PhD Thesis
On Conciseness and P o ini e
RAAGs
Ma eo Pin onello
Supe iso s:
Mon se a Casals-Ruiz
Gus a o A. Fe nández-Alcobe
Decembe 2023
(cc) 2023 Ma eo Pin onello (cc by-nc 4.0)
This hesis has been ca ied ou a he Uni e si y o he Basque Coun y
(UPV/EHU) unde he inancial suppo o he g an FPI-2018 o he Spanish
Go e nmen . In addi ion, he au ho was suppo ed by he Basque Go e n-
men , p ojec s IT974-16 and IT483-22, and he Spanish Go e nmen , p ojec s
MTM2017-86802-P and PID2020-117281GB-I00, pa ly wi h ERDF unds.
ii
Con en s
Acknowledgmen s ii
In oduc ion 1
Resumen de la esis en cas ellano 7
1 P oblems on g oup wo ds 13
1.1 Wo ds and e bal subg oups . . . . . . . . . . . . . . . . . . . . . 14
1.2 Va ie ies and ela i ely ee g oups . . . . . . . . . . . . . . . . . 15
1.3 On h ee ques ions o P.Hall . . . . . . . . . . . . . . . . . . . . . 17
1.4 Conciseness .............................. 18
1.5 Conciseness in o he classes o g oups . . . . . . . . . . . . . . . . 20
1.6 A comp ehensi e lis o known concise wo ds . . . . . . . . . . . . 22
1.7 Tables o concise wo ds . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Conciseness on no mal subg oups 29
2.1 P elimina ies ............................. 30
2.2 Lowe cen al wo ds . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 An example: he wo d δ2....................... 37
2.4 Ou e commu a o wo ds . . . . . . . . . . . . . . . . . . . . . . . 41
3 Coun e examples o Hall’s conjec u e 51
3.1 Elemen s o small cancella ion heo y . . . . . . . . . . . . . . . . 52
3.2 I ano ’s coun e example . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Olshanskii’s coun e example . . . . . . . . . . . . . . . . . . . . . 62
3.4 Olshanskii’s wo d in p o ini e g oups . . . . . . . . . . . . . . . . 65
3.5 On gene a ion o e bal subg oups . . . . . . . . . . . . . . . . . 68
4 Cop ime commu a o s 71
iii
4.1 His o y o cop ime commu a o s . . . . . . . . . . . . . . . . . . . 72
4.2 P elimina ies ............................. 74
4.3 In oduc ion o he p oo s . . . . . . . . . . . . . . . . . . . . . . 77
4.4 The me a-p onilpo en case o γ∗
k.................. 81
4.5 The poly-p onilpo en case o δ∗
k.................. 83
4.6 S ong conciseness o cop ime commu a o s . . . . . . . . . . . . 94
5 P o ini e igh -angled A in g oups 97
5.1 P o ini e g oups ac ing on p o ini e ees . . . . . . . . . . . . . . 98
5.2 Basics on p o-CRAAGs ....................... 103
5.3 Di ec p oduc decomposi ion o p o-CRAAGs . . . . . . . . . . 106
5.4 Cen alise s and no malise s o elemen s . . . . . . . . . . . . . . 109
5.5 Subg oups o p o-Cand p o-pRAAGs ............... 111
6 Abelian spli ings o RAAGs 117
6.1 Abelian spli ings o p o ini e RAAGs . . . . . . . . . . . . . . . . 118
6.2 JSJ decomposi ions . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3 (A,H)-JSJ decomposi ion o p o-CRAAGs . . . . . . . . . . . . 124
6.4 A-JSJ decomposi ion o p o-CRAAGs ............... 127
Bibliog aphy 139
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To my pa en s.
i
Acknowledgmen s
A PhD is no only an academic achie emen , bu a s age o li e in which we ha e
bo h g ea and ha d imes. Looking back a my ime in Bilbao, I am o e whelmed
by he amoun o people ha ha e helped me du ing hese yea s.
Fi s o all, he wo people wi hou whom I would no be he e, my ad iso s.
Gus a o, hank you o wan ing me he e, o helping me and pushing me, and o
always being posi i e. I canno say I’m a ull belie e ye , bu you augh , and
showed me wi h ac s, ha i ’s be e no o be a non-belie e . You we e igh ,
and I owe you a lo ! To Mon se, I canno pu in o wo ds how hank ul I am o
all he help you ga e me! You s epped up in he momen s I was s uggling, you
suppo ed me a lo , and you ha e always been pa ien . Among all he g ea and
amazing people I me he e in Bilbao, you a e he kindes one!
Apa om my ad iso s, I’ll o e e be g a e ul o Ilya o suppo ing me, no
only economically. Hal o he ma hema icians ha I me in Bilbao we e he e
hanks o you, you mus be p oud o he ne wo k you a e c ea ing!
I hank e e y membe o he commi ee ha g ace ully accep ed o ead and
co ec his hesis: Ma a, En ic, and Lei e. Thanks also o all he subs i u e
membe s, in pa icula o Eloisa o w i ing he i s epo .
I is undeniable he impo ance ha my coau ho s ha e had in his wo k, each
one o hem was essen ial in my PhD. Fi s o all o Pa el Zalesskii, hanks o
always answe ing all my silly ques ions wi h you eno mous knowledge and expe -
ience, and o all you pa ience. Then, a gigan ic hank you o Pa el Shumya sky.
You posi i e a i ude eally inspi ed me, and I’m g a e ul o collabo a ing wi h
a ma hema ician I deeply admi e. Ike , we wen h ough a lo o ad en u es (and
disad en u es) oge he . I eally owe you a lo , hanks o he mo i a ion ha you
ga e me, and o always belie ing! I also deeply hank C is ina o he oppo uni y
o isi ing he in Modena.
In hese yea s in Bilbao I me many impo an people in my li e, and I can’
desc ibe how happy I am o ha ing had hem in my li e. I’ll y o hank hem
all, oughly in ch onological o de . Fi s o all Ma ialau a: I will be in deb
ii
ha his same wo d is s ongly concise in p o ini e g oups, se ling ha hese
p oblems di e subs an ially om he classical ques ions in abs ac g oups.
Then, he hesis pu sues he s udy o p oblems in p o ini e g oups, beginning
om some esul s ela ed o s ong conciseness. As we ema ked, his p oblem
could be spli in o wo di e en sub-p oblems: p o ing ha i |w{G}| <2ℵ0 o
a wo d win a g oup G, hen w{G}is ini e, and hen p o ing conciseness in
esidually ini e g oups o w. Fo his eason, se e al esul s on s ongly concise
wo ds elied on he addi ional hypo hesis ha , i a e bal subg oup o a p o ini e
g oup is opologically ini ely gene a ed, hen i can be gene a ed by ini ely many
wo d alues. We p o ide an example, wi h lowe cen al wo ds, ha shows ha
his addi ional condi ion is no always sa is ied.
We hen s udy s ong conciseness o highe o de cop ime commu a o s, ha
a e maps s ongly esembling g oup wo ds. They a e a use ul ool o gene a e some
impo an cha ac e is ic subg oups o p o ini e g oups, like p onilpo en esiduals,
wi h an accu a ely chosen gene a ing se . Simila ly o usual wo ds, we can ask
whe he hey a e (s ongly) concise, in he sense ha in any g oup wi h ini ely
many (o less han 2ℵ0) cop ime commu a o s, hese elemen s gene a e a ini e
subg oup. I was shown by Accia i, Shumya sky and Thillaisunda am ha highe
o de cop ime commu a o s a e concise in esidually ini e g oups, while De omi,
Mo igi and Shumya sky p o ed ha he basic cop ime commu a o map γ∗
2is
s ongly concise. In a join wo k wi h de las He as and Shumya sky, he au ho
p o ed in [39] ha highe o de cop ime commu a o s γ∗
kand δ∗
ka e s ongly
concise in p o ini e g oups, and we p o ide a ull de ailed p oo o hese esul s.
In he second pa o he hesis, we ini ia e he s udy o p o ini e igh angled
A in g oups. Abs ac igh angled A in g oups (RAAGs) a e ini ely gene a ed
g oups whose only ela ions a e commu a o s in he gene a o s. These g oups ha e
a ini e g aph associa ed o hei p esen a ion, and hey include, among o he s,
ee g oups, ee abelian g oups and ee o di ec p oduc s o hem.
The cen al idea in geome ic g oup heo y is o s udy g oups ia ac ions on
spaces. Fo example, ee ac ion o g oups on a space should p o ide a connec ion
be ween he geome y o he space and he algeb a o he g oup. This is he case
wi h ac ions on ees: a g oup ac s eely i and only i he g oup is ee. I we
do no equi e he ac ion o be ee, Bass-Se e heo y gi es a desc ip ion o he
s uc u e o g oups ac ing on ees h ough HNN ex ensions and amalgama ed
p oduc s.
I , a he han on a single ee, we equi e ou g oup G o ac on a di ec p oduc
o wo ees, hen he si ua ion is di e en . Indeed Bu ge and Mozes cons uc ed
in ini e simple g oups ac ing eely and cocompac ly on hem. Howe e , B idson,
4
Howie, Mille and Sho p o ed ha i we equi e some addi ional esidual p ope -
ies, hen such a g oup Gis i ually a di ec p oduc o ee g oups. These esul s
we e gene alised by Haglund and Wise who p o ed ha g oups ac ing eely, and
wi h some addi ional condi ions, on CAT(0) cube complexes a e subg oups o
RAAGs.
As p o ini e g oups sa is y good esidual p ope ies, one can asked i no u he
condi ions a e equi ed in his se ing, namely a p o ini e g oup ac s on a di ec
p oduc o wo p o ini e ees (o , e en mo e ambi iously, on a p o ini e cubing)
i and only i i is i ually a subg oup o a p o ini e RAAG. In o de o app oach
his line o esea ch, we mus i s s udy sys ema ically p o ini e RAAGs. Fo a
gene ic pseudo a ie y Co ini e g oups, p o-CRAAGs a e he p o-Ccomple ion
o abs ac RAAGs and ha e been s udied by Wilkes, K opholle , Snopce and
Zalesskii.
In acco dance o he con en s o he a icle [16], join wi h Casals-Ruiz and
Zalesskii and cu en ly in p epa a ion, we s udy p o-CRAAGs using p o ini e
Bass-Se e heo y as he main ool. This heo y is an analogue o he abs ac
one de eloped mainly by Mel’niko , Ribes and Zalesskii. We use hese me hods o
ob ain s anda d p ope ies o p o-CRAAGs, like he s uc u e o hei cen alize s,
s udying a Ti s al e na i e o hei subg oups, and cha ac e izing 2-gene a ed
subg oups o p o-pRAAGs.
We hen desc ibe some p ope ies o a p o-CRAAG ha a e immedia ely de-
ec able by s udying hei unde lying g aph. Fo example, K ophoplle and Wilkes
al eady obse ed ha a p o ini e RAAG spli s as a ee p oduc i and only i he
unde lying g aph is disconnec ed. We p o e ha p o-CRAAGs a e di ec ly de-
composable i and only i hei unde lying g aph is a join, and we hen ob ain a
cha ac e iza ion o hei spli ings, as p o-Camalgams o HNN ex ensions, o e
abelian subg oups.
We hen con inue he in es iga ion o hei abelian spli ings by de ining JSJ
decomposi ions. These cons uc ions a e a desc ip ion o all he ways a g oup G
can spli o e a ce ain class Ao subg oups, and hey can be ei he gene al (so
A-JSJ decomposi ions) o ela i e o ano he class Ho subg oups ( he so-called
(A,H)-JSJ decomposi ions), in he sense ha we equi e all he subg oups o G
in he class H o be ellip ic.
We gi e a cons uc i e p oo o he exis ence o he (A,H)-JSJ decomposi ion
o a p o-CRAAG Gchoosing A o be he class o abelian subg oups, and wi h
he assump ion ha canonical gene a o s o Gac ellip ically. We hen conclude
by ob aining he gene al A-JSJ decomposi ion o he p o-CRAAG G.
5
S uc u e o he Thesis
In Chap e 1 we gi e an o e iew o he known heo y o conciseness, gi ing a
conside able impo ance o he his o ical de elopmen o he heo y.
In Chap e 2 we p o e ha ou e commu a o wo ds a e concise on no mal
subg oups. This will be ob ained i s in he case o lowe cen al wo ds, and we
will hen app oach he gene al p oo by gi ing an explici desc ip ion o w=δ2,
and hen concluding wi h he p oo o he gene al case.
Chap e 3 will be de o ed o he desc ip ion o he coun e examples on con-
ciseness, and hen o he p oo ha Olshanskii’s wo d is boundedly concise in
esidually ini e g oups and s ongly concise in p o ini e g oups. We conclude he
chap e gi ing an example o a p o ini e g oup wi h p ocyclic de i ed subg oup,
bu whose subg oup canno be gene a ed by ini ely many commu a o s.
In Chap e 4 we p o e ha highe o de cop ime commu a o s γ∗
kand δ∗
ka e
s ongly concise in p o ini e g oups.
In Chap e 5, a e an o e iew o p o ini e Bass-Se e heo y, we ocus on
p o ing basic p ope ies o p o ini e RAAGs, like he s uc u e o hei cen alize s,
and on cha ac e izing hei abelian spli ings.
We conclude he in es iga ion o hei abelian spli ings in Chap e 6, whe e
we explici ly cons uc hei gene al and ela i e abelian JSJ decomposi ions.
6
Resumen de la esis en cas ellano
Aunque el o igen de la eo ía de g upos suele a ibui se a los abajos de Galois,
Jo dan y Klein, odos es os abajos es u ie on mo i ados po la conexión que es a
disciplina iene con la eo ía de núme os o con la geome ía. La eo ía de g upos
abs ac os disc e os ob u o un in e és independien e, sin inspi ación geomé ica,
p incipalmen e a p incipios del siglo XX, y un hi o pa a ello se debe a los aba-
jos de William Bu nside. En 1902 él p egun ó si un g upo de o sión ini amen e
gene ado es necesa iamen e ini o [15], ac ualmen e nos e e imos a es a cues ión
como el “P oblema de Bu nside”. Es e abajo despe ó el in e és po p oblemas
aún más p o undos, como el es udio de la ini ud de los g upos ini amen e ge-
ne ados de exponen e ini o, ambién llamado “P oblema de Bu nside aco ado”.
Explíci amen e, G ün [36] se p egun ó si un g upo G ini amen e gene ado que
sa is ace gn= 1 pa a odo g∈Ges necesa iamen e ini o.
Pod íamos obse a que es e p oblema se puede encuad a en el con ex o más
amplio de una de las p egun as más na u ales que se pueden hace sob e una
es uc u a algeb aica, que es “¿Qué podemos deci sob e un g upo si es e g upo
sigue una egla ija?”
Po supues o, la p egun a es ex emadamen e heu ís ica, pe o podemos e
muchos de los p ime os esul ados en eo ía de g upos a a és de es e en oque,
que puede se in e p e ado como un ejemplo de un p oblema de palab as en g upos.
Una palab a de g upo wes una conca enación ini a de a iables y de sus in e -
sas, que puede e se como un elemen o del g upo lib e gene ado po n a iables
7
x1, . . . , xn. Pa a cualquie g upo G, la palab a wde ine na u almen e una aplica-
ción de GnaG, simplemen e sus i uyendo los elemen os del g upo en las a iables
de odas las o mas posibles. La imagen de es a aplicación es el conjun o de alo es
de la palab a en G, no malmen e deno ado po w{G}, y el subg upo w(G)que
gene an se llama subg upo e bal. De especial in e és es el es udio de las a ieda-
des de g upos, que son las clases de g upos en las que una de e minada palab a
wes una ley, en el sen ido de que oma sólo el alo i ial.
