Ordering the space of finitely generated groups

0.46 MB·English

by  Laurent Bartholdi

ORDERING THE SPACE OF FINITELY GENERATED GROUPS
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2 LAURENTBARTHOLDIANDANNAERSCHLER
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0 ABSTRACT. Weconsidertheorientedgraphwhoseverticesareisomorphismclassesof
2 finitelygeneratedgroups,withanedgefromGtoH if,forsomegeneratingsetT inH
n andsomesequenceofgeneratingsetsSiinG,themarkedballsofradiusiinpG,Siqand
a pH,Tqcoincide.
J GivenanilpotentgroupG,wecharacterizeitsconnectedcomponentinthisgraph: if
thatconnectedcomponentcontainsatleastonetorsion-freegroup,thenitconsistsofthose
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groupswhichgeneratethesamevarietyofgroupsasG.
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Thearrowsinthegraphdefineapreorderonthesetofisomorphismclassesoffinitely

generated groups. Weshowthatapartialordercanbeimbeddedinthispreorderifand
]
R onlyifitisrealizablebysubsetsofacountablesetunderinclusion.
Weshowthateverycountable groupimbeds inagroupofnon-uniform exponential
G
growth. Inparticular, thereexistgroupsofnon-uniformexponential growththatarenot
. residually ofsubexponential growthanddonotadmitauniformimbeddingintoHilbert
h
space.
t
a
m
[

1 3 1. INTRODUCTION
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Our aim, in this paper, is to relate the following preorder on the set of isomorphism
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6 5 classesoffinitelygeneratedgroupswithasymptoticandalgebraicpropertiesofgroups.
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4 6 Definition 1.1. Let G,H be finitely generated groups. We write G ă H, and say that
. 7 G preforms H, if the following holds. There exist a finite generating set T of H and a
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0 8 sequenceoffinitegeneratingsetsS1,S2,... ofG,withbijectionsSn Ñ T suchthat,for
3 9 allR P N,ifnislargeenoughthentheballsofradiusRinthemarkedCayleygraphsof
1 10 pG,SnqandpH,Tqareisomorphic.
v: 11 WedenotebyCpG,SqtheCayleygraphofthegroupGwithrespecttothegenerating
i 12 setS. Itsedgesaremarkedwiththegeneratortheycorrespondto.
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IfGpreformsH,thenwealsosaythatH ispreformedbyG.
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a Definition 1.1 can be interpreted in terms of the Chabauty-Grigorchuktopology, also
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calledthe Cayley topology,definedasfollows. Thespace ofmarked groupsis the setG
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16 of pairs pG,Sq with G a finitely generated group and S a finite ordered generating set,
considereduptogroupisomorphismpreservingthegeneratingset. Thisisequippedwith
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18 a natural topology, two marked groups pG,Sq and pG1,S1q being close to each other if
19 markedballsoflargeradiusintheCayleygraphsCpG,SqandCpG,S1qareisomorphic.
Chabautyconsideredthistopologicalspacein[18,§3];heusedittodescribethespace
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of lattices in locally compact groups. Gromov [28, pages 71–72] used it to derive an
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effectiveversionofhistheoremongroupsofpolynomialgrowth.Grigorchuk[27]wasthe
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firstto study thistopologysystematically; in particular, he used it to constructgroupsof
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Date:January20,2013.
TheworkissupportedbytheERCstartinggrant257110“RaWG”,theANR“DiscGroup:facettesdesgroupes
discrets”andtheCourantResearchCentre“HigherOrderStructures”oftheUniversityofGo¨ttingen.
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wildly-oscillatingintermediategrowth, by approximatingthem in G by solvable groups.
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Forgeneralitiesonthethespaceofmarkedgroups,see[19].
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26 Definition1.1maythenbeformulatedasfollows: G ă H ifandonlyiftheclosureof
theisomorphismclassofGintheChabauty-GrigorchuktopologycontainsH.
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It is essentialforourdefinitionthat we considerlimits in the space of markedgroups
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of a fixed group, letting only its generating set vary. Various authors have already con-
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sidered limits in the space of marked groups, not necessarily restricting to limits within
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one isomorphismclass. Limits of one fixed grouphave been studied when this groupis
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free: theycoincidewithlimitsgroups,asshownbyChampetierandGuirardel[19,Theo-
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rem1.1];see§6.1formorereferences. Zarzycki[61]considersgroupsthatarepreformed
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byThompson’sgroupF,andgivessomenecessaryconditionsforHNNextensionstoap-
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pear in this manner; Guyot [31,32] considers groups that are preformed by G for some
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metabelian groups G, and identifies their closure in G. On the other hand, groups that
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preformfreegroupsaregroupsthathaveinfinitegirthforgeneratingsetsoffixedcardinal-
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ity.OlshanskyandSapircharacterizethemin[45]asgroupswithoutalmost-identities,see
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also§6.2.
