingtherethea[Euclideanconvexa[hullfortheorbitofthedistinguishedpGoint,d]followingideas ff xsUU1991MathematicsSubvjectClassication:%Primary30F35. . * 46 7p0J cmsl10M.UUNaatanenhGinA[3]and[4].ThentwoAconditions,calledthefaceconditionandtheclosing condition,+were instrumental.t.(TherstconditionisderivedinChapter3.)t.Bypro 8jectingPintothePoincarGemoGdelitturnsoutthattheabovePconditionshavePsim-pleIgeometricinterpretations.m(F*orthefacecondition,L$itisgiveninNote3.4,L$andtheyclosingconditionamountstotheanglesofcertaintrianglesatthedistinguishedpGointUUaddinguptoUU2[٫.)&荑Since thePoincarGe moGdelismorecommonlyused,6wehavechosentowritethis[npapGersothatitcanalsobereadwithoutneedtogetintothecalculationsofChapterz3,QstartingfromNote3.4.(ThesubsequentchaptersarewrittenintermsoftheIKPoincarGemoGdel,KbutintheformulasIKL -lengths|whichinitiallyrefertoedgesoftheconvexhull|areusedsincetheymaketheformulassimplerthantheuseofhypGerboliclengthswould.GuT*otranslatetheformulasintohypGerbolicmetric,]itisonlyneededtoreplaceeachLbyPp G