Las “ eglas”mencionadas en la p egun a de G ün son simplemen e leyes en el
g upo, así que en é minos mode nos la cues ión es cómo es udia la a iedad
de g upos gene ada po la ley xn. O os p oblemas que pueden e se desde es a
óp ica son el es udio de los g upos abelianos, nilpo en es y esolubles de clase
aco ada, que son las a iedades gene adas po la palab a conmu ado [x1, x2], po
una palab a cen al in e io o po una palab a de i ada espec i amen e.
En luga de es udia sólo los g upos en los que una palab a wes una ley,
ambién pod íamos p egun a nos si el hecho de que w ome un núme o ini o de
alo es en un g upo G iene alguna implicación en la es uc u a de G. Es ácil
da se cuen a de que cualquie g upo con un núme o ini o de conmu ado es es
ini o-po -abeliano, o, en o as palab as, si el conjun o de alo es de γ2es ini o
en un g upo G, en onces el subg upo e bal co espondien e es ini o. Philip Hall
se dio cuen a de que lo mismo es cie o pa a odas las palab as po encia xn, y
las cen ales in e io es γk, no sólo pa a γ2. Como consecuencia, Hall conje u ó que
pa a cualquie palab a de g upo, si el conjun o w{G}de alo es en un cie o g upo
Ges ini o, en onces el subg upo e bal w(G) ambién es ini o. Si una palab a
sa is ace es a p opiedad pa a cualquie g upo G, se llama concisa y, si lo hace pa a
odos los g upos de una clase dada C, se dice que es concisa en C.
Se ha demos ado que muchas palab as son concisas, además Me zljako demos-
ó que odas las palab as son concisas en g upos lineales, pe o I ano cons uyó
un con aejemplo pa a el caso gene al u ilizando la Teo ía de la cancelación pe-
queña. Más a de, Olshanskii y S o ozhe ob u ie on o os con aejemplos con
mé odos simila es. El es udio de las palab as concisas p og esó de odos modos,
an o buscando nue as palab as que ue an concisas en odos los g upos, como
es udiando el mismo p oblema en o as clases de g upos. Como los g upos linea-
les ini amen e gene ados son esidualmen e ini os, el candida o na u al pa a la
mayo clase de g upos en los que odas las palab as son concisas es la clase de los
g upos esidualmen e ini os. Es in e esan e obse a que una palab a es concisa
en g upos esidualmen e ini os si y sólo si es concisa en g upos p o ini os, po lo
que ecien emen e se ha p opues o o o a ance impo an e.
Cada g upo p o ini o de ca dinalidad meno que 2ℵ0es ini o, y se sugi ió que
8
un enómeno simila ocu e ambién pa a los alo es de las palab as, lle ando a la
conje u a de que cada conjun o de alo es de palab as con menos de 2ℵ0 alo es es
ini o. Uniendo es e p oblema abie o con la conje u a de que odas las palab as
son concisas en g upos esidualmen e ini os, iene sen ido de ini que una palab a
es ue emen e concisa en g upos p o ini os si, siemp e que ome menos de 2ℵ0
alo es, su subg upo (ce ado) e bal es ini o.
En la p ime a pa e de es a esis se discu en a ias con ibuciones que el au o
ha apo ado a la eo ía de los p oblemas de concisión.
La p ime a con ibución se e ie e a la e sión más gene al del p oblema, que
consis e en busca nue as palab as concisas en odos los g upos. Una de las p ime-
as clases de palab as que Philip Hall demos ó que son concisas son las palab as
no conmu ado as, es deci , las palab as que no se encuen an en el subg upo
de i ado del g upo lib e gene ado po las a iables. Más ecien emen e, Delizia,
Shumya sky, To o a y To a demos a on que lo mismo es cie o pa a la pala-
b a γ2(u1, u2), donde u1, u2son palab as no conmu ado as disjun as (es deci , en
conjun os disjun os de a iables). Es e esul ado ue gene alizado en 2022 po
Aze edo y Shumya sky, quienes demos a on que la palab a γ3(u1, u2, u3), pa a
uipalab as no conmu ado as disjun as, es concisa.
En [34], Fe nández-Alcobe y el au o demos a on que w(u1, . . . , uk), con ui
palab as no conmu ado as disjun as, es concisa en el caso de que wsea una pala-
b a cen al in e io (demos ando una conje u a de Aze edo y Shumya sky), y en
el caso de que wsea una palab a de i ada. Los a gumen os del a ículo mencio-
nado ambién uncionan, con algunas pequeñas modi icaciones, cuando wes un
conmu ado ex e no, po lo que p obamos comple amen e es e caso, que incluye
y gene aliza el caso de las palab as cen ales in e io es y de i adas. En ealidad
ob enemos una p opiedad más ue e y demos amos que odos los conmu ado es
ex e nos son concisos en subg upos no males, en el sen ido de que siemp e que el
conjun o de alo es que oma la palab a en una upla Nde subg upos no males
sea ini o, en onces el subg upo que gene an ambién lo es.
Es as nue as palab as concisas in en an ace ca se al lími e en e las palab as
concisas y las que no lo son. De hecho, ac ualmen e se desconocen condiciones
gene ales pa a que una palab a no sea concisa. Las écnicas u ilizadas pa a cons-
ui los es con aejemplos conocidos, el de I ano , de Olshanskii y de S o ozhe
espec i amen e, han sido desa olladas den o de la Teo ía de la cancelación pe-
queña. Es a á ea de la eo ía geomé ica de g upos se basa en la idea de que, si
las elaciones de una p esen ación ija G=hS|Ride un g upo sa is acen algu-
nas condiciones adicionales, es posible deduci algunas p opiedades geomé icas
y algeb aicas de los g upos. Es o se consigue obse ando diag amas, cons uidos
9
u ilizando las elaciones de G, que ep esen an elemen os i iales en el g upo.
La cons ucción comple a de las es palab as no concisas, y de los g upos en los
que es as palab as no son concisas, es bas an e écnica. Po ello, sólo a a emos
de da una idea gene al del esul ado de I ano y, a con inuación, des aca emos
algunas di e encias en e las es palab as no concisas.
A con inuación, nos cen amos en el con aejemplo de Olshanskii. Como demos-
a on Shumya sky y el au o en [68], la palab a de Olshanskii, que no es concisa
en gene al, es en ealidad concisa en g upos esidualmen e ini os. Es e es el p ime
ejemplo de una palab a que no es concisa en odos los g upos, pe o es concisa en
g upos esidualmen e ini os. Luego mos amos ambién que es a misma palab a
es ue emen e concisa en g upos p o ini os, es ableciendo que es os p oblemas
di ie en sus ancialmen e de las cues iones clásicas en g upos abs ac os.
La esis p osigue con el es udio de p oblemas en g upos p o ini os, pa ien-
do de algunos esul ados elacionados con la concisión ue e. Como comen a-
mos, es e p oblema pod ía di idi se en dos subp oblemas di e en es: p oba que
si |w{G}| <2ℵ0pa a una palab a wen un g upo G, en onces w{G}es ini o, y
luego p oba la concisión en g upos esidualmen e ini os pa a w. Po es a azón,
a ios esul ados sob e palab as ue emen e concisas se basaban en la hipó esis
adicional de que, si un subg upo e bal de un g upo p o ini o es opológicamen e
ini amen e gene ado, en onces puede se gene ado po un núme o ini o de alo-
es de la palab a. Apo amos un ejemplo, con palab as cen ales in e io es, que
mues a que es a condición adicional no siemp e se cumple.
A con inuación, es udiamos la concisión ue e pa a conmu ado es cop imos de
o den supe io , que son aplicaciones muy simila es a las palab as de g upo. Son
una he amien a ú il pa a gene a algunos subg upos ca ac e ís icos impo an-
es de los g upos p o ini os, como los esiduales p onilpo en es, con un conjun o
gene ado elegido con cuidado. De o ma simila a las palab as usuales, pode-
mos p egun a nos si son ( ue emen e) concisas, en el sen ido de que en cualquie
g upo con un núme o ini o (o meno que 2ℵ0) de conmu ado es cop imos, es os
elemen os gene an un subg upo ini o. Accia i, Shumya sky y Thillaisunda am
demos a on que los conmu ado es cop imos de o den supe io son concisos en
g upos esidualmen e ini os, mien as que De omi, Mo igi y Shumya sky demos-
a on que el conmu ado cop imo básico γ∗
2es ue emen e conciso. En un abajo
conjun o con de las He as y Shumya sky, el au o demos ó en [39] que los con-
mu ado es cop imos de o den supe io γ∗
kyδ∗
kson ue emen e concisos en g upos
p o ini os, y noso os p opo cionamos una demos ación de allada comple a de
es os esul ados.
En la segunda pa e de la esis, iniciamos el es udio de los g upos de A in de án-
10
gulos ec os p o ini os. Los g upos abs ac os de A in de ángulos ec os (RAAGs)
son g upos ini amen e gene ados cuyas únicas elaciones son conmu ado es en los
gene ado es. Es os g upos ienen un g a o ini o asociado a su p esen ación, e in-
cluyen, en e o os, los g upos lib es, los g upos abelianos lib es y los p oduc os
lib es o di ec os de ellos.
La idea cen al de la eo ía geomé ica de g upos es es udia los g upos median e
acciones en espacios. Po ejemplo, la acción lib e de g upos en un espacio debe ía
p opo ciona una conexión en e la geome ía del espacio y el álgeb a del g upo.
És e es el caso de las acciones en á boles: un g upo ac úa lib emen e en un á bol
si y sólo si el g upo es lib e. Si no exigimos que la acción sea lib e, la eo ía de
Bass-Se e p opo ciona una desc ipción de la es uc u a de los g upos que ac úan
en á boles en é minos de ex ensiones HNN y p oduc os amalgamados.
En luga de en un único á bol, si eque imos que nues o g upo Gac úe en
un p oduc o di ec o de dos á boles, en onces la si uación es di e en e. En e ec o,
Bu ge y Mozes cons uye on g upos simples in ini os que ac úan lib e y cocom-
pac amen e en ellos. Sin emba go, B idson, Howie, Mille y Sho demos a on
que si exigimos algunas p opiedades esiduales adicionales, en onces al g upo G
es i ualmen e un p oduc o di ec o de g upos lib es. Es os esul ados ue on ge-
ne alizados po Haglund y Wise, quienes demos a on que los g upos que ac úan
lib emen e, y con algunas condiciones adicionales, en complejos cúbicos CAT(0)
son subg upos de los RAAG.
Como los g upos p o ini os sa is acen buenas p opiedades esiduales, cabe p e-
gun a se si no se equie en más condiciones en es e con ex o, a sabe , que un g upo
p o ini o ac úa en un p oduc o di ec o de dos á boles p o ini os (o, aún más am-
bicioso, en una cubicación p o ini a) si y sólo si es i ualmen e un subg upo de
un RAAG p o ini o. Pa a abo da es a línea de in es igación, p ime o debemos
es udia sis emá icamen e los RAAG p o ini os. Pa a una pseudo a iedad gené-
ica Cde g upos ini os, los RAAG p o-Cson la compleción p o-Cde los RAAG
abs ac os y han sido es udiados po Wilkes, K opholle , Snopce y Zalesskii.
De acue do con el con enido del a ículo [16], conjun o con Casals-Ruiz y Zaless-
kii y ac ualmen e en p epa ación, es udiamos RAAGs p o-Cu ilizando la eo ía
p o ini a de Bass-Se e como he amien a p incipal. Es a eo ía es un análogo de
la abs ac a desa ollada p incipalmen e po Mel’niko , Ribes y Zalesskii. U ili-
za emos es os mé odos pa a ob ene p opiedades es ánda de los RAAGs p o-C,
como la es uc u a de sus cen alizado es, es udiando una al e na i a de Ti s pa a
sus subg upos, y ca ac e izando subg upos 2-gene ados de RAAGs p o-p.
A con inuación, desc ibi emos algunas p opiedades de un RAAG p o-Cque se
pueden de ec a inmedia amen e a pa i de su g a o subyacen e. Po ejemplo,
11
K ophoplle y Wilkes ya obse a on que un RAAG p o ini o se descompone como
p oduc o lib e si y sólo si el g a o subyacen e es disconexo. De mane a dual,
demos a emos que un RAAG p o-Cse descompone como p oduc o di ec o si y
sólo si su g a o subyacen e es una suma de g a os, y a con inuación ob end emos
una ca ac e ización de sus decomposiciones, como amalgamas p o-Co ex ensiones
HNN, sob e subg upos abelianos.
Pos e io men e, con inuamos con la in es igación de las decomposiciones abe-
lianas de un RAAG p o-C, es a ez en el con ex o de las decomposiciones JSJ.
Es as cons ucciones son una desc ipción de odas las o mas en que un g upo G
puede decompone se sob e una cie a clase Ade subg upos, y pueden se gene a-
les (po an o descomposiciones A-JSJ) o ela i as a o a clase Hde subg upos
(las llamadas descomposiciones (A,H)-JSJ), en el sen ido de que eque imos que
odos los subg upos de Gen la clase Hsean elíp icos.
Da emos una p ueba cons uc i a de la exis encia de la decomposición (A,H)-
JSJ de un RAAG Gp o-Celigiendo Acomo la clase de subg upos abelianos,
y con el supues o de que los gene ado es canónicos de Gac úen elíp icamen e.
Conclui emos ob eniendo la descomposición gene al A-JSJ del p o-CRAAG G.
12
1
P oblems on g oup wo ds
In his chap e we se he ounda ions o he heo y o concise wo ds.
Ini ially we gi e he basic de ini ions o wo d maps and e bal subg oups. We
hen desc ibe a ie ies o g oups, ha a e one o he main mo i a ions d i ing he
de elopmen o wo d p oblems in g oups.
In Sec ion 3, we gi e he o mula ion o h ee conjec u es o Philip Hall. We
b ie ly analyse he pa ial answe o he i s wo o hem, and hen we discuss
he ollow-up o he hi d p oblem in he ou h sec ion. Indeed, he las ques ion
o Hall consis ed in p o ing ha , i a wo d w akes ini ely many alues in a
g oup G, he associa ed e bal subg oup is ini e. A wo d sa is ying his is said
o be concise. We desc ibe he pa ial posi i e answe s and hen men ion he
coun e examples o Hall conjec u e.
In Sec ion 5, we desc ibe he mo e ecen d i ing a eas in conciseness, namely
he s udy o wo ds ha a e concise in esidually ini e g oups. A u he in es -
iga ion is due o he conjec u e ha e e y wo d wis s ongly concise in p o ini e
g oups, meaning ha whene e he se o w- alues has less han 2ℵ0elemen s,
hen he e bal subg oup is ini e.
In Sec ion 6 we glide o e all he esul s o conciseness, add essing in which
h eads o in es iga ion he ma hema ical communi y was able o make imp o e-
men s, and hen conclude wi h a summa y o he bes esul s ob ained so a in
each di ec ion.
13
We can he e o e es ic ou sea ch o commu a o wo ds. In [81], Tu ne -Smi h
p o ed ha lowe cen al wo ds γka e concise ( his was al eady known o P. Hall),
mo eo e he ex ended he esul o de i ed wo ds δk, bu he a gumen s in his
case a e al eady mo e ad anced. Fo se e al yea s he p oblem was un ouched,
un il Wilson p o ed in [82] ha all ou e commu a o wo ds a e concise.
The d eams o ob aining an affi ma i e answe o conciseness p oblems we e
sha e ed by a coun e example, ob ained by I ano in 1989 [43]. We will gi e
some ideas o he cons uc ion o his coun e example in Sec ion 3.2.