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40 We recall that a preorder is a binary relation - such that A - C whenever A - B
41 andB - C andsuchthatA - A forall A. If furthermore‘A - B andB - A’ imply
42 A“B,thenitisanorder. Apreorderisdirectedifeveryfinitesubsethasanupperbound.
43 Itiseasyto seethattherelation‘ ă’isa preorder,andthat G ă H doesnotdependon
thechoiceofafinitegeneratingsetinH (seeLemmas2.2and2.1inthenextsection). It
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is also notdifficultto see thatthe restriction ofthis relationto someclasses ofgroupsis
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anorder;thishappens,forexample,forresiduallyfinitefinitelypresentedgroups,suchas
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polycyclicgroups(seeCorollary2.7). Forsomeotherclassesofgroupsthisisnottrue:for
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example,thereexistsolvablegroupsGadmittingacontinuumofnon-isomorphicsolvable
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groupswhichareequivalentto Gunderourpreorder,thatis,whichbothpreformandare
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preformedby G. Nekrashevychgave in [43] examples of groups acting on rooted trees
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whichareequivalentunderourpreorder.
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In many cases, if A preforms B, then A “looks smaller” than B. Simple examples
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53 of this kind include: Zm ă Zn if and only if m ď n; free groups satisfy Fm ă Fn
54 if and only if m ď n; and the n-generatedfree groups Vn in the variety generatedby a
55 torsion-freenilpotentgroupofnilpotencyclass csatisfy, form,n ě c, thesame relation
56 Vm ă Vn ifandonlym ď n, see TheoremA.On theotherhand,itmayhappenfor A
that preform B that the growth of A is largerthan the growth of B; we consider this in
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moredetailin§1.3.
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59 1.1. Thestructureofcomponents. Weview ă asspecifyingtheedgesetofanoriented
graph with vertex set the isomorphism classes of finitely generated groups. In studying
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this graph, we may consider independently the connected components of its underlying
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unorientedgraph. Whatdotheylooklike? Whichcomponentsadmitaninitialvertex? a
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terminal vertex? Given a connected component, does it have an upper bound? What is
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thegroupofpreorderpreservingbijectionsofagivencomponent?Whichgroups’strongly
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connectedcomponentarereducedtopoints,orhavethecardinalityofthecontinuum?
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Unlike some other natural preorders, such as “being a subgroup”, “being a quotient
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67 group”, or “being larger” in the sence of Pride (G %p H if H1 is a quotient of G1, for
respectivequotientsG ,H offinite-indexsubgroupsofG,H byfinitenormalsubgroups,
68 1 1
see[49,54]),thepreorderthatweconsiderinthispaperhasinfinitelymanyconnectedcom-
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ponents.AneasyexampleistheconnectedcomponentofZ: itcontainsallinfiniteabelian
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ORDERINGTHESPACEOFFINITELYGENERATEDGROUPS 3
any partial order real-
izable by subsets of a
countable set under in-
clusion
Limitgroups
A group of non-uniform exponential
F F ¨¨¨
growth,containingagivengroup 2 3
Surfacegroups
Grigorchuk’sgroupG
012
Groupswithoutalmost-identities
Somesolvablegroups
TheBaumslag-solitargroupBp1,2q
FIGURE 1. Someclassesofgroupsandtheirrelationshipunder ă
groups,andwedescribethegroupoftheorderpreservingbijectionsofthiscomponentin
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Proposition3.7.
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For a nilpotent torsion-free G group, its connected component is closely related to
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groupsthatgeneratedthesamevarietyasG.
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TheoremA(=Proposition4.6). LetGbeafinitelygeneratednilpotentgroupsuchthatG
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76 andG{TorsionpGq generatethe same variety (i.e. satisfy the same identities). Then, for
77 allk PNlargeenough,Gpreformstherelativelyfreegroupofrankkinthatvariety.
Inparticular,everyfinite set ofsuchnilpotentgroupshasa supremumwith respectto
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ourpreorder. We believe, in fact, that this last statement holdsfor all virtually nilpotent
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groups.However,ifanilpotentgroupGisnottorsion-free,theconnectedcomponentofG
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canbemuchsmallerthatthesetof(isomorphismclasses) ofgroupsgeneratingthesame
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varitetyasG,seeCorollary4.12.
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We show, on the other hand, that the preordertypes that can occur are quite general,
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evenwithinsolvablegroupsofclass3,orwithingroupsthatpreformfreegroups:
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85 TheoremB(=Corollary5.2andRemark6.9). LetpX,-qbeapreorder.ThenpG{–, ăq
86 containspX,-qasasubpreorderifandonlyifX hascardinalityatmostthecontinuum,
andallthepartialordersitcontainsare imbeddablein thepartialorderofsubsetsofB
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underinclusion,foracountablesetB.
