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cmcsc10RealizingCount32ableGroups\2AsAfutomorphismGroupsj٩ofRiemannSurfaces%R̍}#-
cmcsc10Ji*orgWinkelmann~:|{Y cmr8ReceivÎed: JulyX19,2001fnCommÎunicatedXbyUlfRehmann+ Abstract.[EveryjcountablegroupcanbGerealizedasthefullau- tomorphismΏgroupofaRiemannsurfaceaswellasthefullgroupof isometriesUUofaRiemannianmanifold. 2000WMathematicsSub 8jectClassication:vPrimary30F99,X{Secondary 20F29,20B27,32M05*Z KeywordsvandPhrases:CountableGroups,pF*ullautomorphismgroup, RiemannUUsurfacep0J
cmsl10Acknowledgement._bThe&authorwhishestoexpresshisgratitudetotheUniver-sityWofT*okyo.yThismanuscriptwaswrittenandisbasedonworkdoneduringthemstayoftheauthorattheUniversityofT*okyoasvisitingAssoGciateProfessor.*ZSaerensandZame[html:4 html: ],2 html:]provedthateverycompactrealLiegroup
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cmmi10K.canbGerealizedasthegroupofholomorphicautomorphismsofacomplexmanifoldaswellasthegroupofisometriesofaRiemannianUUmanifold.HereUUwededuceasimilarresultforcountablediscretegroups.ThusUUthepurpGoseofthispaperistoproveUUthefollowingtheorem:
Theorem. ':
cmti10L}'etGbea(niteorinnite)countablegroup.ThenLFther}'eexistsa(connected)RiemannsurfaceMcasuchthatGisisomorphictothegr}'oupAutO! cmsy7O(M)ofal lholomorphicautomorphismsofM.Mor}'eover,
thereexistsaRiemannianmetrichonMsuchthatAutZO繲(M)e}'qualsthegr}'oupofal lisometriesof(M ;h).
OurKstrategyisasfollows:mUsingGaloistheoryofcoverings,MwerstconstructaBCRiemannsurfaceMٓR cmr71onwhichGacts.8ThenweremoveadiscretesubsetSZ
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cmsy10M1toIkillexcessautomorphisms.mHowever,KweIhavetoshowthatpassingfromuM1~toM1u!nSwedonotriskenlargingtheautomorphismgroup,i.e., DocumentuUa#Mathematica7(2002)413{417 *e̍6XIhtml: html:414}#Ji*orgWinkelmann+4e̍6XIweΤwillshowthateveryautomorphismfromM12nSb1extendstoM1|s.ݵF*orthis 6XIpurpGoseUUweemploytheF*reudenthal'stheoryoftopGologicalends.6XIFinally*,PhypGerbolicityOoftheRiemannsurfaceisexploitedtoensurethatthere6XIis:Kahermitianmetricofconstantnegativecurvqaturesuchthatthegroupofall6XIholomorphicUUautomorphismscoincideswiththegroupofallisometries.6XILethusremarkthatbyuniformizationtheoryitiswell-knownthatthefollowing6XIisVrthelistofallRiemannsurfacewithpGositive-dimensionalautomorphismgroup6XIandUUthattheirautomorphismgroupsarewell-known:iEXJOXK
msbm10P1|s(C),EXJOXKC,EXJOXKC^,EXJOXKH ^+
:=fz72C:=(zp)>0g,EXJOXKE 0er cmmi7M=C=h71;!Ǹidpqy msbm7Z mwithUU߸2H ^+s,EXJOXKA(r;1)=fz72C:r5