S ill, many mo e wo ds ha e been p o ed o be concise. E en u he , many
wo ds ha e been p o ed o be boundedly concise.
De ini ion 1.14 (Boundedly concise wo ds).A wo d wis boundedly concise in a
class Co g oups i he e exis s a unc ion :N→Nsuch ha , i he e is a g oup
G∈ C wi h |w{G}| ≤ m, hen |w(G)| ≤ (m).
In 2009, Fe nández-Alcobe and Mo igi ob ained a di e en p oo o conciseness
o ou e commu a o wo ds in [31]. In he same a icle, he e a e wo p oo s o
he ollowing esul , one by he au ho s and one ha was communica ed o hem
by Mann.
Theo em 1.15. Any wo d w ha is concise is boundedly concise.
1.5 Conciseness in o he classes o g oups
The coun e example o I ano did no impede pu suing be e and u he esul s
on conciseness. In pa icula , a huge de elopmen o he heo y shi ed owa d
p o ing in which classes o g oups e e y wo d is concise.
De ini ion 1.16 (Ve bal conciseness).We will say ha a class o g oups Cis
e bally concise i , o e e y g oup G∈ C and any g oup wo d wwe ha e ha , i
w{G}is ini e, hen w(G)is ini e oo.
Some classes o g oups ha a e ob iously e bally concise a e abelian g oups
(because, i Gis abelian, w(G) = w{G}) o ini e g oups. Tu ne -Smi h p o ed
ha each wo d is concise in esidually ini e g oups such ha all o hei quo ien s
a e esidually ini e [81].
The mos impo an open conjec u e ega ding conciseness is he ollowing.
This conjec u e was discussed by se e al au ho s, bu i is usually a ibu ed o
ei he Jaikin-Zapi ain o Segal.
Conjec u e 1.17. The class o esidually ini e g oups is e bally concise.
20
S udying conciseness in esidually ini e g oups in ol es a di e en machine y
compa ed o he analogous p oblem in gene al abs ac g oups. These addi ional
ools made i possible o p o e ha some wo ds, which a e unknown o be concise
o no in gene al, a e ac ually concise in esidually ini e g oups. Conside , as
an example, Engel wo ds, ha a e de ined i e a i ely as e1(x, y)=[x, y]and
en= [en−1, y] = [x, y, n
. . ., y]. I is known ha hese wo ds a e concise only in
he cases o n≤4(see [1] [32]), bu i is unknown whe he hey a e in gene al.
Howe e , all hese wo ds a e concise in esidually ini e g oups ([26]).
Any esidually ini e g oup embeds in i s p o ini e comple ion, so i is a na u al
ques ion whe he he s udy o conciseness in p o ini e g oups can yield an affi m-
a i e answe o he p e ious conjec u e. An impo an ema k is ha in p o ini e
g oups we will deno e by w(G) he closu e o he abs ac subg oup gene a ed by
he se w{G}. In his se ing, i is ac ually possible o p o e ha i is equi alen
o o mula e Conjec u e 1.17 o p o ini e g oups.
P oposi ion 1.18. A wo d wis concise in all esidually ini e g oups i and only
i i is concise in all p o ini e g oups.
P oo . Le wbe a wo d ha is concise in esidually ini e g oups and suppose ha
w{G}is ini e in a p o ini e g oup G. As Gis esidually ini e, he abs ac sub-
g oup gene a ed by w{G}is ini e oo, bu ini e subse s a e closed, and he e o e
w(G)is ini e oo.
Suppose now ha wis concise in p o ini e g oups and assume ha i akes
ini ely many alues in a esidually ini e g oup G. Each esidually ini e g oups
embeds in i s p o ini e comple ion b
G. The i s s ep is o p o e ha w akes
ini ely many alues in b
G. Le g1, ..., gk∈b
G. Fo each j= 1, . . . , k we can ind a
ne o elemen s gj,i ∈G, indexed by a se I, such ha limi∈Igj,i =gjand he e o e
w(g1, ..., gk) = limi∈Iw(g1,i, ..., gk,i)∈w{G}=w{G}, whe e he las equali y is
ue because w{G}is ini e hence closed. By hypo hesis w(b
G)≤b
Gis ini e and
so w(G)is ini e oo.
Any p o ini e g oup is ei he ini e o uncoun able. De omi, Mo igi and Shumy-
a sky ealized ha a simila duali y could be alid also o wo d maps. Fo his
eason hey conjec u ed in [25] ha any wo d aking coun ably many alues in
a p o ini e g oup has a ini e e bal subg oup, p o ing he conjec u e o ou e
commu a o s and o he speci ic wo ds. An imp o emen o his was ob ained
in [24], whe e he au ho s managed o a oid he dependence on he con inuum
hypo hesis.
21
De ini ion 1.19 (S ongly concise wo ds).A wo d wis said o be s ongly concise
i , whene e |w{G}| <2ℵ0in a p o ini e g oup G, hen w(G)is ini e.
De omi, Klopsch and Shumhya sky p o ed ha ou e commu a o s and o he
speci ic wo ds a e indeed s ongly concise, leading o a s eng hening o Conjec u e
1.17.
Conjec u e 1.20. E e y wo d is s ongly concise.
In iew o Theo em 1.15, we could ask whe he wo ds ha a e concise in esid-
ually ini e g oups a e also boundedly concise in esidually ini e g oups. This is
cu en ly unknown, because one essen ial ool in he p oo s o Fe nández-Alcobe
and Mo igi o Mann in [31] was cons uc ing an ul ap oduc o g oups. We can-
no gene alize hei p oo o esidually ini e g oups because he ul ap oduc o
esidually ini e g oups is no necessa ily esidually ini e. Fo his eason, his is
cu en ly an open p oblem.
Conjec u e 1.21. E e y wo d ha is concise in esidually ini e g oups is also
boundedly concise in esidually ini e g oups.
1.6 A comp ehensi e lis o known concise wo ds
We will gi e a comp ehensi e lis o all esul s ega ding conciseness o wo ds.
As al eady men ioned, he i s a icle ha men ioned he p oblem is by Tu ne -
Smi h [80] in 1964, in which he p o ed ha non-commu a o wo ds, lowe cen al
wo ds and de i ed wo ds a e concise. Wilson p o ed ha all ou e commu a o
wo ds a e concise in [82] in 1974, bu he p oo is al eady mo e con olu ed. I is
impo an o men ion ha Fe nández-Alcobe and Mo igi ga e a di e en p oo
o his las esul in [31]. This las p oo de eloped new me hods in he s udy
o ou e commu a o wo ds, by applying p oo s by induc ion on he heigh and
de ec o hese wo ds, by ep esen ing hem as ini e ees.
Apa om ou e commu a o wo ds, he i s ype o wo ds o which concise-
ness p oblems we e ex ensi ely s udied a e Engel wo ds. Indeed, in 2011 bo h Ab-
dollahi and Russo [1] and Fe nández-Alcobe , Mo igi and T aus ason [32] p o ed
ha Engel wo ds en= [x,ny]a e concise o n≤4. These esul s ely hea -
ily on he ac ha any g oup in which e4is a law is locally nilpo en , whe eas
i is unknown i he same is ue o he gene al n-Engel wo d en. The p oo
o Fe nández-Alcobe , Mo igi and T aus ason ob ains some s uc u al esul s o
g oups Gsuch ha en{G}is ini e o a ce ain posi i e in ege n. Indeed, hey
p o ed ha in his case [en(G), G]is a ini e subg oup.
22
Ano he class o wo ds ha was s udied a e wo ds ob ained by nes ing non-
commu a o wo ds in o ou e commu a o wo ds. We will say ha some wo ds
u1, . . . , una e disjoin i he se s o a iables appea ing in each o hem a e
pai wise disjoin . In 2019 Delizia, Shumya sky, To o a and To a p o ed in
[22] ha he wo d [u1, u2]is concise o u1, u2disjoin non-commu a o wo ds.
This esul was gene alized by Aze edo and Shumya sky in [9] o commu a -
o s [u1, u2, u3] o u1, u2, u3disjoin non-commu a o wo ds. In he same a icle,
Aze edo and Shumuya sky p o ed ha , i u1, . . . , uka e disjoin copies o he
same non-commu a o wo d uand is ano he non-commu a o wo d disjoin
om u1, . . . , uk, hen bo h [u1, . . . , us]and [ , u1, . . . , us]a e concise. Las ly, hey
p o ed ha i uis an ou e commu a o wo d and is a disjoin non-commu a o
wo d, hen [u, ]is concise.
In Chap e 2 we will gi e a ull p oo o a esul ha gene alizes all o hese. In
[34], Fe nández-Alcobe and he au ho p o ed a s onge e sion o a conjec u e o
Aze edo and Shumya sky, showing ha , whene e u1, . . . , uka e non-commu a o
wo ds, hen he wo ds γk(u1, . . . , uk)and δk(u1, . . . , u2k)a e concise.
Theo em 1.15 assu es ha any wo d ha is concise is also boundedly concise.
Howe e , some esul s p o ed ha some se s o wo ds Wa e uni o mly boundedly
concise, which means ha o e e y w∈ W he same unc ion gi es a bound
as in De ini ion 1.14. In [13] B azil, K asilniko and Shumya sky p o ed ha
all lowe cen al wo ds and de i ed wo ds a e uni o mly boundedly concise. This
esul was gene alized o all ou e commu a o wo ds by Fe nández-Alcobe and
Mo igi in [31].
Mo ing owa ds conciseness in some es ic ed classes o g oups, we al eady
men ioned ha Tu ne -Smi h p o ed ha e e y wo d is concise in esidually ini e
g oups all whose quo ien s a e esidually ini e. In 1967 an ex emely impo an
esul o Me zljako in [57] ex ended e bal conciseness o he class o g oups
such ha , o each in ege m∈N, he e exis s a ini e index no mal subg oup
N(m)such ha N(m)is esidually ( ini e o o de cop ime o m). This esul was
used in Me zljako ’s a icle o p o e ha e e y ini ely gene a ed linea g oup is
e bally concise. In his di ec ion, a ecen esul o Zozaya [89] p o ed ha he
class o compac R-analy ic g oups is also e bally concise.
In a simila way, he e a e o he classes o g oups ha a e e bally concise simply
because no wo d can ake ini ely many alues, like he class o g oups ha do
no sa is y any law. This class o g oups con ains o example ee g oups and, as
shown by Abé in [2], Thompson’s g oup F, weakly b anch g oups o p o ini e
g oups wi h al e na ing composi ion ac o s o unbounded deg ee. Conciseness
o his class o g oups ollows om his easy lemma.
23
Lemma 1.22. I a wo d w akes ini ely many alues in a g oup G, hen Gsa is ies
a law.
P oo . Assume |w{G}| ≤ mand ha wis a wo d in n a iables. Conside n×
(m+ 1) a iables, ha we deno e by xj
i,i∈ {1, . . . , n},j∈ {1, . . . , m + 1}and
de ine wa=w(xa
1, . . . , xa
n),wa,b =w−1
awb. I γ is he simple commu a o o leng h
=m(m−1)/2, he wo d
γ (w1,2, . . . , w1,m+1, w2,3. . . , wm−1,m)
ob ained by compu ing γ on all couples (a, b)∈ {1, . . . , m + 1}2wi h a < b is a
law, because a leas wo o he wi,i∈ {1. . . , m + 1}mus be equal.
We will now discuss conciseness in esidually ini e and p o ini e g oups.
The i s wo ds ha we e p o en o be concise in he class o esidually ini e
g oups, bu which a e no known o be concise in all g oups, a e wo ds o he
ype wq o wan ou e commu a o wo d and qa p ime powe . This was p o ed
by Accia i and Shumya sky in [3], whe e hey also showed ha i wis a lowe
cen al wo d, hen wqis boundedly concise in esidually ini e g oups.
In 2015 Gu alnick and Shumya sky p o ed ha weakly a ional wo ds a e concise
in esidually ini e g oups [38]. A wo d wis weakly a ional i , o all ini e g oups
Gand e e y in ege ecop ime o |G| he se w{G}is closed by aking e- h powe s.
Bu ns and Med ede in [14] de ined ha a wo d wimplies i ual nilpo ency
i e e y ini ely gene a ed me abelian g oup in which wis a law has a nilpo en
subg oup o ini e index. The au ho s p o ed ha wimplies i ual nilpo ency i
and only i o all p imes p,wis no a law in he w ea h p oduc CpoC∞. Some
examples o wo ds ha imply i ual nilpo ency a e u −1 o u, semig oup wo ds
in ini ely many gene a o s, Engel wo ds and some gene aliza ions o Engel wo ds.
In a se ies o wo a icles [26] and [27], De omi, Mo igi and Shumya sky p o ed
bounded conciseness in esidually ini e g oups o wo ds implying i ual nilpo-
ency and se e al wo ds o Engel ype [w,ny] o nposi i e in ege and wan
ou e commu a o wo d. Fo w=γn
k o nposi i e in ege hey showed ha
bo h [w,ny]and [y,nw]a e boundedly concise. I wis a p ime powe o an ou e
commu a o wo d, hey p o ed ha [w,ny]is concise in esidually ini e g oups,
bu i is unknown whe he i is boundedly concise oo. The bes esul in his
di ec ion was ecen ly ob ained by Accia i and Shumya sky in [4], showing ha
wand [w,ny]a e concise in esidually ini e g oups o wan a bi a y powe o
an ou e commu a o wo d and ya a iable no appea ing in w.
24
A mo e ecen esul o Aze edo and Shumya sky [9] s a es ha whene e u,
a e wo disjoin wo ds, i uis concise in esidually ini e g oups and is a non-
commu a o wo d, hen [u, ]is concise in esidually ini e g oups. Mo eo e , i u
is boundedly concise, hen he same is ue o [u, ].
In [25] De omi, Mo igi and Shumya sky p o ed ha i w{G}is coun able in a
p o ini e g oup G o w=x2,w= [x2, y]o wan ou e commu a o wo d, hen
w(G)is ini e. All hese esul s we e gene alized o he case |w{G}| <2ℵ0by
De omi, Klopsch and Shumya sky in [24], whe e hey ob ained he same esul
also o he wo ds w=x2,w=x3,w=x6,w= [x3, y],w= [x, y, y],w=
[x, y, y, z1, . . . , z ],w= [x2, z1, . . . , z ]and w= [x3, z1, . . . , z ]whe e x, y, zia e
di e en a iables. In [48], Khukh o and Shumya sky ob ained s ong conciseness
o all Engel wo ds w= [x,ny]in ini ely gene a ed p o ini e g oups. We can also
ex end he no ion o s ong conciseness o some maps ha a e no wo d maps,
like cop ime and an i-cop ime commu a o s. We will discuss hese maps in de ail
in Chap e 4.
We also men ion some esul s on s ong conciseness unde he addi ional hy-
po hesis ha w(G)is gene a ed by ini ely many w- alues. In [24] he au ho s
p o ed ha in his case, weakly a ional wo ds and wo ds implying i ual nilpo-
ency a e s ongly concise. Unde he same hypo hesis, Aze edo and Shumya sky
p o ed in [8] ha [y,n q]and [ q,n, y]a e s ongly concise o =γk(x1, . . . , xk),
and ex ended his esul o some addi ional speci ic wo ds unde mo e condi ions.
O e all, Conjec u es 1.17 and 1.20 a e s ill widely open, bu hey ha e been
pa ially se led o some speci ic subclasses o p o ini e g oups. Indeed in [23]
De omi p o ed ha e e y wo d is s ongly concise in i ually nilpo en p o ini e
g oups, whe eas in [4] Accia i and Shumuya sky p o ed ha Conjec u e 1.17
educes o p o ing conciseness in he class o i ually p o-pg oups o an a bi a y
p ime p.