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89 Furthermore,the imbeddingof pX,-qcan thenbe chosento bewithin the setof iso-
morphismclassesofsolvablegroupsofsolubilityclass3or,alternatively,withinthesetof
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isomorphismclassesofgroupsthatpreformF .
91 3
Thomasstudiesin[54]thecomplexity,withrespecttotheBorelianstructureon G,of
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Pride’s“largeness”preorderandofthe “beinga quotient”preorder. He showsthatthese
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94 preordersarehighintheBorelhierarchy(namely, Kσ-universal). Thepreorder ă differs
fromthe abovementionedpreorderseven if we forgetthe underlyingBorelian structure:
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the quotients and largeness preorders have chains with cardinality the continuum, while
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97 (byTheoremB)chainsfor ă arecountable.
4 LAURENTBARTHOLDIANDANNAERSCHLER
1.2. Groupslargerorsmallerthanagivengroup. GivenagroupG,howmanygroups
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preformG? HowmanygroupsarepreformedbyG? Howbigistheconnectedcomponent
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ofG? Whatisitsdiameter?
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We notethat,ifagroupGisvirtuallynilpotent,thenitscomponentiscountable. The
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numberofgroupsthatarepreformedbyGiscountablyinfinite.
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If G is a free group, a surface group, or more generally a non-abelian limit group
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(see §6.1), then there are countably many groups that are preformedby G, see [37,52].
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However,theconnectedcomponentofGhasthecardinalityofthecontinuum,seeExam-
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ple6.3.
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We study the groups that preform free groups. Schleimer considered groups of un-
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boundedgirth(therearegeneratingsetssuchthatthesmallercycleintheCayleygraphis
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arbitrarilylong)inanunpublishednote[51],andtheyareintimatelyconnectedtogroups
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that preform free groups, see Question 8.5. The latter are groups that do not satisfy an
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almost-identity[45]: awordwhoseevaluationvanishesoneverygeneratingset. Olshan-
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skii and Sapir show in [45] that there are groupswith non-trivialquasi-identitiesamong
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groupssatisfyingnonon-trivialidentitity.
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In §6.3, we modify a criterion by Abe´rt [1] about groups without identities to deter-
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mine when a group has no almost-identity. This lets us answer negatively a question
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bySchleimer[51,Conjecture6.2]thatgroupsofunboundedgirthhaveexponentialword
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growth(see§1.3forthedefinitionofgrowth):
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TheoremC(=Corollary6.12). ThefirstGrigorchukgroupG preformsF .
118 012 3
ExtendinganargumentbyAkhmedov(see[4]),wegiveacriterionforawreathproduct
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withinfiniteactinggrouptopreformafreegroup:
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PropositionD(=Proposition6.15). LetGandH befinitelygeneratedgroups,andsup-
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122 posethatH isinfinite. ThentherestrictedwreathproductG≀H :“ GpHq ⋊H preforms
afreegroupifandonlyifatleastoneofthefollowingconditionsholds:
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(1) Gdoesnotsatisfyanyidentity;
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(2) H doesnotsatisfyanyalmost-identity.
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Fromthis,wededuce(seeRemark6.21)thattheconnectedcomponentofthefreegroup
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hasdiameteratleast3;thisisincontrastwiththenilpotentcase,seeTheoremA.Thereare
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solvablegroups,andinfinitefreeBurnsidegroups,atdistance2fromafreegroup.
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See also subsection 2.4 where we discuss groups that preform a group containing a
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givensubgroup.
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1.3. Growth of groups. We finally give in §7 some new examples of groups of non-
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uniformexponentialgrowth. Recall that, fora group G generatedby a set S, its growth
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133 functioncountsthenumberνG,SpRqofgroupelementsexpressibleasaproductofatmost
134 R generators. The group has exponential growth if λG,S :“ lim aR νG,SpRq ą 1 and
subexponentialgrowthotherwise;itthenhaspolynomialgrowthifν isdominatedbya
135 G,S
polynomial,andintermediategrowthotherwise. Theexistenceofgroupsofintermediate
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growthwasaskedbyMilnorin[42],andansweredbyGrigorchukin[27],bymeansofhis
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groupG .
138 012
If G has exponential growth, then it has uniform exponential growth if furthermore
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140 infSλG,S ą1. Theexistenceofgroupsofnon-uniformexponentialgrowthwasaskedby
Gromov in [29, Remarque 5.12]; see also [36]. The first examples were constructed by
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Wilson[59];seealso[8,43,60].
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ORDERINGTHESPACEOFFINITELYGENERATEDGROUPS 5
TheoremE(=Corollary7.3). EverycountablegroupmaybeimbeddedinagroupGof
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non-uniformexponentialgrowth.
