conjugacy growth of commutators peter s. park资料.pdf

该【conjugacy growth of commutators peter s. park资料 】是由【dt83088549】上传分享，文档一共【36】页，该文档可以免费在线阅读，需要了解更多关于【conjugacy growth of commutators peter s. park资料 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..JournalofAlgebra526(2019)423–458ContentslistsavailableatScienceDirectJournalofAlgebraate/,HarvardUniversity,Cambridge,MA02138,UnitedStatesofAmericaarticleinfoabstractArticlehistory:ForthefreegroupFronr>1generators(respectively,theReceived26November2018freeproductG1?G2oftwonontrivial?nitegroupsG1andG2),Availableonline27February2019weobtaintheasymptoticforthenumberofconjugacyclassesmutatorsinFr(respectively,G1?G2)withagivenwordMSC:lengthina?xedsetoffreegenerators(respectively,theset20F65ofgeneratorsgivenbythenontrivialelementsofG1andG2).mutators20E05infreegroupsandinfreeproductsbyWicks,andbuildson20E06theworksofRivinandSharp,whoasymptoticallycountedmutator-?:GrowthCommutatorsFreegroupsFreeproductsE-mailaddress:******@:///.-8693/?.:..(2019)423–?nitelygeneratedgroupwitha?∈GcanthenbewrittenasawordinthelettersofS,helengthofgbyinf{k|?c1,...,ck∈=c1...ck}.ConsidertheclosedballBk(G,S)?Gofradiuskinthewordmetric,de?nedasthesubsetconsistingofelementswithlength≤:howlargeis|Bk(G,S)|ask→∞,andmoregenerally,whatconnectionscanbemadebetweenthepropertiesofGandthisnotionofitsgrowthrate?Forexample,oneofthepioneeringresultsonthisquestionisthatofGromov[14],whoclassi?,thegrowthofgroupshasbeenwidelystudiedinvariouscontextslargelyarisingfromgeometricmotivations,,inadditiontoapplyingknowledgeonthegrowthofgroupstodescribegrowthinsuchgeometricsettings,onecanalsopassinformationintheotherdirection,.,,Milnor[23]usedinequalitiesrelatingthevolumeandcurvatureofRieman-pact,negatively->1generators,whichalsohasexponentialgrowth;moreprecisely,after?xingasymmetricgeneratingsetS·={x,...,x,x?1,...,x?1},itiseasytoseethat·1r1rkk|B(G,S)|=1+|?B(G,S)|=1+2r(2r?1)i?1kii=1i=1kr(2r?1)?1=1+,r?1where?Bk(G,S)denotesthesubsetoflength-,,helengthofCbyinflength(g),g∈C:..(2019)423–458425andde?ne?Bconj(G,S),theminimal-lengthelementsofaconjugacyclassarepreciselyitscyclicallyreducedelements,?Bconj(G,S)～(2r?1)k/k,whichagreeswiththeintuitionofkonjugatesamongthe2r(2r?1)k?1wordsoflengthk;forthefullexplicitformula,see[19,].Onecontextforwhichconjugacygrowthmaybeamorenaturalquantitytostudythanthegrowthrateintermsofelementsiswhencharacterizingthefrequencywithwhichaconjugacy-[G,G].Onthisfront,Rivin[putedthenumberckoflength-mutatorsubgroup(.,havetrivialabelianization)tobetheconstanttermintheexpressionr√k11(22r?1)Tk√xi+,22r?1i=1xiwhereTkdenotesthekthChebyshevpolynomialofthe?,Sharp[32]obtainedtheasymptoticsofck,givenby4r(2r?1)2m?1c2m～rrr,(2π)2σm2where√1121r+2r?12σ····=√1+√;2r?1r?2r?1notealsothatwhenkisodd,wehaveck=,onecanderivethegrowthofconjugacyclasseswithtrivialabelianizationbyusingM?biusinversion,duetothefollowingrelationships:ck=pd,d|kwherepddenotesthenumberofprimitive(.,notaproperpowerofanysubword)length-dwordswithtrivialabelianization,andconjpd|?Bk(G,S)∩[G,G]|=,dd|kwhichtogetherimplybyM?biusinversionthat:..