25
1.7 Tables o concise wo ds
We conclude he chap e wi h some ables summa izing he esul s we desc ibed,
highligh ing only he mos gene al esul s.
Concise wo ds
Wo ds Re e ences No es
Non-commu a o s [81] (P. Hall)
Ou e commu a o s [82], [31] Uni o mly concise
[31]
Engel wo ds en,n≤4[1], [32] [en(G), G]is ini e o
e e y n[32]
γk(u1, . . . , uk),δk(u1, . . . , u2k)
uidisjoin non-commu a o s [34]
Ve bally concise classes o g oups
Class o g oups Re e ences
Res. ini e wi h all
quo ien s es. ini e [81]
Linea g oups [57]
Compac R-analy ic [89]
G oups wi hou
any law Lemma 1.22
We also ema k ha e e y wo d is s ongly concise in i ually nilpo en p o ini e
g oups ([23]).
26
Wo ds concise in esidually ini e g oups
Wo ds Re e ences No es
wq,[wq,ny]
wou e comm., q∈N[4]
Weakly a ional [38]
Wo ds implying
i ual nilpo ency [26]
[wq,ny],[y,nwq]
wou e comm., q∈N[27] boundedly concise o
[γq
k,ny],[y,nγq
k]
[u, ],u, disjoin ,
uconcise in es. ini e
non-commu a o
[9] boundedly concise i
uboundedly concise
We will w i e (FG) o “w(G)is gene a ed by ini ely many w- alues”.
S ongly concise wo ds
Wo ds Re e ences No es
Ou e commu a o [24]
w=xq,q= 2,3,6
and some speci ic wo ds [24]
Engel wo ds en,n∈N[48] Fo ini ely gene a ed
p o ini e g oups
Cop ime commu a o s γ∗
k,δ∗
k[39] No g oup wo ds
see Chap e 4
S ongly concise wo ds unde addi ional condi ions
Weakly a ional,
implying i ual nilpo ency [24] Condi ion (FG)
[y,nγq
k],[γq
k,ny]
y, γkdisjoin , q∈N
and some speci ic wo ds
[8] Condi ion (FG)
27
28
2
Conciseness on no mal subg oups
In his chap e we desc ibe some con ibu ions o he lis o known concise wo ds.
Delizia, Shumya sky, To o a, and To a p o ed in [22] ha , i u1and u2a e non-
commu a o wo ds in disjoin se s o a iables, hen [u1, u2]is concise oo. This
esul has been ex ended o he case when u1is an ou e commu a o wo d and
u2is a non-commu a o and o commu a o s [u1, u2, u3]o non-commu a o s in
[9]. Fo longe commu a o s, he only pa ial esul was ob ained by Aze edo
and Shumya sky in [9], who p o ed ha i u1, . . . , uka e copies o he same non-
commu a o wo d in di e en a iables, hen [u1, . . . , uk]is concise.
Aze edo and Shumya sky conjec u ed ha , i uia e non-commu a o wo ds in
disjoin se s o a iables and w=γk, hen w(u1, . . . , uk)is concise. The aim o
his chap e is o p o e his conjec u e, and mo eo e o ex end i o he case
o a gene ic ou e commu a o wo d w. We will oughly ollow he a icle [34] o
Fe nández-Alcobe and he au ho , whe e we p o ed hese esul s o lowe cen al
wo ds and de i ed wo ds.
In he i s sec ion we will de elop some p elimina y lemmas. These will be
sufficien o se le he conjec u e o Aze edo and Shumya sky, o w=γk, in he
second sec ion. The main idea o he p oo is o ind a se ies o e bal subg oups
such ha each sec ion o his se ies has some linea i y p ope ies. This could be
ob ained as a co olla y o he case o gene ic ou e commu a o wo ds, bu he
p oo in his case is mo e s aigh o wa d and easie , so i makes sense o ha e a
29
1. Siis a no mal subse o G.
2. The e exis s ni∈Nsuch ha all ni h powe s o elemen s o Nia e con ained
in Si.
I o he uple S= (S1, . . . , S ) he se o alues γ {S}is ini e o o de m, hen
he subg oup γ (N)is also ini e, o (m, , n1, . . . , n )-bounded o de .
P oo . We ollow he no a ion Niand P
i, in oduced in he s a emen o The-
o em 2.11. We a e going o p o e ha P
iis ini e o bounded o de o i=
1, . . . , + 1 by e e se induc ion on i. Since P
1=γ (N), his p o es he esul .
The basis o he induc ion ollows om Lemma 2.12, since we ha e ha P
+1 =
[γ (N), γ (N)]. Le us assume ha P
i+1 is ini e o bounded o de and p o e ha
he same holds o P
i. Recall ha he quo ien P
i/P
i+1 is he image o γ (Ni),
and hen, by a sui able applica ion o Lemma 2.3, i can be gene a ed by he
images o he se To commu a o s
[s1, . . . , si−1, xi, si+1, . . . , s ],
wi h sj∈Sj o 1≤j≤ ,j6=i, and xi∈γi{Si}, whe e Si= (S1, . . . , Si). By
Lemma 2.7, we ha e γi{Si} ⊆ S∗2i−1
i, and hen Lemma 2.8 implies ha
[s1, . . . , si−1, xi, si+1, . . . , s ]∈γ {S}∗2i−1⊆γ {S}∗2 −1.
F om he assump ion ha |γ {S}| =m, we ge
|T| ≤ (2m+ 1)2 −1,
and consequen ly P
i/P
i+1 can be gene a ed by an (m, )-bounded numbe o
elemen s. Since P
i/P
i+1 is abelian, he p oo will be comple e once we show
ha all elemen s in Tha e bounded ini e o de modulo P
i+1.
By Theo em 2.11, he wo d γ is linea in posi ion io he uple Nimodulo
P
i+1. In pa icula ,
[s1, . . . , si−1, xi, si+1, . . . , s ]λni≡[s1, . . . , si−1, xλni
i, si+1, . . . , s ] (mod P
i+1),
(2.3)
o e e y λ∈Z. Since xi∈γi(N1, . . . , Ni)≤Ni, i ollows om (ii) in he
s a emen o he heo em ha xλni
i∈Si o all λ∈Z. Thus we ge
[s1, . . . , si−1, xλni
i, si+1, . . . , s ]∈γ {S}.
36
Since γ {S}is ini e o o de m, i ollows ha he e exis λ, µ ∈ {0, . . . , m},
λ6=µ, such ha
[s1, . . . , si−1, xi, si+1, . . . , s ]λni≡[s1, . . . , si−1, xi, si+1, . . . , s ]µni(mod P
i+1).
This implies ha [s1, . . . , si−1, xi, si+1, . . . , s ]has (m, ni)-bounded ini e o de
modulo P
i+1, as desi ed.
I we ake Si=Ni, we ge Theo em 2.2 o he lowe cen al wo ds.
Co olla y 2.14. Le ∈Nand le N= (N1, . . . , N )be a uple o no mal
subg oups o a g oup G. I γ {N}is ini e o o de m, hen he subg oup γ (N)is
also ini e, o (m, )-bounded o de .
Now we deduce Theo em 2.1 o lowe cen al wo ds.
Co olla y 2.15. Le ∈Nand le u1, . . . , u be disjoin non-commu a o wo ds.
Then he wo d γ (u1, . . . , u )is boundedly concise. In pa icula , γ (xn1
1, . . . , xn
)
is boundedly concise o all ni∈Z {0}.
P oo . Le us conside he wo d w=γ (u1, . . . , u ), and le Gbe a g oup in
which w akes ini ely many alues, say |w{G}| =m. By Co olla y 2.5, we ha e
w(G) = γ (u1(G), . . . , u (G)). No e ha ui(G) = hSii, whe e Si=ui{G}, and
ha w{G}=γ {S}, whe e S= (S1, . . . , S ). Now obse e ha Siis a no mal
subse o Gand ha , since uiis a non-commu a o wo d, o some ni∈Z {0}
we ha e {gni|g∈G} ⊆ ui{G}. Hence w(G)is ini e o (m, , n1, . . . , n )-bounded
o de by Theo em 2.13.
2.3 An example: he wo d δ2
We now wan o p o e Theo ems he analogues o Theo ems 2.1 and 2.2 o a gen-
e ic ou e commu a o wo d w. The gene al s a egy is s ill he same as o lowe
cen al wo ds: we a e going o ob ain a sui able se ies o no mal subg oups o G,
going om [w(N), w(N)] o w(N), wi h he p ope y ha each o he ac o s o
he se ies can be gene a ed by a e bal subg oup on a uple o no mal subg oups
ha is closely ela ed o w(N)and linea in one componen . This is basically The-
o em 2.20 below. Fo simplici y, le us e e o such a se ies as a linea se ies. The
a gumen needed o ob ain a linea se ies o de i ed wo ds p esen s difficul ies
and sub le ies ha did no a ise wi h lowe cen al wo ds, and is also signi ican ly
37
mo e echnical. Fo he con enience o he eade and in o de o make he p o-
cedu e o a gene al wmo e unde s andable, i s o all we a e going o p o ide a
ske ch o i in he pa icula case o δ2.
O cou se, δ1=γ2and, acco ding o Theo em 2.11, we ha e he ollowing linea
se ies o δ1(N1, N2):
[N1, N2]
N1,[N1, N2]
[N1, N2],[N1, N2]
Figu e 2.1: Se ies o [N1, N2]
In his and in he nex diag ams, a ed box indica es he componen in which we
ha e linea i y.
Le us see how we can cons uc a linea se ies o δ2(N1, N2, N3, N4) om he
se ies abo e o δ1. To his pu pose, we will use Lemma 2.10, which ensu es ha
linea i y is p ese ed a e aking sui able commu a o s, and also he ema k made
be o e ha lemma, saying ha linea i y is p ese ed a e mul iplying by a no mal
subg oup. To s a wi h, we ake he commu a o o he e ms o he p e ious
se ies wi h [N3, N4], ob aining he se ies
[N1, N2],[N3, N4]
hN1,[N1, N2],[N3, N4]i
h[N1, N2],[N1, N2],[N3, N4]i
38
Now we mul iply his se ies by [N1, N2],[[N1, N2],[N3, N4]], which con ains he
subg oup [N1, N2],[N1, N2],[N3, N4]by P. Hall’s Th ee Subg oup Lemma, and
we ge he ollowing diag am:
[N1, N2],[N3, N4]
hN1,[N1, N2],[N3, N4]i
h[N1, N2],[N1, N2],[N3, N4]i
Figu e 2.2: Fi s diag am o [N1, N2],[N3, N4]
He e, and in he emaining diag ams, ins ead o he subg oups o he se ies, we
a e showing e bal subg oups on no mal subg oups whose images coincide wi h
he co esponding ac o s o he se ies. A e all, i is in hese subg oups whe e
we a e going o ob ain he linea i y condi ions. Be awa e hen ha e ical lines
in he diag ams do no deno e inclusions om his poin onwa ds.
By swapping he oles o (N1, N2)and (N3, N4), we can ob ain his o he dia-
g am:
[N1, N2],[N3, N4]
h[N1, N2],N3,[N3, N4]i
h[N1, N2],[N3, N4],[N3, N4]i
Figu e 2.3: Second diag am o [N1, N2],[N3, N4]
Now we ake he commu a o o [N1, N2]wi h he e ms o his las diag am, and
we add he ex a e m δ2(N1, N2, N3, N4)′a he bo om:
39
h[N1, N2],[N1, N2],[N3, N4]i
h[N1, N2],[N1, N2],N3,[N3, N4]i
h[N1, N2],[N1, N2],[N3, N4],[N3, N4]i
h[N1, N2],[N3, N4],[N1, N2],[N3, N4]i
Figu e 2.4: Se ies o h[N1, N2],[N1, N2],[N3, N4]i
Finally, by gluing he diag ams in Figu es 2 and 4 oge he , we ob ain a linea
se ies o he subg oup δ2(N1, N2, N3, N4).
O cou se, his is simply a ske ch wi hou p oo s, bu we a e going o ollow
he same p ocedu e in he p oo o Theo em 2.20, in o de o ge a linea se ies
o w= [α, β] om he se ies o he ou e commu a o wo ds αand β. A his
poin , i is wo h no ing an impo an di e ence wi h he si ua ion o a lowe
cen al wo d γ . In ha case, e e y ac o o he linea se ies is o he ollowing
o m (again we show he linea componen in ed):
N1, . . . , Ni−1,[N1, . . . , Ni], Ni+1, . . . , N .
We obse e ha his subg oup is o he o m γ (M), whe e he j h componen Mj
o Mis ei he Njo a commu a o o he e ms o N ha in ol es Nj, and he
linea i y happens in Mi. Howe e , i we look a he se ies o δ2ob ained abo e,
he i s wo subg oups in Figu e 2.4 a e
δ2(N1;N2; [N1, N2]; [ N3, N4]) (2.4)
and
δ2(N1;N2; [N1, N2]; [N3,[N3, N4]]),(2.5)
which a e no o he o m δ2(M)wi h e e y Mja commu a o om Nin ol ing
Nj, as we can see by looking a he hi d componen o δ2. Also, he linea i y does
40
no happen in a componen o δ2, bu in a mo e in e io posi ion. Ne e heless,
we can w i e hese subg oups as e bal subg oups on no mal subg oups o ou e
commu a o wo ds di e en om δ2. Mo e speci ically, i
(x1, x2, x3, x4, y1, y2) = [x1, x2],[[y1, y2],[x3, x4]]
hen he subg oups in (2.4) and (2.5) a e (M1)and (M2), whe e
M1= (N1, N2, N3, N4, N1, N2)and M2= (N1, N2, N3,[N3, N4], N1, N2),(2.6)
whe e again we ha e ma ked he linea componen s in ed.
2.4 Ou e commu a o wo ds
A e ha ing illus a ed he p ocedu e wi h he case o δ2, le us p oceed o sys-
ema ically de elop he ools ha a e necessa y o he p oo o Theo em 2.20.
We s a by in oducing a special ype o wo ds ha we can de i e om a gi en
ou e commu a o wo d w, which we call ex ended wo ds o w. Be o e gi ing he
de ini ion, we show he idea behind ex ended wo ds wi h an example. Conside
he wo d δ2= [[x1, x2],[x3, x4]]. This is o med by aking he commu a o o x1
and x2, aking he commu a o o x3and x4, and hen aking he commu a o o
hese wo commu a o s. Now suppose ha on some occasions, be o e pe o ming
one o hese commu a o s, we in oduce a change by aking i s he commu a o
o one (o bo h) o he componen s wi h an ou e commu a o wo d no in ol ing
he a iables x1, . . . , x4appea ing in δ2. Fo example, be o e p oducing [x1, x2], we
ake he commu a o [[y1, y2], x1]and now we ollow as in δ2 aking he commu a o
wi h x2, ob aining [[y1, y2], x1], x2. We could con inue wi h he p ocess o aking
commu a o s wi hou making any o he changes, so ge ing
[[y1, y2], x1], x2,[x3, x4],
bu we could also make some simila changes in he p ocess, as in he wo ds
h[[y1, y2], x1], x2,x3,[y3, x4]i
and h[[y1, y2], x1], x2,x3,[y3, x4], y4i.
Ano he possibili y is o make a commu a o a he e y end, a e ha ing com-
ple ed δ2, as in y1,[[x1, x2],[x3, x4]].
41
Obse e ha all hese ex ended wo ds a e again ou e commu a o wo ds, because
we ne e epea a a iable when we make changes in he cons uc ion o δ2.
Le us now gi e he o mal de ini ion o ex ended wo ds. No ice ha his
de ini ion di e s om he one o ex ensions o ou e commu a o wo ds gi en in
De ini ion 3.1 o [33].