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145 Furthermore, letα « 0.7674be the positive root of23´3{α `22´2{α `21´1{α “ 2.
ThenGmayberequiredtohavethefollowingproperty: thereisaconstantK suchthat,
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147 foranyRą0,thereexistsageneratingsetS ofGwith
ν prqďexppKrαqforallr ďR.
G,S
Theorem E implies the existence of groups of non-uniform exponential growth that
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do notimbeduniformlyinto Hilbertspace; this answers a question by Brieussel [15, af-
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ter Proposition 2.5], who asked whether there exist groups of non-uniform exponential
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growthwithouttheHaagerupproperty.Wealsoconstructgroupsofnon-uniformexponen-
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tial growth that admit infinitely many distinct intermediate growth functions at different
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scales. Moreover, these examples can be constructed among groups that preform free
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groupsandgroupsofintermediategrowth.
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The idea of the proof of Theorem E is as follows. We denote by G the first Grig-
155 012
156 orchuk group. It acts on the infinite binary tree t0,1u˚ and its boundary t0,1u8. We
157 denote by X the orbit G012 ¨18. We prove in Corollary 7.2 that the group G≀X G012
has non-uniform exponential growth whenever G is a group of exponential growth. To
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proveCorollary7.2weshowthatG≀ G preformsagroupofintermediategrowth.(In
159 X 012
fact,allknownexamplesofgroupsofnon-uniformexponentialgrowthpreformgroupsof
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intermediategrowth,thoughthecorrespondinggroupofintermediategrowthisnotalways
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givenexplicitlybytheirconstruction;formoreonthisseeQuestion8.7).
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1.4. Acknowledgments. TheauthorsaregratefultoYvesdeCornulier,SlavaGrigorchuk,
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Fre´de´ricPaulinandPierredelaHarpefortheircommentsonanearlierversionofthisman-
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uscript; to Olga Kharlampovichforhavingcorrectedan inaccuracyin ourunderstanding
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oflimitgroups;toMishaGavrilovichforenlighteningdiscussions;andtoSimonThomas
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andTodorTsankovfortheirgenerousexplanationsonBorelrelations.
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168 2. FIRST PROPERTIES AND EXAMPLES
Lemma2.1(Aspecialcaseof[19,Proposition2.20]). The“forsomegeneratingsetT”
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inDefinition1.1maybechangedto“foreverygeneratingsetT”.
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171 Proof. AssumeG ă H,thatT generatesH andthatCpG,SnqcoincideswithCpH,Tq
172 oneverlargerballs. Writeτn :T ÑSnthebijections.
173 LetT1beanothergeneratingsetofH;writeeverytPT1asawordwtoverT.Letkbe
174 themaximumofthelengthsofthewt. ConsiderthegeneratingsetsSn1 “twtpτnq: tPTu
175 ofGobtainedbyreplacingeachT-letterinwtbyitscorrespondingelementτnptqPG.
176 Then,ifCpG,SnqXBp1,RqisisomorphictoCpH,TqXBp1,Rq,thenCpG,Sn1qX
177 Bp1,tR{kuqis isomorphicto CpH,T1qXBp1,tR{kuq, since theyare respectivesubsets
178 intheisomorphicgraphsCpG,SnqXBp1,RqandCpH,TqXBp1,Rq. (cid:3)
179 Lemma2.2. Therelation ă isapreorder.
180 Proof. ItisclearthatG ă GholdsforallgroupsG.
181 Consider now G ă H ă K, and let U be a generating set for K. There are then
182 generatingsetsTn forH,inbijectionwithU,suchthatCpH,TnqandCpK,Uqagreein
everlargerballs. For each n, there are generatingsets S for G, in bijection with T ,
183 mn n
184 suchthatCpG,SmnqandCpH,Tnqagreeineverlargerballs.
185 Therefore,thegeneratingsetsSnn,whichareinbijectionwithU,aresuchthatCpG,Snnq
186 andCpK,Uqagreeineverlargerballs,whichshowsG ă K. (cid:3)
6 LAURENTBARTHOLDIANDANNAERSCHLER
LetFbethefreegrouponinfinitelymanygeneratorsx ,x ,...,andconsiderthespace
187 1 2
188 G offinitelygeneratedgroupspG,Tqwithmarkedgeneratingset. Thismarkingmaybe
189 givenbyahomomorphismF ։ Gsuchthatalmostallxn mapto1;andthisidentifiesG
with the set of normalsubgroupsof F containingalmost all the x . This turns G into a
190 n
locallycompactPolishspace. Inthisalternativeterminology,wehavetheobvious
191
192 Lemma 2.3. Let G,H be finitely generated groups. Then G ă H if and only if for
193 some (hence all) generating set T, the marked group pH,Tq belongs to the closure of
194 tpG,Sq: S generatesGuinG.
