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1、TopicsSurroundingtheCombinatorialAnabelianGeometryofHyperbolicCurvesI:InertiaGroupsandProfiniteDehnTwistsYuichiroHoshiandShinichiMochizukiReceivedMarch31,2011.RevisedDecember28,2011.2010MathematicsSubjectClassification.Primary14H30;Secondary14H10.Keywordsandphrases.anabeliangeometry,combin
2、atorialanabeliangeom-etry,profiniteDehntwist,semi-graphofanabelioids,inertiagroup,hyperboliccurve,configurationaspace.ThefirstauthorwassupportedbyGrant-in-AidforYoungScientists(B),No.22740012,JapanSocietyforthePromotionofScience.2Abstract.LetΣbeanonemptysetofprimenumbers.Inthepresentpa-per,
3、wecontinueourstudyofthepro-Σfundamentalgroupsofhyper-boliccurvesandtheirassociatedconfigurationspacesoveralgebraicallyclosedfieldsofcharacteristiczero.Ourfirstmainresultasserts,roughlyspeaking,thatifanF-admissibleautomorphism[i.e.,anautomorphismthatpreservesthefibersubgroupsthatariseaskernel
4、sassociatedtothevariousnaturalprojectionsoftheconfigurationspaceunderconsider-ationtoconfigurationspacesoflowerdimension]ofaconfigurationspacegrouparisesfromanF-admissibleautomorphismofaconfigura-tionspacegroup[arisingfromaconfigurationspace]ofstrictlyhigherdimension,thenitisnecessarilyFC-adm
5、issible,i.e.,preservesthecus-pidalinertiasubgroupsofthevarioussubquotientscorrespondingtosurfacegroups.Afterdiscussingvariousabstractprofinitecombinato-rialtechnicaltoolsinvolvingsemi-graphsofanabelioidsofPSC-typethataremotivatedbythewell-knownclassicaltheoryoftopologicalsurfaces,weprocee
6、dtodevelopatheoryofprofiniteDehntwists,i.e.,anabstractprofinitecombinatorialanalogueofclassicalDehntwistsassociatedtocyclesontopologicalsurfaces.ThistheoryofprofiniteDehntwistsleadsnaturallytocomparisonresultsbetweentheabstractcombinatorialmachinerydevelopedinthepresentpaperandmoreclas-sica
7、lscheme-theoreticconstructions.Inparticular,weobtainapurelycombinatorialdescriptionoftheGaloisactionassociatedtoa[scheme-theoretic!]degeneratingfamilyofhyperboliccurvesoveracompleteequicharacteristicdiscretevaluationringofcharacteristiczero.Finally,weapplythetheoryofprofin