De ini ion 2.16 (Ex ended wo ds).Le w=w(x1, . . . , x )be an ou e commu -
a o , and le Y={yn}n∈Nbe a se o a iables ha a e disjoin om X. Fo
e e y k∈N∪ {0}, we de ine ecu si ely he se ex k(w)o k h ex ended wo ds o
was ollows:
1. ex 0(w) = {w}.
2. Fo k≥1,ex k(w)consis s o he se
{[p, q],[q, p]|pou e commu a o in Y,q∈ex k−1(w),pand qdisjoin }
={[p, q]|pou e commu a o in Y,q∈ex k−1(w),pand qdisjoin }±1,
and, i w= [α, β], also o he se
S
ℓ+m=k
{[p, q]|p∈ex ℓ(α),q∈ex m(β),pand qdisjoin }.
I ∈ex k(w) hen we say ha wis an ex ended wo d o deg ee ko wby ou e
commu a o s.
Fo b e i y, in he emainde we will simply speak o ex ended wo ds when we
mean ex ended wo ds by ou e commu a o s. Obse e ha an ex ended wo d
o an ou e commu a o w=w(x1, . . . , x )is again an ou e commu a o , in
he a iables {x1, . . . , x } ∪ Y. Whene e i is con enien we will assume, a e
enaming he a iables, ha = (x1, . . . , x , y +1, . . . , ys).
Nex we gene alize Lemma 2.8 o ex ended wo ds o an ou e commu a o wo d.
Lemma 2.17. Le = (x1, . . . , x , y +1, . . . , ys)be an ex ended wo d o deg ee k
o an ou e commu a o wo d w=w(x1, . . . , x ). Assume ha S= (S1, . . . , S )is
a uple o no mal subse s o a g oup G. I = ( 1, . . . , s)is a uple o elemen s
o Gsuch ha i∈S∗mi
i o e e y i= 1, . . . , , hen
( )∈w{S}∗m1...m 2k.
42
P oo . We use induc ion on k+ . I k= 0 hen =wand he esul is Lemma 2.8.
This gi es in pa icula he basis o he induc ion. Suppose now ha he esul
holds o smalle alues o k+ , and ha k≥1. Acco ding o De ini ion 2.16,
we may assume ha ( ) = [p( ′), q( ′′)], whe e pand qa e disjoin and
1. ei he pis an ou e commu a o wo d in Yand q∈ex k−1(w),
2. o p∈ex ℓ(α),q∈ex m(β), wi h w= [α, β]and ℓ+m=k.
In case (i), all elemen s 1, . . . , appea in he ec o ′′, and by he induc-
ion hypo hesis we ha e q( ′′)∈w{S}∗m1...m 2k−1. Then he esul ollows by
applying Lemma 2.7 o he commu a o wo d [x1, x2]and he no mal subse
w{S}∗m1...m 2k−1.
Suppose now ha we a e in case (ii), and assume wi hou loss o gene ali y
ha α=α(x1, . . . , xq)and β=β(xq+1, . . . , x ). Se S′= (S1, . . . , Sq)and S′′ =
(Sq+1, . . . , S ). Since αand βin ol e less a iables han w, he esul is ue o
pand q, and so
p( ′)∈α{S′}∗m1...mq2ℓand q( ′′)∈β{S′′}∗mq+1...m 2m.
Now he esul ollows by applying Lemma 2.8 o he commu a o wo d [x1, x2]
and he pai o no mal subse s (α{S′}, β{S′′}).
We also need o de ine a ype o ex ensions o uples o no mal subg oups and
o e bal subg oups on no mal subg oups. The idea behind he de ini ion is o be
able o deal wi h uples like he ones appea ing in (2.6) and wi h he co esponding
e bal subg oups on no mal subg oups in ha pa ag aph.
De ini ion 2.18 (Ou e commu a o ex ension).Le Gbe a g oup and conside
wo uples N= (N1, . . . , N )and M= (M1, . . . , Ms)o no mal subg oups o G.
We say ha Mis an ou e commu a o ex ension o Ni he ollowing condi ions
hold:
1. s≥ .
2. Fo e e y i= 1, . . . , s, we ha e Mi=wi(Ni), whe e wiis an ou e commu -
a o wo d and all componen s o Nibelong o N.
3. Fo e e y i= 1, . . . , , he subg oup Niis a componen o Ni, and con-
sequen ly Mi≤Ni.
43
De ini ion 2.19 (Ex ensions o w(N)).Le w=w(x1, . . . , x )be a wo d and
le Nbe an - uple o no mal subg oups o a g oup G. An ex ension o deg ee
ko w(N)by ou e commu a o s is a subg oup o he o m (M), whe e is an
ex ended wo d o deg ee ko wand Mis an ou e commu a o ex ension o N.
Fo example, we can see he subg oup in (2.5) as an ex ension o δ2(N) =
δ2(N1, N2, N3, N4)by aking =[x1, x2],[[y1, y2],[x3, x4]]and he uple M=
(N1, N2, N3,[N3, N4], N1, N2). No e ha (M)is linea in he ou h componen
modulo he subg oup ha appea s below i in Figu e 4.
We now p o e he exis ence o a linea se ies o ou e commu a o wo ds.
We ecall ha he heigh o an ou e commu a o wo d w= [α, β]is de ined
induc i ely, wi h a single a iable ha ing heigh 0, and wi h he heigh o wbeing
1 + max{heigh (α),heigh (β)}. No ice ha he heigh o an ou e commu a o
wo d in s a iables will always be a mos s−1.
Theo em 2.20. Le ∈Nand le N= (N1, . . . , N )be a uple o no mal
subg oups o a g oup G. Conside an ou e commu a o wo d w= [α, β]in
a iables, say o heigh h. Then he e exis s a se ies
[w(N), w(N)] = V0≤V1≤ · · · ≤ V =w(N)
o no mal subg oups o Gsuch ha , o e e y i= 1, . . . , , he ollowing hold:
1. The sec ion Vi/Vi−1is he image o an ex ension i(Mi)o w(N)o deg ee
a mos h−1.
2. In he sec ion Vi/Vi−1, he wo d iis linea in one componen o he uple
Mi.
Fu he mo e, he wo ds iand he wo ds appea ing in he ou e commu a o ex-
ensions Midepend only on wand , and no on he g oup Go on he uple
N.
P oo . We p o e he heo em by induc ion on he heigh o he ou e commu a o
wo d w, wi h he base case being a single a iable, which is ob ious. We can hen
assume ha he e exis wo se ies o subg oups sa is ying he condi ions o he
heo em o he ou e commu a o wo ds o smalle heigh αand β. Assume ha
x1, . . . , xqand xq+1, . . . , xq+ma e he a iables in ol ed in αand β espec i ely,
and in pa icula =q+m.
Se N1= (N1, . . . , Nq)and N2= (Nq+1, . . . , Nq+m). By he induc ion hypo-
hesis, he e exis wo se ies o leng h sand espec i ely
A0= [α(N1), α(N1)] ≤ · · · ≤ Ai≤ · · · ≤ As=α(N1)(2.7)
44
and
B0= [β(N2), β(N2)] ≤ · · · ≤ Bi≤ · · · ≤ B =β(N2)(2.8)
such ha , o e e y i= 1, . . . , s, he ac o s Ai/Ai−1and Bi/Bi−1a e he images
o α
i(Mα
i)and β
i(Mβ
i), espec i ely, whe e:
(a) α
iand β
ia e ex ended wo ds o αand β espec i ely, each o deg ee a
mos h−2.
(b) Mα
iis an ou e commu a o ex ension o N1.
(c) Mβ
iis an ou e commu a o ex ension o N2.
(d) In he sec ions Ai/Ai−1and Bi/Bi−1, he wo ds α
iand β
ia e linea in one
componen o he uples Mα
iand Mβ
i, espec i ely.
Le us now see how o ob ain he se ies o wand o he uple N om he wo
se ies (2.7) and (2.8). We will ha e ha he leng h o he se ies we a e looking
o depends on he leng h o hese wo se ies, in he o m ha = +s+ 1. We
s a by aking he commu a o o all e ms o he se ies (2.7) wi h β(N2). This
way we ob ain he se ies
[A0, β(N2)] ≤ · · · ≤ [Ai, β(N2)] ≤ · · · ≤ [α(N1), β(N2)] = w(N).(2.9)
By P. Hall’s Th ee Subg oup Lemma, we ha e
[A0, β(N2)] = [α(N1), α(N1), β(N2)]
≤[α(N1), β(N2), α(N1)] = [α(N1), w(N)].
Now we mul iply all e ms o he se ies (2.9) by [α(N1), w(N)], and his is he
igh mos pa o he se ies we a e seeking (whe e = +s+ 1, as abo e):
V −s= [α(N1), w(N)] ≤ · · · ≤ V −s+i= [Ai, β(N2)] [α(N1), w(N)]
≤ · · · ≤ V =w(N).(2.10)
No e ha −s= + 1. The ac o s in his se ies a e he images o he subg oups
[ α
i(Mα
i), β(N2)],
which can be ep esen ed in he o m i(Mi)by aking
i= [ α
i, β(xq+1, . . . , xq+m)]
45
3.1 Elemen s o small cancella ion heo y
In his sec ion we in oduce some basics in small cancella ion heo y, which a e
c ucial o he cons uc ion o he coun e examples o Hall’s conjec u e. We s a
wi h some no a ion.
Gi en a su ace o a polygon X, we deno e by ∂(X) he bounda y o Xand
by ι(X) = X ∂(X) he in e io o X. I we iew an edge Xo a polygon as a
polygon i sel , hen ∂(X)consis s o he wo endpoin s.
When de ining diag ams o e g oups, i a g oup Ghas a se So gene a o s, i
will be use ul o conside he se S∗o abs ac wo ds in he alphabe S∪S−1.
In acco dance o he no a ion in oduced by Olshanskii, in his chap e we will
deno e wo ds in S∗by capi al le e s C, L, M, X, Y, Z. We will w i e |X| o deno e
he leng h o he wo d X∈S∗and, i X, Y ∈S∗, we will w i e X≡Y(and say
ha Xand Ya e isually equal) i |X|=|Y|and we ha e a le e -by-le e
equali y.
De ini ion 3.1 (Cells).Conside a n-gon Pin a plane wi h edges e1, . . . , en.
Conside a map :P→X, whe e Xis any su ace, such ha :
• |ι(P)is an embedding;
• |ι(ei)is an embedding o each i∈ {1, . . . , n};
• i a, b ∈Pa e dis inc poin s wi h (a) = (b), hen a, b ∈∂(P). I ais a
e ex, so is b, o he wise i a∈ei,b∈ej, hen (ei) = (ej).
The image (P)o such a map is called a cell on X.
In o mally, a cell is an iden i ica ion o he n-agon Pin X, bu we allow e ices
and edges o be pas ed oge he by , s ill p ese ing he s uc u e o open disc
o ι(P).
De ini ion 3.2 (Cell decomposi ion).Acell decomposi ion o a su ace Xis a
ini e se {(Pi, i)|i= 1, . . . , m}o cells such ha X=Sm
i=1 i(Pi)and such ha
i(Pi)∩ j(Pj),i6=jis ei he emp y o i is a se o e ices and/o edges.
A cell decomposi ion can be hough as a pa i ion o Xin o a ini e se o
cells, bu allowing hese cells o in e sec in edges and/o e ices. The images
o e ices o edges o any o he Piwill be called e ices and edges o he cell
decomposi ion. No mally, we will deno e a cell i(Pi)wi h a single le e C.
E en i he heo y can be de eloped o a bi a y su aces, we will only wo k
wi h o ien able su aces. I will be use ul o gi e an o ien a ion o edges o a
52
cell decomposi ion, by assuming ha any edge eadmi s an in e se e−1, which
geome ically co esponds o he same elemen , bu wi h in e se o ien a ion.
Fix now an alphabe Sand assign o each o ien ed edge eo he cell decompos-
i ion a label ϕ(e)∈S∪S−1. I hese labels a e chosen such ha ϕ(e−1) = ϕ(e)−1
o each edge eo he cell decomposi ion, we will mo eo e say ha he decom-
posi ion is a diag am. I pis a pa h, ob ained by conca ena ing some o ien ed
edges p=e1· · · ek he label o pis he wo d ϕ(p) = ϕ(e1)· · · ϕ(ek)∈S∗, whe e
he endpoin o eicoincides wi h he beginning o ei+1.
Whene e he su ace Xunde lying a diag am is a disc, i will be called a
ci cula diag am. No ice ha i Xhas a bounda y, hen i mus consis o edges
and e ices o he diag am, and he e o e each o i s connec ed componen s will
ha e a label as a pa h. We will say ha any connec ed componen o he bounda y
∂X o Xis a con ou o X. Mo eo e any cell Co a diag am can be seen as a
disc (possibly wi h some pa s o he bounda y pas ed oge he ), and in his case
he con ou is equal o he bounda y, and hus we will use he same no a ion ∂C.
We will use he con en ion ha he label o he con ou o e e y cell o a diag am
o e an o ien able su ace will be ead clockwise, and simila ly o he label o he
con ou o a ci cula diag am.
I we ha e a cell Cin a diag am, he bounda y ∂Cis a pa h (induced by he
o ien a ion o he polygons), so we can always conside he label ϕ(∂C)o he
con ou . The leng h o he con ou o a cell o o a su ace Xis he numbe o
edges o ∂X as a ini e pa h and will be deno ed by |∂X|.
No mally, we simply s udy g oups gi en by a p esen a ion G=hS| Ri. In ou
case, we will need o conside g oups wi h g aded ela ions, in he sense ha we
pa i ion he se Ro ela ions as R=S∞
i=1 Riin such a way ha no ela o in
Rican coincide wi h a cyclic conjuga e o a wo d in Rjo i s in e se i j6=i. In
his se ing, we will conside he g aded p esen a ion
G=S| R =
∞
[
i=1
Ri.(3.1)
A cell in a diag am ∆is an R-cell i i s label is isually equal (up o cyclic
conjuga ion) o a ela o o an in e se o a ela o in R. I such ela o is in he
se Ri, we will say ha he cell is an i-cell, o a cell o ank i. Mo eo e we will
say ha i is a 0-cell (o a cell o ank 0) i i s label is isually equal o a wo d
ss−1o s−1s o s∈S.
De ini ion 3.3 (Diag am o e a g oup).I Gis a g oup gi en by he p esen a ion
(3.1), a diag am o e Gis a diag am ∆o e he alphabe Ssuch ha all cells
53
a e ei he R-cells o 0-cells. The ank o he diag am is he maximum among he
anks o i s cells.
No ice ha his de ini ion depends on he p esen a ion (3.1) chosen o G, no
only on he g oup Gi sel . When s udying diag ams o e a g oup G, we wan o
s udy he simples possible e sion o hem. In ou case, we will say ha a ci cula
subdiag am o ank jcan be simpli ied i we can subs i u e i wi h a subdiag am
o smalle ank wi h he same coun ou label. As i is shown in Sec ion 13.2 o
[66], a sequence o hese ope a ions can always lead o a educed diag am, ha is
a diag am wi hou subg aphs ha can be simpli ied.
In 1933 an Kampen p o ed ha some undamen al p oblems in g oup heo y,
like unde s anding i a wo d in he gene a o s is he i ial elemen in he g oup,
can be sol ed h ough he use o diag ams o g oups. We will gi e a e sion o his
esul o educed diag ams o e g aded g oups, and e e o Theo em 13.1 o [66]
o he p oo .
Theo em 3.4 ( an Kampen).Le Wbe a non-emp y wo d in he alphabe S∪S−1.