195 WeobservethatifG ă H andeitherGorH arefinite,thenG “ H. Wethusrestrict
ourselvestoinfinite,finitelygeneratedgroups.
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Lemma2.4. LetGbeafinitelygeneratedgroup,andletH beafinitelypresentedgroup.
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198 IfG ă H,thenGisaquotientofH.
Proof. LetT beageneratingsetofH,andletRbethemaximallengthofH’srelatorsin
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200 thatgeneratingset. IfG ă H,thenthereexistsageneratingsetSforGsuchthatCpG,Sq
201 andCpH,TqcoincideinaballofradiusR;soallrelationsofH holdinT. (cid:3)
We note ([19, Example 2.4(e)]) that every residually finite group is a limit of finite
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groups; however, the closure of the set of finite groups in G has not been convincingly
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identified.
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IthasbeenshownbyShalom[53]thateverygroupGwithKazhdan’sproperty(T)isa
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quotientofafinitelypresentedgroupwithKazhdan’sproperty(T).Therefore,
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207 Lemma2.5([19,Proposition2.15]). IfG ă H andGdoesnothaveKazhdan’sproperty
208 (T),thenneitherdoesH. (cid:3)
There are isolated points in the space of groups; they are studied in [23]. Clearly,
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210 isolatedgroupsareminimalelementsfor ă;buttheconverseisnottrue. Forexample,Z
211 andZ‘Z{pZareminimal,butnoneofthemisisolated.
212 2.1. Partialorders. Onsomeclassesofgroups,therelation ă isalsoantisymmetric,and
thereforedefines a partial order. Recall that a group G is Hopfian if everyepimorphism
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214 G։Gisanautomorphism.
215 Lemma 2.6. Among Hopfian, finitely presented groups, ă is an order relation. More
216 generally,ifGandH arefinitelypresentedgroupswithG ă H ă GandGisHopfian,
thenGandH areisomorphic.
217
218 Proof. FromG ă H andLemma2.4wededucethatGisaquotientofH;andsimilarly
219 H is a quotient of G. Therefore we have epimorphisms G ։ H ։ G, and since G is
220 Hopfiantheseepimorphismsareisomorphisms. (cid:3)
221 Corollary 2.7. The relation ă is an order relation on polycyclic groups, and on limit
groups.
222
Proof. Polycyclicgroupsareknowntobefinitelypresentedandresiduallyfinite. Wewill
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recallsomeknownfactsaboutlimitgroupsin§6.1;fortheproofofthecorollaryitsuffices
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toknowthatlimitgroupsareresiduallyfreeandthereforeresiduallyfinite; andthatthey
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arefinitelypresented.
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SinceresiduallyfinitegroupsareHopfian(see[40]),thecorollaryfollowsfromLemma2.6.
227
228 (cid:3)
ORDERINGTHESPACEOFFINITELYGENERATEDGROUPS 7
2.2. Identitiesanduniversalstatements. LetGbeagroup. AnidentityforGisanon-
229
230 trivial word wpx1,x2,...q in the free group on countably many generators, such that
231 wpg1,g2,...q “ 1 for every choice of gi P G. Note that w is really a word in finitely
232 manyofthexi’s,namelyw “wpx1,...,xnqforsomenPN.
233 AnidentityforGisreallythefollowinguniversalsentence.‘@g1,g2...pw “1q’.More
generally,anywell-formedexpressionmadeofconjunctions,disjunctions,equalities,and
234
universalquantifiers,isapositiveuniversalsentence.Iffurthermorenegationsareallowed,
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itisauniversalsentence.ThevarietygeneratedbyagroupGisthesetofidentitiesthatit
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satisfies;andits(positive)universaltheoryisthesetof(positive)universalsentencesthat
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itsatisfies.
238
239 Forexample,considerthegroupG “ xx,y,z | rx,ysz´1,z2,rx,zs,ry,zsy. Itsatisfies
240 theidentityrx1,x2s2. Italsosatisfiesthepositiveuniversalstatement
@x ,...,x prx ,x s“1_rx ,x s“1_¨¨¨_rx ,x s“1q.
1 4 1 2 1 3 3 4
As a last example, limits groups are known to be “commutative-transitive”; this is the
241
universalstatement
242
(1) @x,y,zprx,ys“1^ry,zs“1ñrx,zs“1q.
243 Note that this statement is not positive; rewriting it in terms of the primitives _,^,
244 gives @x,y,zp prx,ys “ 1 ^ ry,zs “ 1q _ rx,zs “ 1q. An example of a positive
statementappearsinExample4.13. Formoredetailsrelatinglogictothespaceofmarked
245
groups, see §6.1 and [19, §5]. In particular, the first assertion of the following lemma
246
is[19,Proposition5.2].
