combinatorial anabelian topics iii

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1、TOPICSSURROUNDINGTHECOMBINATORIALANABELIANGEOMETRYOFHYPERBOLICCURVESIII:TRIPODSANDTEMPEREDFUNDAMENTALGROUPSYUICHIROHOSHIANDSHINICHIMOCHIZUKIFEBRUARY2013Abstract.LetΣbeasubsetofthesetofprimenumberswhichiseitherequaltotheentiresetofprimenumbersorofcardinal-ityone.Inthepresentpaper,w

2、econtinueourstudyofthepro-Σfundamentalgroupsofhyperboliccurvesandtheirassociatedcon-figurationspacesoveralgebraicallyclosedfieldsinwhichtheprimesofΣareinvertible.Thefocusofthepresentpaperisonappli-cationsofthetheorydevelopedinpreviouspaperstothetheoryoftemperedfundamentalgroups,inth

3、estyleofAndr´e.Theseapplicationsaremotivatedbythegoalofsurmountingtwofunda-mentaltechnicaldifficultiesthatappearinpreviousworkofAndr´e,namely:(a)thefactthatthecharacterizationofthelocalGaloisgroupsintheglobalGaloisimageassociatedtoahyperboliccurvethatisgiveninearlierworkofAndr´eison

4、lyprovenforaquitelimitedclassofhyperboliccurves,i.e.,aclassthatis“farfromgeneric”;(b)theproofgiveninearlierworkofAndr´eofacertainkeyinjectivityresult,whichisofcentralimportanceinestablishingthetheoryofa“p-adiclocalanalogue”ofthewell-known“global”theoryoftheGrothendieck-Teichm¨ulle

5、rgroup,containsafundamentalgap.Inthepresentpaper,wesurmountthesetechnicaldifficultiesbyintroduc-ingthenotionofan“M-admissible”,or“metric-admissible”,outerautomorphismoftheprofinitegeometricfundamentalgroupofap-adichyperboliccurve.Roughlyspeaking,M-admissibleouterautomorphismsareouter

6、automorphismsthatarecompatiblewiththedataconstitutedbytheindicesatthevariousnodesofthespe-cialfiberofthep-adiccurveunderconsideration.Bycombiningthisnotionwithcombinatorialanabelianresultsandtechniquesde-velopedinearlierpapersbytheauthors,togetherwiththetheoryofcyclotomicsynchroniz

7、ation[alsodevelopedinearlierpapersbytheauthors],weobtainageneralizationofAndr´e’scharacterizationofthelocalGaloisgroupsintheglobalGaloisimageassociatedtoahyperboliccurvetothecaseofarbitraryhyperboliccurves[cf.(a)].Moreover,byapplyingthetheoryoflocalcontractibilityofp-adicanalytics

8、pacesdevelopedbyBerkovich,weshowt