(2019)423–458??conj1dce?μ(d)?|?Bk(G,S)∩[G,G]|=μce=deekdd|ke|de|kd|eceφ(k/e)φ(k/e)=·=ce.()ek/eke|ke|kIntheabove,,thereaderisdirectedto[19,Chapter17].Inthispaper,mutatorsratherthanforcommutator-?erentinthatitaimstosolveaDiophantineequationoveragroupG(whether,foragivenW∈G,thereexistX,Y∈GsuchthatXYX?1Y?1=W),ratherthanasubgroup-membershipproblem(whetherWisin[G,G]).Inparticular,mutatorsisnotmultiplicativelyclosed,,weuseatheoremofWicks[35],mutatorifandonlyifitisacyclicallyreducedmutatorsatisfyingthefollowingde??∈FroftheformABCA?1B?1C?,itisawordoftheformABCA?1B?1C?1suchthatthesubwordsA,B,andCarereduced;therearenocancellationsbetweenthesubwordsA,B,C,A?1,B?1,andC?1;andthe??cation,≥-torsinFrwithlengthkisgivenby2k2?1(2r?2)(2r?1)2k+Or(k),(),usingWicks’mutatorsinfreeproducts,toanswertheanalogousquestionforthefreeproductG1?G2oftwonontrivial?,freeproductshaveexponentialgrowth,whichhasbeenstudiedrecentlybyMann[18].WeconsiderthesetofgeneratorsS····=(G1{1})∪(G2\{1})ofG1?,atheoremofWicks[35]analogoustothepreviousoneimpliesthatanelementofG1?onjugatemutatorsatisfyingthefollowingde?nition.:..(2019)423–458427De??G2isawordW∈G1?G2thatisfullycyclicallyreduced,whichistosaythattheadjacentletters(.,nonidentityelementsofG1andG2intheword)areindi?erentfactorsofthefreeproduct,asarethe?rstandlastletters;andinoneofthefollowingforms:∈mutatorofG1,∈mutatorofG2,?1forα,α∈Gthatareconjugates,?1Aforα1,α2∈G1thatareconjugates,?1B?1,?1α3B?1α4forα1,α2,α3,α4∈G2satisfyingα4α3α2α1=1,?1α4B?1forα1,α2,α3,α4∈G1satisfyingα4α3α2α1=1,?1β2B?1α3C?1β3forα1,α2,α3,β1,β2,β3∈G2satisfyingα3α2α1=1andβ3β2β1=1,withA,B,ontrivial,?1β3B?1α3C?1forα1,α2,α3,β1,β2,β3∈G1satisfyingα3α2α1=1andβ3β2β1=1,withA,B,∈G1?G2withlengthk>1alternatesbetweenk/2lettersinG1\{1}andk/2lettersinG2\{1},wherek/,ifk>1isodd,thentherearenofullycyclicallyreducedelementsofG1?,mutatornotoftheform(1)or(2)inDe?,itisnecessarynotonlythatkiseven,butalsothatk/,allbutpossibly?nitelymany(thelength-1exceptions)mutatorsofG1?,weobtainthenumberoflength-kconjugacyclassesinG1?≥?G2withlengthkisgivenby1kk(|G|?1)(|G|?2)2+(|G|?2)2(|G|?1)k2(|G|?1)4?1(|G|?1)4?1121212192kk+O|G1|,|G2|k(|G1|?1)4(|G2|?1)4,wheretheimpliedconstantonlydependson|G1|and|G2|,|k>,thesetofcyclicallyreducedelementsofG1?G2(whoselettersofoddpositionareelementsofG1,withoutlossofgenerality)withlengthkandtrivialabelianizationbijectivelymapstotheCartesianproductoftwosets:thesetofclosedpathsoflengthk/pletegraphK|G1|with?xedbasepointP1,andthesetofclosedpathsoflengthk/pletegraphK|G2|with?xedbasepointP2.nForagraphE,thenumberofclosedpathsonEwithlengthnisgivenbyλ,whereλrangesovertheeigenvalues(countedwithmultiplicity):..