247
248 Lemma2.8. (1) IfG ă H andGsatisfiesauniversalstatement(e.g.,anidentity),
thenH satisfiesittoo;
249
250 (2) If G ă H and H is a finitely presented group satisfying a positive universal
statement,thenGsatisfiesittoo;
251
252 (3) IfG ă H andGistorsion-free,thenH istorsionfree. Moregenerally,ifF isa
finitesubgroupofH,thenF imbedsinG.
253
Remark2.9. Itisessentialnottoallownegationsin(2): agroupwithtorsion,andmore-
254
over a torsion group, can preform a finitely presented torsion-free group — e.g., Grig-
255
256 orchuk’s group G preforms F3. In fact, if G ă Fn for some n, then G has the same
positiveuniversaltheoryasF. However,GisuniversallyequivalenttoH ifandonlyifG
257
258 isanon-abelianlimitgroupofSela(see§6.1,thatis,ifF ă G.
ThelemmaimpliesinparticularthatifGisvirtuallynilpotent,theneverygroupinthe
259
sameconnectedcomponenthasthesamelanguageofpositiveuniversalstatements. How-
260
ever,inanysuchconnectedcomponenttherearegroupsthatarenotuniversallyequivalent
261
toG.
262
263 Proof. Ad(1):considerauniversalstatementsatisfiedinG;itisoftheform@x1,...,xnpEq
fora booleanexpression E made ofidentitiesw ,...,w . LetR be the maximallength
264 1 ℓ
w ,...,w .
265 1 ℓ
266 Considerarbitrary h1,...,hn P H. Extendth1,...,hnu to a generatingset T of H,
267 andfindageneratingsetS ofGsuchthattheballsofradiusRinCpG,SqandCpH,Tq
coincide.Letg ,...,g bethegeneratorsofGthatcorrespondtoh ,...,h respectively.
268 1 n 1 n
269 Then wi traces a path in CpG,Sq that remains in an R-neighbourhoodof the origin, so
270 wi traces a closed loop in CpG,Sq if and only if it traces a closed loop in CpH,Tq;
271 therefore, wiph1,...,hnq “ 1 ô wipg1,...,gnq “ 1, so Eph1,...,hnq follows from
272 Epg1,...,gnq.
8 LAURENTBARTHOLDIANDANNAERSCHLER
Ad(2): Lemma2.4showsthatGisaquotientofH;andpositiveuniversalstatements
273
arepreservedbytakingquotients.
274
Ad (3): consider a finite group F. Then the fact that F is not a subgroup of G
275
is a universal statement: writing f ,...,f the elements of F, with multiplication ta-
276 1 k
277 ble fifj “ fmpi,jq, the statement is @g1,...,gkpgi “ gj forsomei ‰ j _ gigj ‰
278 gmpi,jq forsomei,jq. Therefore(3)followsfrom(1). (cid:3)
2.3. Varieties. We defined varieties in §2.2 as collections of identities. Alternatively
279
(see [44]), it is a family of groups closed under taking subgroups, quotients and carte-
280
sian products, namely the class V of all the groups that satisfy these identities. The va-
281
riety V is finitely based if it may be defined by finitely many identities. It is finite if all
282
283 finitely generated groups in the variety are finite. For a group G, one defines VpGq “
284 xwipg1,g2,...q: i ě 1, g1,g2,¨¨¨ P Gy, the verbal subgroup of G correspondingto V;
285 thus VpGq “ 1 if and only if G belongs to the variety. The k-generated relatively free
286 groupisVk :“Fk{VpFkq;itbelongstoV,andeveryk-generatedgroupisV isaquotient
ofV . AdirectconsequenceofLemma2.8(1)isthe
287 k
288 Lemma2.10. IfG ă H andGbelongstoV,thenH belongstoV. (cid:3)
289 Wewillconsider,inlatersections,therestrictionoftherelation ă togroupsbelonging
to a variety. Just as G is a topologyon the normalsubgroupsof F , there is a topology
290 k
291 GpVqonthenormalsubgroupsofVk,orequivalentlyonthenormalsubgroupsofFk that
292 containVpFkq. Directlyfromthedefinitions,
293 Lemma2.11([19,Lemma2.2]). ThenaturalmapGpVqãÑG isahomeomorphismonits
294 image,andthattheimageisclosedifandonlyifVk isfinitelypresentedforallk PN. (cid:3)
295 Lemma2.12. LetV beafinitevariety. IfG ă H,thenVpGq ă VpHq.
296 Proof. LetH begeneratedbyasetT “ th1,...,hkuofcardinalityk,andletFk denote
297 thefreegrouponkgeneratorsx1,...,xk. ThenVpFkqadmitsageneratingsetoftheform
298 wpv1,...qforsomeidentitiesw inV andsomev1,¨¨¨ P Fk. ThenVpHqisgeneratedby
299 thesetT1ofallcorrespondingwpv1ph1,...,hkq,...q.
