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On power subgroups of mapping class groups? Louis Funar Institut Fourier BP 74, UMR 5582 University of Grenoble I 38402 Saint-Martin-d'Heres cedex, France e-mail: September 21, 2011 Abstract In the first part of this paper we prove that the mapping class subgroups generated by the D-th powers of Dehn twists (with D ≥ 2) along a sparse collection of simple closed curves on an orientable surface are right angled Artin groups. The second part is devoted to power quotients, i.e. quotients by the normal subgroup generated by the D-th powers of all elements of the mapping class groups. We show first that for infinitely many D the power quotient groups are non-trivial. On the other hand, if 4g + 2 does not divide D then the associated power quotient of the mapping class group of the genus g ≥ 3 closed surface is trivial. Eventually, an elementary argument shows that in genus 2 there are infinitely many power quotients which are infinite torsion groups. 2000 MSC Classification: 57 M 07, 20 F 36, 20 F 38, 57 N 05. Keywords: Mapping class group, Dehn twist, power subgroup, symplectic group. 1 Introduction and statements The aim of this paper is to give a sample of results concerning power subgroups of mapping class groups.

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Onpowersubgroupsofmappingclassgroups∗LouisFunarInstitutFourierBP74,UMR5582UniversityofGrenobleI38402Saint-Martin-d’He`rescedex,Francee-mail:funar@fourier.ujf-grenoble.frSeptember21,2011AbstractIntheﬁrstpartofthispaperweprovethatthemappingclasssubgroupsgeneratedbytheD-thpowersofDehntwists(withD≥2)alongasparsecollectionofsimpleclosedcurvesonanorientablesurfacearerightangledArtingroups.Thesecondpartisdevotedtopowerquotients,i.e.quotientsbythenormalsubgroupgeneratedbytheD-thpowersofallelementsofthemappingclassgroups.WeshowﬁrstthatforinﬁnitelymanyDthepowerquotientgroupsarenon-trivial.Ontheotherhand,if4g+2doesnotdivideDthentheassociatedpowerquotientofthemappingclassgroupofthegenusg≥3closedsurfaceistrivial.Eventually,anelementaryargumentshowsthatingenus2thereareinﬁnitelymanypowerquotientswhichareinﬁnitetorsiongroups.2000MSCClassiﬁcation:57M07,20F36,20F38,57N05.Keywords:Mappingclassgroup,Dehntwist,powersubgroup,symplecticgroup.1IntroductionandstatementsTheaimofthispaperistogiveasampleofresultsconcerningpowersubgroupsofmappingclassgroups.WedenotebyM(S)themappingclassgroupoftheorientablesurfaceS,namelythegroupofisotopyclassesofhomeomorphismsthatﬁxpoint-wisetheboundarycomponents.SetΣgr,kfortheorientablesurfaceofgenusgwithkboundarycomponentsandrpunctures.WewillomittheindiceskandrinΣgr,kwhentheyare.orezDeﬁnition1.1.LetAbeacollectionof(isotopyclassesof)simpleclosedcurvesonthesurfaceS.WedenotebyM(S)(A;D)thesubgroupgeneratedbyD-thpowersofDehntwistsalongcurvesinA.WhenAisasetSCC(S)ofrepresentativesforallsimpleclosedcurvesuptohomotopyonthesurfaceSthegroupM(S)(SCC(S);D)willbedenotedbyM(S)[D].WewillomittheindiceskandrinM(Σgr,k)[D]andM(Σgr,k)(A;D)whentheyarezero.Forsimplicity,whenwedonotneedtospecifythesurfaceΣgr,kwewillusethenotationMgr,kforM(Σgr,k)andrespectivelyMgr,k[D]forM(Σgr,k)[D],withthesameconventionconcerningtheindiceskandr,whichweomitwhentheyarezero.ObservethatMg[D]isanormalsubgroupofMg,whosedeﬁnitionissimilartothatofthecongruencesubgroupsofthesymplecticgroups.Infact,letTadenotetheDehntwistalongthesimpleclosedcurvea.Thenforeveryh∈MgwehavehTaDh−1=ThD(a)∈Mg[D].AsMg[D]isgeneratedbytheTaD,forarunningoverthesetofallsimpleclosedcurves,itfollowsthatMg[D]isanormalsubgroup.TheﬁrstresultsonMg[D]wereobtainedbyHumphries([19])whoprovedthatMg/Mg[2],foreachg≥1,M2/M2[3]andM3/M3[3]areﬁnite,whileM2/M2[D]isinﬁnitewhenD≥4.Ontheotherhand,usingquantumtopologytechniquesweprovedin[13]thatthegroupsMg[D]areofinﬁniteindexinMg,ifg≥3,andD6∈{1,2,3,4,6,8,12}.∗Thisversion:December2010.Preprintavailableelectronicallyathttp://www-fourier.ujf-grenoble.fr/˜funar1

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