(2019)423–458m≥2,theadjacencymatrixofKmisthem×mmatrixwithdiagonalentries0andallotherentries1;itseigenvaluesarem?1(withmultiplicity1)and?1(withmultiplicitym?1).Consequently,forevenintegersn>0,thenumberofclosedpathsonKmwithlengthnisgivenby(m?1)n+(m?1)(?1)n=m(m?1)n?,thenumberofsuchclosedpathswithagivenbasepointis(m?1)n?,thenumberofcyclicallyreducedwordswithlengthkandtrivialabelianizationinG1?G2isgivenbyk2?1k2?1(|G1|?1)(|G2|?1).ApplyingM?biusinversionasdonein(),weseethatthenumberofconjugacyclassesmutatorsinG1?parabletothesquarerootofthenumberofalllength-(X),andletCbeaconjugacyclassofπ1(X)withtrivialabelianization,correspondingtothefreehomotopyclassofaho-mologicallytrivialloopγ:S1→,mutatorlengthofC,de?nedasthemutatorswhoseproductisequaltoanelementofC,isalsotheminimumgenusofanorientablesurface(ponent)thatcontinu-ouslymapstoXsothattheboundaryofthesurfacemapstoγ[3,].Thus,aconjugacyclassofπ1(X)mutatorsifandonlyifitscorrespondingfreehomotopyclassγ:S1→Xsatis?esthefollowingtopologicalproperty:ponent()andacontinuousmapf:Y→Xsatisfyingf(?Y)=,,thenumberoffreehomotopyclassesofloopsγ:S1→Xwithlengthk(inthegeneratorsofS)satisfyingProperty()isgivenby2k2?1(2r?2)(2r?1)2k+Or(k),96rputable.:..(2019)423–?G2withthesetofgeneratorsS····=(G1\{1})∪(G2\{1}).Then,thenumberoffreehomotopyclassesofloopsγ:S1→Xwithlengthk(inthegeneratorsofS)satisfyingProperty()isgivenby1kk(|G|?1)(|G|?2)2+(|G|?2)2(|G|?1)k2(|G|?1)4?1(|G|?1)4?1192121212kk+O|G|,|G|k(|G1|?1)4(|G2|?1)4,12wheretheimpliedconstantdependsonlyon|G1|and|G2|,-?nitionoflengthforloops,oneexpectsthewordlengthofafreehomotopyclassofloopstocorrelatewiththegeometriclengthofthefreehomotopyclass,takentobethein?mumofthegeometriclengthsofitsloops,r1awedgej=1Sofrunitcircles,forwhichthereisperfectcorrelation;～=～=Γ\Hbeahyperbolicorbifold,whereΓ?PSL2(R),everyfreehomotopyclassofloopsinXthatdoesnotwraparoundacusphasauniquegeodesicrepresentative,sothegeometriclengthoftheclassisrealizedasthegeometriclengthofthisuniqueclosedgeodesic[2,].Inparticular,wehaveacanonicalcorrespondencebetweenhyperbolicfreehomotopyclassesofloops(.,onesthatdonotwraparoundacusp)(surfaceobtainedbyremovingthreedisjointopendisksfromasphere).WehavethatX～=～=Γ\HforΓ～=～=F2,so?,Chas,Li,andMaskit[4](Chas–Li–Maskit).Foranarbitraryhyperbolicmetricρonthepairofpantsthatmakesitsboundarygeodesic,letDkdenotethesetoffreehomotopyclasseswithwordlengthkina?,thereexistconstantsμandσ(dependingonρ)suchthatforanya<b,theproportionofC∈Dksuchthatthegeometriclengthh(C)oftheuniquegeodesicrepresentativeofCsatis?esh(C)?μk√∈[a,b]k:..(2019)423–458convergestob1?x2√e2σ2dxσ2πaask→∞.Inotherwords,ask→∞,thedistributionofgeometriclengthsg(C)forC∈,theresultingdistri-,≤LsatisfyingProperty().ForageneralhyperbolicorbifoldΓ\H,wehavetheaforementionednaturalcorrespon-dencebetweentheclosedgeodesicsof