300 Consider a generating set S “ tg1,...,gku of G, such that CpG,Sq coincides with
301 CpH,Tqinalargeball;thenS1 “twpv1pg1,...,gkq,...q,...ugeneratesVpGq,andthe
302 CayleygraphsCpVpGq,S1qcoincideswithCpVpHq,T1qinalargeball. (cid:3)
GivenavarietyV,theverbalproductofgroupsG ,G ,...,G isdefinedasfollows:
303 1 2 n
304 firstsetG“G1˚G2¨¨¨˚Gnthefreeproduct;then
G
źGi “ VpGqXxrg ,g s: g PGG,g PGG,i‰jy.
V i j i i j j
Forexample,ifV isthevarietyofallgroups,then isthefreeproduct;whileifV isthe
305 śV
varietyofabeliangroups,then isthedirectproduct.
306 śV
RecallthatthewreathproductoftwogroupsG ,G is
307 1 2
G ≀G “tf :G ÑG offinitesupportu⋊G ,
1 2 2 1 2
308 whereG2actsbyshiftonfunctionsG2 ÑG1.
309 Lemma2.13. LetG1,G2,H1,H2begroups,andassumeG1 ă H1andG2 ă H2. Then
310 (1) G1ˆG2 ă H1ˆH2;
311 (2) G1˚G2 ă H1˚H2;
312 (3) LetV beavarietyofgroups. ThenśVGi ă śVHi;
ORDERINGTHESPACEOFFINITELYGENERATEDGROUPS 9
313 (4) G1≀G2 ă H1≀H2.
Proof. Westartby(2),andarguethat,forarbitrarilylarge R,wecanmakeballsofradius
314
315 R agree in respective Cayley graphs. For all i P t1,2u, let Ti generate Hi, and let Si
316 generateGi insuchamannerthatballsofradiusRcoincideinCpGi,SiqandCpHi,Tiq.
317 ThenT :“ ŮTi generatesH :“ ˚iHi, andthecorrespondingset S :“ ŮSi generates
318 ˚iGi. BallsofradiusRcoincideinCpG,SqandCpH,Tq.
319 Ad(3),therelationsimposedon˚iGi and˚iHi areformallydefinedby V,soagain
320 ballsofradiusRinCpśVGi,SqandCpśVHi,Tqcoincide.
(1)isaspecialcaseof(3).
321
322 Ad(4),notethattherelationsgivingG1≀G2 fromG:“G1˚G2 arerxx12,y1y2sforall
323 x1,y1 PG1andx2,y2 PG2zt1u. Theserelationsdonotexactlydefineavarietalproduct;
324 butneverthelessthereisabijectionbetweennon-trivialelementsofnormď RinG2 and
325 H2,andbetweenelementsofnormďRinG1andH1. Theresultagainfollows. (cid:3)
326 Note thatin(1)we can haveG1 ˆC ă H1 ˆC withouthavingG1 ă H1. We we
examinemorecarefullythisforabeliangroupsin§3:
327
328 Example2.14. Wehave1ˆZ ă ZˆZ,yet1doesn’tpreformZ.
329 For A “ Z{6ˆZ, B “ Z{35ˆZ, C “ Z{10ˆZ, D “ Z{21ˆZ, we also have
330 AˆB ă CˆDwhileA,B,C,Daremutuallyincomparable.
331 Proof. Consider tp1,0q,p0,1qu a generatingset of ZˆZ, and, forarbitrary R P N, the
332 generatingsettp0,1q,p0,2R`1quof1ˆZ.TheirCayleygraphsagreeonaballofradius
R.
333
334 Forthesecondclaim,notethatAˆB isisomorphictoCˆD,butforanytwogroups
335 amongA,B,C,D,noneisaquotientoftheother. (cid:3)
336 Similarly, in (2)we canhave G1 ˚C ă H1 ˚C withouthavingG1 ă H1. We will
examinemore closely the situation of freegroupsin §6.1; here and in the sequelwe use
337
thenotationF forfreegroupsonkgenerators.Fornow,wejustmentionthe
338 k
339 Example 2.15. Let G bea k-generatedgroup. Then, forevery m ě 2, thefree product
340 G˚Fm preformsFk`m “Fk˚Fm;yetGneednotpreformFk,forexampleifGsatisfies
anidentity.
