Some results on Teichmuller spaces of Klein surfac

arXiv:math/9507207v1[math.GT]21Jul1995SOMERESULTSONTEICHM¨ULLERSPACESOFKLEINSURFACESPABLOAR´ESGASTESIJuly20,1995Abstract.Inthispaper,weprovesomeisomorphismstheoremsbetweenTe-ichm¨ullerspacesofnon-orientablecompactsurfaces.Wealsodevelopatechnique,basedonsimilarresultsforRiemannsurfaces,togiveexplicitexamplesofTe-ichm¨ullerspacesofKleinsurfaces.1.BackgroundandstatementofmainresultsThedeformationtheoryofnon-orientablesurfacesdealswiththeproblemofstudy-ingparameterspacesforthedifferentdianalyticstructuresthatasurfacecanhave.ItisanextensionoftheclassicaltheoryofTeichm¨ullerspacesofRiemannsurfaces,andassuch,itisquiterich.InthispaperwestudysomebasicpropertiesoftheTeichm¨ullerspacesofnon-orientablesurfaces,whoseparallelsintheorientablesitu-ationarewellknown.Moreprecisely,weproveanuniformizationtheorem,similartothecaseofRiemannsurfaces,whichshowsthatanon-orientablecompactsurfacecanberepresentedasthequotientofasimplyconnecteddomainoftheRiemannsphere,byadiscretegroupofM¨obiusandanti-M¨obiustransformation(mappingswhoseconjugatesareM¨obiustransformations).ThisuniformizationresultallowsustogiveexplicitexamplesofTeichm¨ullerspacesofnon-orientablesurfaces,assubsetsofdeformationspacesoforientablesurfaces.Wealsoprovetwoisomorphismtheo-rems:inthefirstplace,weshowthattheTeichm¨ullerspacesofsurfacesofdifferenttopologicaltypearenot,ingeneral,equivalent.Wethenshowthat,ifthetopolog-1991MathematicsSubjectClassification.30F12.Keywordsandphrases.Riemannsurface,Kleinsurface,Kleiniangroup,Teichm¨ullerspace.12PABLOAR´ESGASTESIicaltypeispreserved,butthesignaturechanges,thenthedeformationsspacesareisomorphic.ThesearegeneralizationsofthePattersonandBers-GreenbergtheoremsforTeichm¨ullerspacesofRiemannsurfaces,respectively.ARiemannsurface(Σ,X)isatopologicalsurfaceΣwithacomplexstructureX,thatis,acoveringofΣbychartswithholomorphicchangesofcoordinates.SinceholomorphicfunctionshavepositiveJacobian,itturnsoutthatRiemannsurfacesareorientable.Thenaturalgeneralisationtothecaseofnon-orientablesurfacesisthatofadianalyticstructure,wherewerequirethatthechangesofcoordinatesareeitherholomorphicoranti-holomorphic(thecomplexconjugateisholomorphic).Apair(Σ,X),whereΣisasurfaceandXisadianalyticstructure,iscalledaKleinsurface.Inparticular,RiemannsurfacesareKleinsurfaces.ItisclassicalfactthatanyKleinsurfacecanberepresentedas˜X/Γ,where˜XiseithertheRiemannsphere,thecomplexplaneortheupperhalfplane,andΓisagroupofdianalyticbijectionsof˜X.Exceptforafew(finitenumberof)cases,Kleinsurfacesarecoveredbytheupperhalfplane;thesearecalledhyperbolicsurfaces.Acompactnon-orientablesurfaceΣistheconnectedsumofg(real)projectiveplanes;giscalledthegenusofthesurface.Observethathereweusethegenusinthetopologicalsense;someauthors(inparticular,[14])usetheso-calledarithmeticgenus,whichisequaltog−1.Anon-orientablesurfaceishyperbolicifandonlyifg≥3.Inthefirstresultofthispaper,weproveauniformizationtheorem,bygroupswhicharemoresuitableforcomputationsthatgroupsactingontheupperhalfplane.Theorem1.1.LetΣbeacompactnon-orientablesurfaceofgenusbiggerthan2.ThenthereexistsaKleiniangroupG,actingdiscontinuouslyonasimplyconnectedsetΔofˆC,andanantiholomorphicfunctionr,suchthat:1.g(Δ)=Δforallg∈G;r(Δ)=Δ;2.Δ/GisisomorphictothecomplexdoubleΣcofΣ;3.risoftheformr:z→az+bcz+d,withad−bc6=0;4.Δ/Γ∼=Σ,whereΓisthegroupgeneratedbyGandr;5.ΓisuniqueuptoconjugationbyM¨obiustransformations.HerebyaKleiniangroupwemeanagroupofM¨obiustransformationsthatactsdiscontnuouslyonanon-emptyopensetoftheRiemannsphere.ThecomplexTEICHM¨ULLERSPACESOFKLEINSURFACES3doubleofΣisaRiemannsurfaceΣc,togetherwithaunramifieddoublecoverπ:Σc→Σ.IfΣishyperbolic,thenΣcisalsohyperbolic(see§2below).LetM(Σ)denotethesetofdianalyticstructures,onthenon-orientablesurfaceΣ,thatarecompatiblewiththedifferentialstructureinducedbyX.ThequotientofM(Σ)bythegroupofdiffeomorphismshomotopictotheidentity(actingbypullback,see§3),istheTeichm¨ullerspaceT(Σ)ofΣ.IthasanaturalrealanalyticstructuregivenbyprojectingthenaturalstructureofM(Σ).ItisnothardtoprovethatT(Σ)embeddsintheTeichm¨ullerspaceofΣc(see§3).Combiningthisembeddingwiththeorem1.1andtheresultsofI.Krain[9],wecangivepresentationsforthedeformationspacesofsomenon-orientablesurfaces.Asanexample,wecomputetheTeichm¨ullerspaceofasurfaceofgenus3.Theorem1.2.ThespaceT(Σ)ofanon-orientablesurfaceofgenus3canbeiden-tifiedwiththesetofpoints(τ1,τ2,τ3)ofT(Σc),suchthatRe(τ2)=0Re(τ1)=Im(τ3)Re(τ1)+Re(τ3)=0Weintroducetheconceptofpunctureonanon-orientablesurfaceasagenerali-sationofthecorrespondingideaonRiemannsurfaces:apunctureisadomainonΣ,homeomorphictotheunitdiscminustheorigin,thatcannotbecompletedtobehomeomorphictotheunitdisc,andsuchthatanychangeofcoordinatesinthedomainisholomorphic.Theabovetheoremsextendseasilytothecaseofsurfaceswithpunctures.Forexample,wecanidentifythedeformationspaceofasurfaceofgenus1withtwopunctures.Theorem1.3.ThespaceT(Σ),whereΣisthe(real)projectiveplanewithtwopun-tures,canbeidentifiedwiththesetofpointsoftheupperhalfplanewithimaginarypartbiggerthen1.OnecandefineaKleinhyperbolicorbifoldasanon-orientablesurfaceΣ,withfinitelym