Then W= 1 in a g oup Gwi h g aded p esen a ion (3.1) i and only i he e exis s
a educed ci cula diag am o e Gsuch ha he label o i s con ou is isually equal
o W.
w
1
2
Figu e 3.1: Van Kampen’s Lemma
The name “small cancella ion heo y” is due o he ac ha we o en equi e
ha di e en ela o s ha e a small o e lapping. This can be made mo e p ecise
wi h he ollowing de ini ion.
De ini ion 3.5 (Pieces and Condi ion C′(λ)).Le Gbe a g oup wi h g aded
p esen a ion (3.1), and le R1, R2,R16=R2, be wo cyclic conjuga es o wo
ela o s o in e ses o ela o s in R. A wo d Xin he alphabe S∪S−1is a piece
i R1and R2a e isually equal o wo ds o he o m XY1and XY2 espec i ely.
54
The p esen a ion (3.1) sa is ies small cancella ion condi ion C′(λ) o a numbe
0< λ ≤1i , whene e Ris a cyclic conjuga e o a ela o , o o an in e se o a
ela o , such ha i is isually equal o XY o a piece X, hen |X|< λ|R|.
Example 3.6. The g oup ha, b |aba−1b−1isa is ies C′(λ) o all λ > 1/4. Pieces
consis o single le e s o hei in e ses.
The su ace g oup ha, b, c, d |[a, b][c, d]]isa is ies C′(λ) o all λ > 1/8, and as
be o e pieces consis o single le e s o hei in e ses.
I is possible o see ha any g oup can ha e a p esen a ion sa is ying condi ion
C′(λ) o λ > 1
5(Gol’be g, see Sec ion 12.4 o [66]), bu i we ask λ o be smalle ,
i allows us o ob ain in e es ing condi ions on he g oups.
Theo em 3.7 (G eendlinge ’s, Theo em 12.1 [66]).Le ∆be a educed ci cula
diag am o e a p esen a ion o a g oup G ha sa is ies C′(λ) o λ≤1
6wi h a
leas one R-cell. Suppose ha he label ϕ(∂∆) is cyclically educed and has no
p ope subwo d equal o he iden i y. Then he e exis s an ex e io a c p(i.e. a
pa h p∈∂C ∩∂∆) o some R-cell Csa is ying |p|>1
2|∂C|.
This heo em has c ucial implica ions in combina o ial g oup heo y because,
i he e exis s a p esen a ion (3.1) o a g oup Gsa is ying C′(λ) o λ≤1
6, i is
possible o de ine an algo i hm (called Dehn’s algo i hm) ha in a ini e numbe
o s eps can ecognize i a wo d W∈S∗is equal o he iden i y in G. Mo eo e
his combina o ial condi ion has s ong geome ic implica ions, namely he g oup
Gis an hype bolic g oup.
E en i he p esen a ions we will use will no sa is y any C′(λ)condi ion, we
will ind an an analogue o G eendlinge ’s Theo em in g oups sa is ying weake
small cancella ion condi ions.
3.2 I ano ’s coun e example
In 1989 I ano ob ained he i s coun e example o P.Hall’s conjec u e s a ing
ha all wo ds a e concise in he class o all g oups. Mo e p ecisely, he p o ided a
wo d wIand a g oup Iin which wIis no concise.
Theo em 3.8 ([43]).The wo d
wI(x, y) = [[xpn, ypn]n, ypn]n
o n > 1010 odd and p > 5000 p ime, akes only wo alues {1, z}in a o sion ee
2-gene a ed g oup Ibu he e bal subg oup wI(I) = hzi, ha co esponds o he
cen e o he g oup I, is in ini e cyclic.
55
This g oup is a cen al ex ension o an in ini e wo gene a ed g oup GI(∞)o
bounded exponen , which is cons uc ed using small cancella ion heo y.
We i s need a c ucial esul in cen al ex ensions. We ecall ha a se Ro
ela ions o a g oup G=hS| Ri is independen i no p ope se R′⊆ R o
ela ions gi es he same g oup G(wi h he iden i y map on S).
Theo em 3.9. Suppose ha he g oup G=hS| Ri can be conside ed as G∼
=
F/N, wi h Fbeing he ee g oup wi h basis Sand N=hRi. Then
•G=F/[F, N]is a cen al ex ension o G=F/N, i.e. N=N[F, N]is
con ained in he cen e o G. Mo eo e , i Gis cen e less, hen N=Z(G);
•G=hS|[ , s] o ∈ R, s ∈Si;
•i Ris an independen se o ela ions o G, hen Nis a ee abelian g oup
wi h basis R=R[F, N].
Fo he p oo , we e e o Chap e 31 o [66], in pa icula o Theo em 31.1 and
he discussion abo e.
In he ollowing we cons uc he g oup I, gi ing an idea o he a gumen s in-
ol ed in he p oo ha wI akes a single non- i ial alue in I. In o de o do
so, we i s cons uc he g oup GI(∞), which is a a ia ion o he ee Bu nside
g oup cons uc ed by Olshanskii in [64] by induc i ely imposing (possibly di e -
en ) o sion o elemen s o a ee g oup, and by ob aining a o sion g oup as he
limi o all o hese quo ien s.
Le F2=F(a, b) he ee g oup in wo le e s and le V, W ∈F2be elemen s o
he ee g oup. Deno e by |V| he minimal leng h o Vas a wo d in he alphabe
{a, b, a−1, b−1}. Fix an o de ing in F2such ha i |V|<|W|, hen V < W (bu we
do no necessa ily ha e o choose lexicog aphic o de o wo ds o he same leng h).
Fo each i≥1we induc i ely cons uc he g oups GI(i). De ine GI(0) = F2,
hen assume we al eady cons uc ed GI(i−1). Le Ci∈F2be he smalles wo d
(wi h espec o he ixed o de ing o F2) co esponding o an elemen o in ini e
o de in GI(i−1), such a wo d will be called pe iod o ank i. De ine
GI(i) = GI(i−1)/hCni
iiGI(i−1),
whe e hCni
iiGI(i−1) is he no mal closu e o Cni
iand niis a odd numbe g ea e
han n= 1010. The limi o hese quo ien s is he g oup
GI(∞) = F2/hCni
i|i∈NiF2.(3.2)
When ni=n o e e y iwe ob ain he ee Bu nside g oup B(2, n). In his
cons uc ion, by using diag ams on g oups, Olshanskii p o ed ha he se {Cni
i|i∈
56
N}is an independen se o ela ions (i.e. no p ope subse s o ela ions de ines
he same g oup GI(∞)), ha e e y wo d o ini e o de in GI(i)is conjuga e o
a powe o a pe iod Cj o j≤i(so in he o sion g oup GI(∞)all wo ds a e
conjuga e o a powe o a pe iod), and mos impo an ly ha he g oup ob ained
in his way is in ini e and wi h i ial cen e .
I ano ’s g oup Iwill be ob ained as a cen al ex ension o GI(∞) o some
speci ic choices o he exponen s ni. In his case, we mus choose some ela o s in
a di e en way and we will impose some speci ic pe iods o ha e di e en o de s.
In de ail, o n > 1010 odd and p > 5000 p ime we equi e ha he wo ds Ci
sa is y ha :
• he smalles wo d o leng h 1 is C1=B1=aand we impose i o ha e o de
p2n;
• he smalles wo d Cio leng h 4(pn+1) is B2= [bapnb−1, apn]and we impose
i o ha e o de pn;
• he 8 smalles wo ds Cio leng h 8n(pn + 1) will be B3, . . . , B10
[[bε1aε2pnb−ε1, aε3pn]n, aε3pn]
o ε1, ε2, ε3∈ {±1}. We impose hese 8 wo ds o ha e o de n;
• all he o he wo ds Ciwill ha e o de n.
In Lemma 2 o [43] i is p o ed ha he wo d Cihas in ini e o de in GI(i−1)
(and hence i can be chosen o be a pe iod o app op ia e ank) and ha he
g oup GI(∞)wi h p esen a ion (3.2) ob ained by imposing hese es ic ions is
in ini e.
The g oup Iis ob ained as he quo ien
I=F2/h[Cni
i, T], Cni
i=Cnj
j|i, j ∈N, T ∈F2iF2.
In acco dance o Theo em 3.9, i we conside ed only he i s se o ela o s
{[Cni
i, T]}we would ob ain a cen al ex ension o GI(∞). The cen e o such
a g oup would be a ee abelian g oup wi h in ini e basis {Cni
i|n∈N}, bu by
adding he ela o s Cni
i=Cnj
j o all i, j ∈N, we ob ain a cyclic cen e , gene a ed
by a single elemen z=Cni
i o e e y i∈N. Now we wan o show ha he only
non- i ial alue aken by he wo d wIin he g oup Iis exac ly z.
Following he s eps o a p oo in [64], I ano p o ed he ollowing esul (Lemma
1 o [43]), which has a clea analogy o Theo em 3.7.
57
Lemma 3.10. Le ∆be a educed annula diag am o diag am on a disc wi h wo
holes o e GI(∞)(wi h p esen a ion 3.2, Cias in he p e ious pa ag aph) such
ha he labels o he con ou segmen s a e cyclically unsho enable. I ∆con ains
a leas an R-cell, hen he e exis s a cell Cwi h ∂C ∩∂∆ = p o a pa h psuch
ha |p| ≥ 10−4|∂C|.
To show ha zis indeed he only alue assumed by he wo d, we need o
“ unnel” he alues o some subwo ds o he wo d wI. We will explici ly explain
he i s s eps o show he use o diag ams o e g oups and o unde s and why we
need he di e si ica ion o he exponen s.
Suppose i s ha Xand Ya e wo wo ds in he alphabe S={a, b}. We
i s assume ha [Xpn, Y pn]n= 1 in GI(∞), in which case [Xpn, Y pn]nis in he
cen e o Iso (X, Y ) = 1. As all wo ds a e conjuga e o a powe o a pe iod in
GI(∞), i Xo Ya e conjuga e o a pe iod di e en han C1=a, hen ei he
Xpn o Ypn a e equal o he iden i y in GI(∞)(as all he pe iods di e en om
C1=aha e o de di iding pn) and we a e in he case [Xpn, Y pn]n= 1 in GI(∞),
ha we al eady conside ed. We can he e o e assume ha X=L−1a 1Land
Y=M−1a 2M o some wo ds L, M ∈S∗, 1, 2∈Z, and ha [Xpn, Y pn]n6= 1 in
GI(∞).
In his case, he wo d [Xpn, Y pn]mus be conjuga e in GI(∞) o a powe o
B1=ao B2, being he only pe iods wi h o de no di iding nin GI(∞). We wan
o p o e ha i canno be conjuga e o a powe o a. Suppose by con adic ion i
is possible. Then, o a ce ain N∈S∗, we would ha e
[L−1a 1pnL, M−1a 2pnM] = N−1a 3N.
By Theo em 3.4, and hen pas ing he pa hs o he con ou wi h label Nand
N−1, we can cons uc a diag am ∆o e GI(∞)on a disc wi h an hole such ha
he ex e io con ou has label a 3and he in e io con ou , ead clockwise, has
label [L−1a 1pnL, M−1a 2pnM](see Figu e 3.2). We can now pas e oge he he
wo segmen s o he in e nal con ou wi h label LM−1a 2pnML−1and i s in e se
espec i ely, so we ge a diag am on a disc wi h wo holes and he con ou s a e
a 3,a 1pn and a− 1pn (Figu e 3.3). By e ining i i necessa y, we can assume he
diag am o be educed.
58
a 3
[L−1a 1pnL, M−1a 2pnM]
N
Figu e 3.2: Pas ing con ou s wi h label N
a 3
a− 1pn
LM−1a 2pnML−1
a 1pn
Figu e 3.3: Pas ing con ou s wi h label LM−1a 2pnML−1
Now use small cancella ion heo y: by Lemma 3.10, i ∆con ains a leas an
R-cell, he e exis s a cell Cwi h con ou label Cnj
j o a ce ain j∈Nsuch ha i
has a bounda y a c po leng h a leas 10−4|Cnj
j|. As 10−4ni≥2 o e e y i∈N,
C2
jmus be a subwo d o ak o k=± 1pn o k=± 3, so Cj=C1=a. We can
now excise he cell C, in he sense ha we emo e Cand, i ∂C=pq wi h pbeing
he bounda y a c C ∩ δ∆, he new con ou o ∆will ollow he pa h qin place o
he p e ious bounda y a c p(Figu e 3.4).
59
ak1ak2ak2
ak4
ak3ak3
ap2nExcision
Figu e 3.4: Excision o a cell
By excising all he cells o his ype, wi h labels a±p2n, we change he exponen s
o some labels o he con ou , bu no hei esidual class modulo p2n. A e
ha ing excised all hese cells, we ha e a diag am on a disc wi h wo holes and
by Lemma 3.10 i canno con ain any R-cell (and in pa icula he disc wi h wo
holes is degene a e, wi h no in e io , Figu e 3.5). Looking a he inal diag am,
we can no ice ha he label o he ex e io bounda y is equal o he label o he
pa h ob ained by conca ena ing he wo in e io bounda ies. This implies ha
3≡pn( 1− 1)≡0 (mod p2n)so [L−1a 1pnL, M−1a 2pnM] = N−1a 3N= 1 in
GI(∞), and his con adic s ou assump ions.
ak1ak2
ak3
Figu e 3.5: Final diag am, a e exicisions
Wi h simila a gumen s, by means o cong uences p ese ed by cell excision in
diag ams, I ano p o ed ha i [Xpn, Y pn]n6= 1, hen his wo d is conjuga e o
ei he B2o B−1
2, no o a p ope powe o hem. S ill applying he same ideas, bu
wi h mo e complica e cong uences, he also p o ed ha [[Xpn, Y pn]n, Y pn]mus be
a conjuga e o exac ly one o he wo ds B3, . . . , B10. We e e o he las pa o
60
Lemma 3 in [43] o he explici compu a ions. This is sufficien o conclude ha
he only non-iden ical alue o he wo d mus be z.
A u he in e es ing ema k is ha , as i is w i en in he acknowledgemen s
o [43], he anonymous e e ee claimed ha , using Adian’s a gumen s o [6], he
wo d wA= [x , y ]n akes exac ly wo alues in a cen al ex ension (wi h cyclic
cen e ) o he ee Bu nside g oup B(2, n) o odd n= 3 ≥1005. This claim
has no been p o ed, bu i would p o ide he i s example o a wo d ha is
concise (in his case [x , y ], see [22], o Theo em 2.13) bu such ha i s powe
is no concise. No ice ha i such a claim was ue, using ha each in e se o a
wA- alue is s ill a wA- alue, he wo d wAwould ha e o ake a leas h ee alues
( he iden i y, a non- i ial elemen o in ini e o de and i s in e se). E en wi h
his co ec ion, his emains only a claim and he p oo is no a s aigh o wa d
adap a ion o I ano ’s me hods.
In iew o Conjec u es 1.17 and 1.20, i is na u al o ask whe he he coun e -
example ob ained by I ano p o ides a nega i e answe o hese ques ions oo.
Howe e , i is well known ha he g oup Iis no esidually ini e, hus canno be
used o ob ain a coun e example o he a o emen ioned conjec u es in a s aigh -
o wa d way. We now p o ide a p oo o his ac .
Lemma 3.11. Le Gbe a esidually ini e g oup and le Nbe he ma ginal subg oup
o a wo d w(x1, . . . , xn). Then G/N is esidually ini e.
