combinatorial anabelian topics ii

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1、TOPICSSURROUNDINGTHECOMBINATORIALANABELIANGEOMETRYOFHYPERBOLICCURVESII:TRIPODSANDCOMBINATORIALCUSPIDALIZATIONYUICHIROHOSHIANDSHINICHIMOCHIZUKIJUNE2014Abstract.LetΣbeasubsetofthesetofprimenumberswhichiseitherequaltotheentiresetofprimenumbersorofcardina

2、l-ityone.Inthepresentpaper,wecontinueourstudyofthepro-Σfundamentalgroupsofhyperboliccurvesandtheirassociatedcon-figurationspacesoveralgebraicallyclosedfieldsinwhichtheprimesofΣareinvertible.Thestartingpointofthetheoryofthepresentpaperisacombinatorialana

3、belianresultwhich,unlikeresultsob-tainedinpreviouspapers,allowsonetoeliminatethehypothesisthatcuspidalinertiasubgroupsarepreservedbytheisomorphisminquestion.Thisresultallowsusto[partially]generalizecombina-torialcuspidalizationresultsobtainedinpreviou

4、spaperstothecaseofouterautomorphismsofpro-Σfundamentalgroupsofconfig-urationspacesthatdonotnecessarilypreservethecuspidalinertiasubgroupsofthevariousone-dimensionalsubquotientsofsuchafun-damentalgroup.Suchpartialcombinatorialcuspidalizationresultsallow

5、oneineffecttoreduceissuesconcerningtheanabeliangeom-etryofconfigurationspacestoissuesconcerningtheanabeliangeometryofhyperboliccurves.Theseresultsalsoallowus,inthecaseofconfigurationspacesofsufficientlylargedimension,togivepurelygroup-theoreticcharacteriza

6、tionsofthecuspidaliner-tiasubgroupsofthevariousone-dimensionalsubquotientsofthepro-Σfundamentalgroupofaconfigurationspace.Wethenturntothestudyoftripodsynchronization,i.e.,roughlyspeaking,thephenomenonthatanouterautomorphismofthepro-Σfundamentalgroupofa

7、logconfigurationspaceassociatedtoalogstablecurvetypicallyinducesthesameouterautomorphismonthevarioussub-quotientsofsuchafundamentalgroupdeterminedbytripods[i.e.,copiesoftheprojectivelineminusthreepoints].Ourstudyoftripodsynchronizationallowsustoshowtha

8、touterautomorphismsofpro-Σfundamentalgroupsofconfigurationspacesexhibitsomewhatdiffer-entbehaviorfromthebehaviorthatmaybeobservedinthecaseofdiscretefundamentalgroups,asaconsequenceoftheclassicalDehn-Nielsen-Baertheorem.Otherapplicationsofthetheo