341
342 Proof. LetSgenerateFk,letT generateFm,andlettg1,...,gkugenerateG. ThenS\T
343 generatesFk ˚Fm. In Fm, there existelements w1,...,wk such thatnorelationamong
344 themandT,oflengthď R, holds;considerthegeneratingset tg1w1,...,gkwku\T of
345 G˚Fm. ThennorelationoflengthďRholdsamongthem. (cid:3)
346 Note finally that in (4) we may have G1 ≀ C ă H1 ≀ C without having G1 ă H1;
see§6.5formoreexamples:
347
348 Example2.16. ConsiderA,B arbitrarygroups,andaninfinitegroupC. ThenpA˚Bq≀
349 C ă pAˆBq≀C.
Ontheotherhand,ifAandB arenon-trivial,finitelypresented,andeachsatisfiesan
350
351 identity, then A˚B does not satisfy the identities of AˆB, so A˚B doesn’t preform
352 AˆB byLemma2.8(2).
Proof. Let S,T,U be generatingset of A,B,C respectively. Then, as generatingset of
353
354 pAˆBq≀C, we considerS1 \T1 \U, inwhichS1 correspondstothe generatorsof A
355 supportedat1PC,andsimilarlyforT1.
10 LAURENTBARTHOLDIANDANNAERSCHLER
356 ForarbitraryRPN,choosexPCofnormąR,andconsiderthefollowinggenerating
357 setS2 \T2 \U ofpA˚Bq≀H. ThecopyS2 ofS correspondstothe generatorsof A
358 supportedat1PC,whilethecopyT2correspondstothegeneratorsofT supportedatx.
359 BothpAˆBq≀CandpA˚Bq≀CarequotientsofA˚B˚C;inbothcases,allrelations
360 oftheformrsh1,s2sandrth1,t2sareimposedforallh‰1andsi PS1,ti PT1,respectively
361 si P S2,ti P T2. However, in the formercase, all relations of the form rsh,ts are also
362 imposed for all h P H and s P S1,t P T1. In the latter case, these relations are only
363 imposedforh ‰ xands P S2,t P T2. However,thisdistinctionisinvisibleintheballof
364 radiusR. (cid:3)
2.4. Limitsandprelimitsofgroupswithagivensubgrouporquotient. Westartbythe
365
followingstraightforwardlemma.
366
367 Lemma2.17. IfA ă BandAisasubgroupofG,thenthereexistsagroupH containing
368 BasasubgroupandsatisfyingG ă H:
G ă H
Y Y
A ă B.
369 Proof. ConsiderfinitegeneratingsetsSn ofAandT ofB suchthatpA,Snqconvergesto
370 pB,TqinthespaceG ofmarkedgroups,asnÑ8. LetS beafinitegeneratingsetofG.
371 SetSn1 “ S \Sn; thesedefinefinitegeneratingsetsofG. Considerasubsequencepnkq
372 suchthatpG,Sn1kqconvergesinG;denoteitslimitbypH,U \Vq.
373 Inparticular,pA,SnkqconvergestothesubgroupxVyofH. SincepA,Snqconverges
374 toB,weconcludethatxVyisisomorphictoB. (cid:3)
375 Lemma 2.18. If A ă B and A is a quotient of G, then there exists a group H with
376 G ă H andB isaquotientofH:
G ă H
ÓÓ ÓÓ
A ă B.
Proof. Let A,B be k-generated, with T a generating set for B. Since A preforms B,
377
378 thereexistsasequenceofgeneratingsetsSnofcardinalityksuchthatpA,SnqÑpB,Tq.
379 Withoutlossofgenerality,wemayassume1PSnforallnPN.
380 Let π: G ։ A be the given epimomorphism. Let G be ℓ-generated. Then for each
381 n P NthereexistsageneratingsetSn1 “ Sn2 \Sn3 ofGsuchthatSn2 mapsbijectivelyto
382 Sn underπ andSn3 mapsto1 P Aandhascardinalityℓ. Indeedfirstchooseagenerating
383 setS1 forGofcardinalityℓ;then,foreachn P N,chooseanarbitrarylift Sn1 ofSn; and
384 multiplyeachg PS1byanappropriatewordinSn1 toobtainSn3 mappingto1.
385 Passingifneedbetoasubsequence,wecanassumethatpG,Sn1qconvergesinthespace
386 G ofmarkedgroups. DenotethelimitofthesubsequencebypH,T1q,againwithdecom-
387 positionT1 “ T2\T3. Letusconstructanepimorphismρ: H ։ B,showingthatB is
aquotientofH. RecallthatT2innaturallyinbijectionwithT,viaS2 andS . Wedefine
388 n n
389 ρonT2bythisbijection,andputρptq“1foralltPT3.
390 Toprovethatρisahomomorphism,considerawordwpx1,...,xk`ℓqwithwpT1q“1
391 inH. SincepG,Sn1qconvergestopH,T1q,forsufficientlylargenPNwehavewpSn1q“1
392 inG.Letvpx1,...,xkqdenotethewordobtainedfromwbedeletingitslettersxk`1,...,xk`ℓ.
393 Sinceπisahomomorphism,wethenhavevpSnq“1,andthereforeinthelimitvpTq“1.
394 ThisispreciselytheresultofcomputingρpwpT1qqletterbyletter.