P oo . Le g∈Gsuch ha g /∈N, in pa icula he e exis s an index i∈
{1, . . . , n}and some elemen s h1, . . . , hn∈Gsuch ha
=w(h1, . . . , hig,...,hn)w(h1, . . . , hi, . . . , hn)−16= 1
By esidually ini eness he e exis s a no mal subg oup Mo ini e index in G
such ha /∈M. We claim ha g /∈MN. I i was, le m∈M, n ∈Nsuch ha
g=mn. As Nis ma ginal o w, we would ha e ha
=w(h1, . . . , himn, . . . , hn)w(h1, . . . , hi, . . . , hn)−1=
w(h1, . . . , him, . . . , hn)w(h1, . . . , hi, . . . , hn)−1
bu his would imply ha ≡1 (mod M), con adic ing ou choice o M. This
p o es ha o e e y g /∈N he e exis s a ini e index subg oup MN such ha
g /∈MN, as desi ed.
Co olla y 3.12. Any ini ely gene a ed g oup which is cen al-by-(in ini e g oup
o ini e exponen ) canno be esidually ini e. In pa icula , I ano ’s g oup Iis
no esidually ini e.
61
3.5 On gene a ion o e bal subg oups
In [24], he au ho s p o ed s ong conciseness o se e al classes o g oup wo ds, like
wo ds implying i ual nilpo ency o weakly a ional wo ds, unde he addi ional
assump ion ha , i he e bal subg oup w(G)is ini ely gene a ed, hen i can be
gene a ed by ini ely many w- alues. I w(G)is a p o-pg oup, hen by looking a
he quo ien w(G)/Φ(w(G)) and using Bu nside basis heo em, i is immedia e o
see ha w(G)is ini ely gene a ed i and only i i is gene a ed by ini ely many
w- alues. The au ho s asked whe he his is always ue.
Conjec u e 3.21. Le Gbe a p o ini e g oup and wbe a wo d. I w(G)is
opologically ini ely gene a ed, i can be gene a ed by ini ely many wo d alues.
We will now show ha his ques ion has a nega i e answe o lowe cen al
wo ds w=γk.
Theo em 3.22. The e is a p o ini e g oup Gsuch ha he subg oup γk(G)is
p ocyclic o e e y k, bu i canno be gene a ed by ini ely many γk- alues.
Clea ly he g oup Gin ou cons uc ion canno be ini ely gene a ed o he wise,
as Nikolo and Segal p o ed in [59][60], all he abs ac subg oups o he lowe
cen al se ies would be closed. In ha case, whene e a e bal subg oup w(G)is
ini ely gene a ed, i is also gene a ed by ini ely many w- alues: since i coincides
wi h an abs ac subg oup ha is ini ely gene a ed, each gene a o is a ini e wo d
in he alphabe w{G}, so he subg oup i sel is also gene a ed by ini ely many
w- alues.
A special case o he ques ion we a e in e es ed in is when he de i ed subg oup
is p ocyclic. Unde his mo e es ic i e hypo hesis, is i ue ha i is gene a ed
by a single commu a o ? This ques ion was s udied, in he se ing o abs ac
g oups and cyclic subg oups, by Macdonald in [56]. He p o ed he ollowing
esul .
Theo em 3.23 (Macdonald).Le Gbe an abs ac g oup and assume G′is cyc-
lic. I ei he Gis nilpo en o G′is in ini e, hen G′is gene a ed by a sui able
commu a o . In gene al, o any gi en posi i e in ege k, he e is a ini e g oup
Mksuch ha M′
kis cyclic bu i canno be gene a ed by less han kcommu a o s.
The main ool in ou p oo is he g oup Mkin he second pa o he p oposi ion,
so we will gi e an idea o i s s uc u e. Fixed k, se m= 22k−1and pick a se
o di e en odd p imes p1, . . . , pm, chosen a bi a ily. The g oup Mkwill be a
semidi ec p oduc CoC2k
2, whe e C2k
2=ha1, . . . , a2kiis he di ec p oduc o
68
2kcopies o he cyclic g oup o o de 2 and C=hciis a cyclic g oup o o de
p1p2· · · pm.
Assume [c, ai] = cαi o some in ege αi. Macdonald p o ed, wi h he use o
some accu a ely chosen cong uences, ha i is possible o selec he in ege s αiin
he cons uc ion o Mkin such a way ha he de i ed subg oup is he whole Cand
ha o any se o k−1commu a o s g1, . . . , gk−1 he e is a p ime pj∈ {p1, . . . , pm}
such ha
hg1, . . . , gk−1i=hcpji.
In [45], Kappe obse ed ha γ2{Mk}=γj{Mk} o all j≥2, hence he ollowing
esul is a di ec consequence o Macdonald’s heo em.
Co olla y 3.24 (Kappe).Fo any gi en posi i e in ege kand any j≥2, he e
is a ini e g oup Mksuch ha γj(Mk)is cyclic bu i canno be gene a ed by less
han kcommu a o s.
P oo o Theo em 3.22. We will p o e he esul o he commu a o subg oup.
The same cons uc ion wo ks o all lowe cen al wo ds by Co olla y 3.24.
Fo e e y k, le Mkbe he g oup cons uc ed by Macdonald in Theo em 3.23.
In he choices o he g oup Mk, we had o choose some odd p imes pk
1, . . . , pk
22k−1,
we can equi e all o hem o be di e en bo h pai wise and om all p imes pj
l o
1< j < k and 1< l < 22j−1.
De ine Bi=Qi
k=1 Mk.By cons uc ion, B′
iis a di ec p oduc o cyclic sub-
g oups o cop ime o de , so i is cyclic oo. As M′
icanno be gene a ed by less
han icommu a o s, he same is ue o B′
i. Mo eo e , by cons uc ion he g oups
Bi, o i∈N, o m an in e se sys em o ini e g oups, so we can de ine he p o ini e
g oup G= lim
←− Bi.
Clea ly G′is p ocyclic as i is he in e se limi o B′
i. I canno be gene a ed by
a ini e se Xo commu a o s, say o ca dinali y n, because o he wise he images
o Xin he quo ien Bn+1 would gene a e B′
n+1 oo, con adic ing he p e ious
pa ag aph.
In he way he au ho s use his p ope y in [24], i is s ill ele an o ask whe he
he same phenomena can happen unde he assump ion ha |w{G}| <2ℵ0. O
cou se, in iew o Conjec u e 1.20, i is possible ha no wo d can ake coun ably
many alues in a g oup unless i akes ini ely many, so his could be conside ed
as an in e media e s ep owa ds he p oo o he s ong conciseness conjec u e.
Conjec u e 3.25. Le Gbe a p o ini e g oup and wbe a wo d. I w(G)is
opologically ini ely gene a ed and |w{G}| <2ℵ0, hen w(G)can be gene a ed
by ini ely many wo d alues.
69
Clea ly he example in Theo em 3.22 canno be used in o de o con adic his
conjec u e because lowe cen al wo ds a e s ongly concise (see [24]).
70
4
Cop ime commu a o s
In his chap e we p o e s ong conciseness o cop ime commu a o s γ∗
kand δ∗
k,
ollowing he a icle [39] by de las He as, Shumya sky and he au ho .
We will i s de ine cop ime commu a o s and gi e an o e iew o hei his o y.
Cop ime commu a o s a e no wo d maps, bu beha e in a simila way, and hey
cons i u e a good gene a ing se o he p onilpo en esidual ( o γ∗
k,k≥2) o o
he k- h p onilpo en esidual ( o δ∗
k,k≥1).
In he second sec ion we will discuss some basic lemmas ha a e necessa y
o de elop ou esul s. Some o hem we e al eady p esen in he li e a u e and
o he s a e o iginal.
In he hi d sec ion we will ou line he s uc u e o he p oo s, wi h a desc ip-
ion o an in e es ing se o p onilpo en subg oups ha is p esen in p osol able
g oups.
We will hen p o e he main heo ems, o s ong conciseness o cop ime com-
mu a o s, bo h in he me a-p onilpo en case o γ∗
kin Sec ion 4, and in he (p o-
sol able o Fi ing heig h k+ 1) case o δ∗
kin Sec ion 5. The gene al s a emen s
will be hen p o ed join ly in Sec ion 6.
71
4.1 His o y o cop ime commu a o s
Highe o de cop ime commu a o s we e in oduced by Pa el Shumya sky in [75]
as a way o ob ain a smalle na u al se o gene a o s o some classical subg oups.
Gi en a p o ini e g oup Gand an elemen x∈G, we deno e by |G|( espec i ely
|x|) he o de o G( espec i ely x) as a supe na u al numbe and π(G)( espec -
i ely π(x)) will s and o he se o p ime numbe s di iding |G|( espec i ely |x|).
We will say ha an elemen g∈Gis a simple cop ime commu a o i and only i
i can be w i en as g= [g1, g2] o g1, g2∈Gwi h (|g1|,|g2|) = 1.
I was al eady well-known ha he se o simple cop ime commu a o s in a
ini e g oup Ggene a es he nilpo en esidual γ∞(G), ha is, he smalles no mal
subg oup Nsuch ha G/N is nilpo en (see Theo em 2.1 o [75]). O cou se, in
p o ini e g oups he p onilpo en esidual γ∞(G) = Tiγi(G)is he in e sec ion o
he e ms o he lowe cen al se ies o G.
Cop ime commu a o s o highe o de we e de ined in [75] o ini e g oups, bu
he de ini ion na u ally ex ends o he p o ini e case.
De ini ion 4.1 (Highe o de cop ime commu a o s).Le
γ∗
1{G}=δ∗
0{G}=G
and, o e e y posi i e in ege ide ine induc i ely he se s
γ∗
i{G}=[xλ, g]|x∈γ∗
i−1{G}, λ ∈b
Z, g ∈G, (|xλ|,|g|) = 1
δ∗
i{G}=[xλ1, yλ2]|x, y ∈δ∗
i−1{G}, λ1, λ2∈b
Z,(|xλ1|,|yλ2|) = 1.
Mo eo e , o he gene a ed subg oups we will w i e γ∗
i(G) = hγ∗
i{G}i and
δ∗
i(G) = hδ∗
i{G}i.
E en i cop ime commu a o s a e no wo d maps, he analogy wi h classical
wo d maps is clea . Indeed, γ∗
i(G)and δ∗
i(G)a e ully in a ian subg oups because
he o de o (x)always di ides he o de o x o e e y homomo phism and
e e y x∈G. Fo his eason, i is in e es ing o desc ibe he subg oups gene a ed
by cop ime commu a o s, and he main esul s o he a icle [75] comple ely sol e
his na u al ques ion.
Theo em 4.2 ([75] Theo ems 2.1, 2.7).Le Gbe a p o ini e g oup.
I k≥2, he subg oup γ∗
k(G)is i ial i and only i Gis p onilpo en .
The subg oup δ∗
k(G)is i ial i and only i Gis p osol able o Fi ing heigh a
mos k.
72
I is in e es ing o poin ou ha a consequence o Theo em 4.2 is ha he e
exis s no wo d w∈F(X∞)such ha w(G) = γ∗
i(G)(i= 2,3, . . .) o w(G) =
δ∗
i(G)(iposi i e in ege ) o e e y p o ini e g oup Gbecause nilpo en g oups o
unbounded class do no o m a a ie y o g oups.
Se e al p oblems, ha we e classical o usual commu a o s, we e hen adap ed
o cop ime commu a o s. An example is O e’s Conjec u e, which s a ed ha e e y
elemen o a ini e simple g oup is a commu a o , and was sol ed in [53]. In [75],
he au ho conjec u ed ha e e y elemen o a ini e simple g oup can be ealized
as a cop ime commu a o and p o ed he conjec u e o he class o al e na ing
g oups. The same conjec u e was la e se led o PSL2(q) o e e y p ime powe
qin [67] and o Suzuki g oups 2B2(q) o e e y odd qin [88].
Ano he na u al consequence o he analogy be ween cop ime commu a o s and
usual commu a o s was he s udy o conciseness p oblems o hem. O cou se we
will say ha γ∗
i( esp δ∗
i) is concise i γ∗
i(G)( esp. δ∗
i(G)) is ini e whene e γ∗
i{G}
( esp. δ∗
i{G}) is ini e.
In [5] he au ho s p o ed ha , i he e exis s a posi i e in ege msuch ha
he wo d γ∗
io δ∗
i akes a mos m alues in a ini e g oup G, hen he gene a ed
subg oup has m-bounded o de . The bound does no depend on i, so ha cop ime
commu a o s a e uni o mly concise in he class o ini e g oups. A s aigh o wa d
consequence is ha cop ime commu a o s o highe o de a e concise in esidually
ini e g oups.
In he a icle [24], ha began he in es iga ion in s ong conciseness, he au ho s
no iced ha he concep o s ong conciseness can be applied in a wide con ex .
Suppose Cis a class o p o ini e g oups and ϕ{G}is a subse o G o e e y G∈ C.
Is he subg oup gene a ed by ϕ{G} ini e whene e |ϕ{G}| <2ℵ0? Such map ϕis
said o be s ongly concise in he class Ci he answe is posi i e. This ques ion
is in e es ing whene e ϕ{G}is de ined in some na u al way and/o p ope ies o
he subg oup hϕ{G}i ha e s ong impac on he s uc u e o G. Fo his eason,
in [28] he au ho s examined s ong conciseness o cop ime commu a o s and
managed o se ha he map γ∗
2is s ongly concise in p o ini e g oups. In his
chap e , which oughly ollows he a icle [39], we will p o e s ong conciseness o
γ∗
iand δ∗
i o e e y posi i e in ege i.
Theo em 4.3. A p o ini e g oup Gis ini e-by-p onilpo en i and only i he e is
ksuch ha he se o γ∗
k- alues in Ghas ca dinali y smalle han 2ℵ0.
Theo em 4.4. A p o ini e g oup Gis ini e-by-(p osol able o Fi ing heigh a
mos k) i and only i he se o δ∗
k- alues in Ghas ca dinali y smalle han 2ℵ0.
73
O cou se he e a e esul s o s ong conciseness because by Theo em 4.2 he
alues o he wo ds γ∗
kand δ∗
kgene a e he ini e subg oups o Theo ems 4.3 and
4.4.
4.2 P elimina ies
We will i s lis some esul s ha we e p esen in he li e a u e, o some small
a ia ions o hem, ha will be use ul in he p oo s o Theo ems 4.3 and 4.4.
The i s one is a undamen al esul in he s udy o s ong conciseness. A di ec
applica ion o his esul is ha conjugacy classes in p o ini e g oups a e ei he
ini e o o ca dinali y a leas 2ℵ0(see Lemma 3.17).
P oposi ion 4.5 ([24] Lemma 2.1).Le φ:X→Ybe a con inuous map be ween
wo non-emp y p o ini e spaces ha is nowhe e locally cons an (i.e. he e is no
non-emp y open subse U⊆oXwhe e φ|Uis cons an ). Then |φ(X)| ≥ 2ℵ0.
A classical esul in he heo y o cop ime au omo phisms is he ollowing.
Lemma 4.6 ([42], Lemma 4.29).Le Abe a g oup o au omo phisms o a ini e
g oup Gwi h (|A|,|G|) = 1. Then, [G, A] = [G, A, A].
The ollowing lemma is a s onge e sion o his esul o he case whe e Gis
a p onilpo en g oup.
Lemma 4.7 ([47] Lemma 4.6).Le φbe an au omo phism o a p onilpo en g oup
Gwi h (|φ|,|G|)=1. De ine he se he se S={[g, φ]|g∈G}. Then he map
θ:S→Sde ined as
θ:x→[x, φ]
is bijec i e.
The ollowing is a p o ini e e sion o Lemma 2.4 in [75].
Lemma 4.8. Le Gbe a p o ini e g oup and le g1, . . . , gkbe δ∗
k−1- alues in G.
