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arXiv:math/9912176v1[math.AG]21Dec1999GENUSZEROACTIONSONRIEMANNSURFACESSADOKKALLELANDDENISSJERVE1.INTRODUCTIONInthispaperwesolvethefollowingproblem:PROBLEM.WhichfinitegroupsGadmitanactiononsomecompactconnectedRiemannsurfaceMsothatifHisanynon-trivialsubgroupofGthentheorbitsurfaceM/Hhasgenuszero,thatis16=H⊆G=⇒M/H=P1(C).(1)AnyfinitegroupGwilladmitinfinitelymanyactionsG×M→MsothatM/G=P1(C).ItwillturnoutthatveryfewgroupsGsatisfy(1).TheactionG×M→Missupposedtobeanalyticandeffective.ThusGisasubgroupofAut(M),thegroupofallanalyticautomorphismsofM.ForanyactionG×M→M,ifM/H1=P1(C)forsomeH1⊆G,thenautomaticallyM/H2=P1(C)ifH1⊆H2⊆G.ThismeansthatwecanreducetheproblemtotheconsiderationofcyclicsubgroupsZp⊆G,wherepisaprimedividingtheorderofG.Ourproblemthusbecomes:PROBLEM.WhichfinitegroupsGadmitanactiononsomecompactconnectedRiemannsurfaceMsothatifHisanycyclicsubgroupofprimeorderpthentheorbitsurfaceM/Hhasgenuszero,thatisM/Zp=P1(C),forallZp⊆G,wherepisaprimedividing|G|.(2)AllRiemannsurfacesconsideredinthispaper,withjustafewexceptions,willbecompactandconnected.Theexceptionsarethecomplexplaneandtheupperhalfplane.Thegroupsbeingstudiedinthispaperwillalwaysbefinite.InoursolutionofthisproblemnotonlydowedescribethegroupsGadmittingsuchactions,butwealsodetermineallpossibleactionsforeachgroup.Thisamountstodescribingalladmissibleepimorphismsθ:Γ→G,whereΓisaFuchsiangroupofsignature(0|n1,...,nr).SeeSection(2).InSection(3)weconsiderthelowgeneracasesg=0,1,wheregisthegenusofM.Otherthantheseexceptionswewillusuallyassumethatg1.Date:February8,2008.ResearchsupportedbyPImsNSERCgrantA7218.12SADOKKALLELANDDENISSJERVEDefinition1.AgroupGissaidtohavegenuszeroifthereexistsaRiemannsurfaceMandanactionofGonMsatisfying(1),orequivalently(2).Wealsosaythattheactionhasgenuszero.Toexplainthesignificanceoftheaboveproblem,andtogiveitsomemotivation,werecallthedefinitionofafixedpointfreelinearactiononaCvectorspaceV.Definition2.AlinearactionG×V→VissaidtobefixedpointfreeifS∈G,S6=1=⇒S(v)6=vforallv∈V,v6=0.NowletVbethevectorspaceofholomorphicdifferentialsonM.VisaCvectorspaceofdimensiong,wheregisthegenusofM.TheactionofGonMinducesalinearactiononthevectorspaceV.IfS∈GisanelementoforderpthentheinducedlineartransformationS∗:V→Valsohasorderp,assumingg1.ItfollowsthattheeigenvaluesofS∗arepthrootsofunity.Inparticular+1mightbeaneigenvalueofT∗.PartoftheproofoftheEichlertraceformula,seeFarkasandKra[2],statesthatthedimensionofthe+1eigenspaceofS∗isthegenusofM/H,whereH∼=ZpisthecyclicgroupoforderpgeneratedbyS.ThereforetheinducedactionG×V→Visfixedpointfreeif,andonlyif,G×M→Misagenus-zeroaction.WecanimposeametriconVsothattheactionbecomesunitary.ThustheactionG×M→Mhasgenuszeroif,andonlyif,S2g−1/Gisanellipticspaceform.Thisequivalenceassumesthatg1.HereS2g−1denotestheunitsphereinV.TheconverseproblemofwhenafixedpointfreelinearactionG×V→Varisesfromagenus-zeroactiononsomeRiemannsurfaceisnotconsideredinthispaper.GroupsGwhichadmitfixedpointfreelinearactionshavebeenclassified,seeWolf[5].InparticulartheymustsatisfyastrongconditionontheirSylowp-subgroups.FirstrecallthatthegeneralizedquaterniongroupQ(2n)isdefinedasfollows:Definition3.Thegeneralizedquaterniongroupisthegroupwiththepresentation:Q(2n)=DA,B|A2n−1=1,B2=A2n−2,BAB−1=A−1E.(3)Wewillalwaysassumen≥3,sinceotherwiseQ(2n)iscyclic.Definition4.WesaythatagroupGsatisfiestheSylowconditionsifthefollowingtwoconditionshold.1.ForanoddprimeptheSylowp-subgroupsarecyclic.2.TheSylow2-subgroupsareeithercyclicorgeneralizedquaternion.TheSylowconditionsareequivalenttothefollowing:Definition5.WesaythatagroupGsatisfiesthep2conditionsifeverysubgroupoforderp2iscyclic,wherepisanyprime.GENUSZEROACTIONSONRIEMANNSURFACES3EverygroupGadmittingafixedpointfreelinearactionsatisfiestheseconditions.Infactthesegroupsmustsatisfytheevenstrongerpqconditions.Definition6.AgroupGsatisfiesthepqconditionsifeverysubgroupoforderpqiscyclic,wherepandqarearbitraryprimes.LetD2ndenotethedihedralgroupoforder2n.IfnisoddthenD2nsatisfiesthep2conditionsbutnotthepqconditions.InfactanygroupGofevenorderwhichadmitsafixedpointfreelinearactionmusthaveexactlyoneelementoforder2,andthiselementgeneratesthecenterofG.Todescribeourresultsweneedanotherdefinitionandsomenotation.Definition7.WeletGm,n(r)denotethegrouppresentedasfollows:generators:A,B;relations:Am=1,Bn=1,BAB−1=Ar;(4)conditions:GCD((r−1)n,m)=1andrn≡1(modm).ThesegroupsarepreciselythegroupshavingallSylowsubgroupscyclic,seeBurnside[1].Toavoidthetrivialcaseswherethegroupiscyclicwewillusuallyassumethatm1,n1.Notethattheconditionsimplyr6≡1(modm).IfddenotestheorderofrmodulomthenZassenhaus[6]provedthatGm,n(r)satisfiesthepqconditionsif,andonlyif,everyprimedivisorofdalsodividesnd.HealsoprovedthatGm,n(r)admitsafixedpointfreelinearrepresentationif,andonlyif,thepqconditionshold.ThegroupsGm,n(r)areknownasZassenhausmetacyclicgroups(abbreviatedtoZMgroups).LetI∗denotethebinaryicosahedralgroup.Ithasorder120andadmitsafixedpointfreelinearrepresentation.I∗isnon-solvable,andifGisanon-solvablegroupadmittingalinearfixedpointfreerepresentationthenGcontainsI∗asasubgroup.Themainresultsofthispaperarecontainedinthefollowin