Suppose ha g1, . . . , gk∈NG(H) o a subg oup H≤Gwi h (|H|,|gi|) = 1 o
e e y i∈ {1, . . . , k}. Then, o e e y h∈H, he elemen [h, g1, . . . , gk]is a
δ∗
k- alue.
Using he p e ious wo lemmas oge he , we will be able o gua an ee ha some
special ypes o long commu a o s a e also alues o δ∗
k.
74
Lemma 4.9. Le G1, . . . , Gkbe p onilpo en subg oups o a p o ini e g oup G
such ha Gj≤NG(Gi) o all j≤i. Le xi∈Gi o e e y iand assume ha
(|xi|,|xi+1|)=1 o all i= 1, . . . , k. Then he elemen g= [x1, . . . , xk]is in
δ∗
k−1{G}and π(g)⊆π(xk).
P oo . We will p o e by induc ion on i ha gi:= [x1, . . . , xi]∈δ∗
i−1{G} o e e y
i∈ {1, . . . , k}and ha π(gi)⊆π(xi). The s a emen o he lemma co esponds
o he case i=k. I i= 1 he esul is ob ious, so assume i > 1and ha gi−1is
aδ∗
i−2- alue wi h π(gi−1)⊆π(xi−1), so in pa icula (|gi−1|,|xi|) = 1. I His he
minimal Hall subg oup o he p onilpo en g oup Gicon aining xi, hen gi−1ac s
as a cop ime au omo phism o H. By Lemma 4.7, he e exis s yi∈Hsuch ha
[xi, gi−1] = [yi, gi−1,i−1
. . ., gi−1],
and Lemma 4.8 shows ha gi= [xi, gi−1]is a δ∗
i−1- alue, as desi ed. As gi∈H,
we immedia ely ha e ha π(gi)⊆π(xi).
The nex esul is a p o ini e e sion o Lemma 2.4 in [5]. We ecall ha by
“me a-p onilpo en ” g oup we mean a p o ini e g oup Gha ing a no mal p onil-
po en subg oup Nsuch ha G/N is p onilpo en .
Lemma 4.10. Le Gbe a me a-p onilpo en g oup. Then γ∞(G) = Qp[Kp, Hp′],
whe e Kpis a Sylow p-subg oup o γ∞(G)and Hp′is a Hall p′-subg oup o G.
Fo a gene al g oup wo d w, he se w{G}o w- alues o a p o ini e g oup G
is always closed in G. We will show ha he same is ue o he se s o γ∗
kand
δ∗
k- alues.
P oposi ion 4.11. Le S1, . . . , Skbe closed subse s o a p o ini e g oup G. Then
he se
C={(g1, . . . , gk)∈S1× · · · × Sk|(|gi|,|gi+1|) = 1 o all i= 1, . . . , k −1}
is closed in S1× · · · × Sk. Fu he mo e, he se s γ∗
k{G}and δ∗
k{G}a e closed in
G.
P oo . Le Pbe he se o all p imes and p∈ P. Fi s no ice ha o e e y closed
subse So G he se
Sp′={g∈S|p /∈π(g)}
is closed. Indeed Sp′=TN⊴oGSp′Nbecause p∈π(g)i and only i he e is a
no mal subg oup Nsuch ha gN has o de di ided by pin G/N. Also, he se
75
S
b
Z={gλ|g∈S, λ ∈b
Z}is he image unde he con inuous map (g, λ) = gλo
he compac se S×b
Z, so i is closed oo.
Le now A, B be subse s o G. We claim ha he se
RA,B =
p∈P (A×Bp′)∪(Ap′×B)(4.1)
is exac ly he se o elemen s (a, b)∈A×Bwi h |a|and |b|cop ime. On he
one hand, i |a|and |b|a e cop ime hen (a, b)∈(A×Bp′)∪(Ap′×B) o e e y
p∈ P, because, i b∈B Bp′, hen a∈Ap′necessa ily. On he o he hand, i
(a, b)∈RA,B and a p ime pdi ides |a|, hen (a, b)∈A×Bp′so pdoes no di ide
|b|, and he claim ollows. No ice now ha i Aand Ba e closed, he se RA,B is
an in e sec ion o closed subse s o G×Gso i is closed oo.
I is now easy o p o e by induc ion on k ha he se s γ∗
k{G},δ∗
k{G}a e closed:
jus no e ha γ∗
k{G}is exac ly he se RA,B in (4.1) wi h A= (γ∗
k−1{G})
b
Z,B=G,
whe eas δ∗
k{G}is he se RA,B in (4.1) wi h A=B= (δ∗
k−1{G})
b
Z.
To p o e ha he se Cis closed in S1× · · · × Sk, i suffices o no ice ha by
he abo e a gumen s he se
Ci=S1× · · · × Si−1×RSi,Si+1 ×Si+2 × · · · × Sk
is closed o e e y i∈ {1, . . . , k −1}and C=Tk−1
i=1 Ci.
As we showed in Lemma 1.12, whene e a g oup wo d w akes ini ely many
alues in a g oup G, he subg oup w(G)is ini e i and only i w(G)/w(G)′is
ini e. I w akes less han 2ℵ0 alues in Gwe canno ob ain he same conclusion
in gene al, bu wi h some sligh ly s onge hypo hesis we can anyway ob ain a
simila esul .
Lemma 4.12. Le ϕbe a map ha associa es o e e y g oup Ga no mal subse
ϕ{G} ⊆ G. Le Gbe a p o ini e g oup wi h |ϕ{G}| <2ℵ0and le Kbe a
p onilpo en subg oup o hϕ{G}i gene a ed by a subse o ϕ(G). I K/K′is ini e,
hen Kis ini e.
P oo . Since Kis p onilpo en , we ha e K′≤Φ(K), whe e Φ(K)s ands o he
F a ini subg oup o K. Thus K/Φ(K)is ini e, and hence we can ind a ini e
subse So ϕ{G}gene a ing K. Since ϕ(G)is a no mal subse o G, by Lemma
3.17 each o hese gene a o s has ini ely many conjuga es in G, so in pa icula
|G:CG(s)|<∞ o e e y s∈S. Since CG(K) = Ts∈SCG(s), his implies ha
Z(K) = K∩CG(K)has ini e index in K, and by Schu ’s heo em K′is ini e.
76
We will use Lemma 4.12 o ϕ=γ∗
ko ϕ=δ∗
k, bu i could be applied o o he
cases, such as any g oup wo d map o uni o m (an i-cop ime) commu a o s (see
[28] o [29]).
4.3 In oduc ion o he p oo s
In o de o ully unde s and he p oo s o Theo ems 4.3 and 4.4, we ha e o
begin om he p oo o De omi, Mo igi and Shumya sky in [28] ha se led he
analogous esul o γ∗
2.
In he a o emen ioned a icle, he au ho s i s p o ed ha γ∗
2is s ongly concise
in me a-p onilpo en g oups and hen used his pa ial esul o se le he gene al
case. We will simila ly spli ou p oo : i s we will p o e s ong conciseness o γ∗
k
in me a-p onilpo en g oups (P oposi ion 4.20 in Sec ion 4.4), hen we will se le
he p oblem o δ∗
kin p osol able g oups o Fi ing heigh k+ 1 (P oposi ion 4.33
in Sec ion 4.5) and we will use hese pa ial esul s in he p oo o Theo ems 4.3
and 4.4, ha will be p o ed join ly in Sec ion 4.6.
The p oo o P oposi ion 4.20 consis s o ex ending he easoning ha was used
in [28] o γ∗
2, wi h a ocal use o Lemma 4.7. The p oo o he gene al case also
pa ially ollows [28], wi h some complica ions in he a gumen s.
The case o δ∗
kin p osol able g oups o Fi ing heigh k+ 1, howe e , in ol ed
a lo o echnical p oblems and is su ely he mo e complex pa o his chap e .
Fo his eason, in his case we gi e a deepe analysis and mo i a ion o he ideas
in ol ed.
An essen ial ool o he p oo is he ollowing collec ion o subg oups.
De ini ion 4.13 (Sylow basis).ASylow basis o a p o ini e g oup Gis a amily
{Pi}o Sylow subg oups o G, one o each p ime in π(G), such ha PiPj=PjPi
o e e y i, j. The no malize o a Sylow basis is T=TiNG(Pi).
Basic p ope ies o Sylow bases o ini e g oups can be ound in Sec ion 9.2 o
[73] and hey ex end na u ally o p o ini e g oups.
Lemma 4.14. Any p osol able g oup admi s a Sulow basis and any wo Sylow
bases a e conjuga e. In his case, he Sylow basis no malize Tis p onilpo en and
G=Tγ∞(G). Mo eo e , i Gis me a-p onilpo en , γ∞(G) = [T, γ∞(G)].
P oo . The i s s a emen is a classical esul , see o example P oposi ion 2.3.9
o [72], whe eas he ac ha G=Tγ∞(G)is Lemma 5.6 o [69]. I Gis me a-
p onilpo en , we ha e ha
γ∞(G) = [G, γ∞(G)] = [Tγ∞(G), γ∞(G)] = [T, γ∞(G)]
77
P oo . Assume i s ha ℓ6= 1, and p oceed by induc ion on −ℓ. I −ℓ= 0,
hen
[g1, . . . , g −1, g′
g ] = [g1, . . . , g′
]g [g1,...,g ]−1[g1, . . . , g ],
and he esul ollows. Assume −ℓ > 0, and we w i e, o he sake o b e i y,
y= [g1, . . . , gℓ−1]. By induc ion, we ha e
[y, g′
ℓgℓ, gℓ+1, . . . , g ] = [[y, g′
ℓ, ghℓ
ℓ+1, . . . , gh −2
−1]h −1[g1, . . . , g −1], g ]
wi h hi∈Gℓ· · · Gi o i∈ {ℓ, . . . , −1}. Now,
[[y, g′
ℓ, ghℓ
ℓ+1, . . . , gh −2
−1]h −1[g1, . . . , g −1], g ]
= [[y, g′
ℓ, ghℓ
ℓ+1, . . . , gh −2
−1]h −1, g ][g1,...,g −1][g1, . . . , g ]
= [y, g′
ℓ, ghℓ
ℓ+1, . . . , gh −2
−1, g(h −1)−1
]h −1[g1,...,g −1][g1, . . . , g ],
and he lemma ollows. I ℓ= 1, a simila a gumen applies.
Lemma 4.22. Le G1, . . . , G be subg oups o a p o ini e g oup Gsuch ha
Gj≤NG(Gi) o e e y j≤i. Le ℓ∈ {1, . . . , }and Y1, Y2⊆Gℓbe such
ha π(y1), π(y2)⊆π(y1y2) o e e y y1∈Y1,y2∈Y2. Le Xi⊆Gi o
i∈ {1, . . . , ℓ −1}, and o i∈ {ℓ+ 1, . . . , }deno e Xi=Gi. Then:
1. I φ∗
{ℓ}(Y1;Xi) = φ∗
{ℓ}(Y2;Xi) = 1, hen φ∗
{ℓ}(Y1Y2;Xi) = 1.
2. I φ∗
{ℓ}(Yj;Xi) = ∅ o some j∈ {1,2}, hen φ∗
{ℓ}(Y1Y2;Xi) = ∅.
P oo . Since π(y1), π(y2)⊆π(y1y2) o e e y y1∈Y1,y2∈Y2, he second s a e-
men is s aigh o wa d. Mo eo e , i φ{ℓ}(y1y2;gi)∈φ∗
{ℓ}(Y1Y2;Xi), hen o
j∈ {1,2}we ha e φ{ℓ}(yj;gi)∈φ∗
{ℓ}(Yj;Xi). The esul ollows now di ec ly om
Lemma 4.21.
In iew o he p eceding lemma, we now in oduce a con enien way o choose
cose ep esen a i es o no mal subg oups. These will play an impo an ole
h oughou he chap e .
De ini ion 4.23 (Good ep esen a i es).Le Gbe a p o ini e g oup and U⊴G.
An elemen g∈Gis a good ep esen a i e o he cose gU i π(g), π(u)⊆π(gu)
o e e y u∈U.
Lemma 4.24. Le Ube an open no mal subg oup o a p onilpo en g oup G. Le
gbe a ep esen a i e o he cose gU and w i e g=Qp∈π(G)gpwi h gpap-elemen
o G. Then he ollowing a e equi alen :
84
(i) gis a good ep esen a i e o he cose gU;
(ii) gp= 1 whene e gp∈U o p∈π(G);
(iii) π(g)is minimal among all ep esen a i es o he cose gU.
In his case, i σ=π(G/U), hen π(g)⊆σ.
P oo . We i s p o e (i)⇒(ii). Assume gis a good ep esen a i e and suppose
ha gp∈U. I gp6= 1, hen π(g·g−1
p)does no con ain p, con adic ing ha
π(g)⊆π(gu) o all u∈U.
(ii)⇒(i). W i e u=Qp∈π(G)up o a ce ain u∈Uand suppose gp= 1
whene e gp∈U. Then, i ei he gp6= 1 o up6= 1, hen gpup6= 1, ha is exac ly
he condi ion o being a good ep esen a i e.
(ii)⇔(iii) is immedia e, and he las ema k ollows om (ii).
The ollowing lemma is an applica ion o P oposi ion 4.5 o a special ype o
cop ime commu a o s.
Lemma 4.25. Le G1, . . . , G be p onilpo en subg oups o a p o ini e g oup Gsuch
ha Gj≤NG(Gi) o all j≤i, and |δ∗
−1{G}| <2ℵ0. Fo e e y i∈ {1, . . . , }, le
Sibe a closed subse o Gi. I φ∗(Si)6=∅, hen, he e exis elemen s xi∈Giand
open subg oups Ui⊴oGisuch ha |φ∗(xiUi∩Si)|= 1.
P oo . Le
C=n(x1, . . . , x )∈S1× · · · × S (|xi|,|xi+1|) = 1 o all i= 1, . . . , o.
As φ(C) = φ∗(Si), we ha e C 6=∅. No e ha Cis closed in G1× · · · × G by
Lemma 4.11.
Fix (x1, . . . , x )∈ C. By Lemma 4.9 he elemen gk:= [x1, . . . , x ]is in δ∗
−1{G}.
Hence, |Imm(φ)|<2ℵ0, and by P oposi ion 4.5, i ollows ha he e exis elemen s
xi∈Giand open no mal subg oups Ui⊴Gisuch ha
C ∩ (x1U1× · · · × x U )6=∅
and |φ∗(xiUi∩Si)|= 1.
Lemma 4.25 will o en p o ide some cose s o open subg oups o Gin which
cop ime commu a o s a e i ial. Lemmas 4.26 and 4.29 below will allow us o
ela e cop ime commu a o s o hese cose s wi h cop ime commu a o s o he open
subg oups hemsel es.
85
Lemma 4.26. Le G1, . . . , G be subg oups o a p o ini e g oup Gsuch ha Gj≤
NG(Gi) o e e y j≤i, and o e e y i∈ {1, . . . , }, le xi∈Giand Ui⊴Gi.
Assume also ha Gj≤NG(Ui) o e e y j≤i. Fix j∈ {1, . . . , }and w i e
J={1, . . . , j −1}, hen:
(i) I φ(xiUi) = 1 hen φJ(xiUi;Ui) = 1.
(ii) I φJ(xiUi;Ui) = 1 hen
φJ∪{j}(xiUi;Ui) = φ(x1U1, . . . , xj−1Uj−1, xj, Uj+1, . . . , U ).
P oo . (i) We will p oceed by e e se induc ion on j∈ {1, . . . , + 1}, whe e he
base case j= + 1 ansla es o φ(xiUi)=1, which is ue by hypo hesis. Le
hus j < + 1 and assume ha φJ∪{j}(xiUi;Ui) = 1.
Le C = 1 and o e e y i∈ {j+1, . . . , −1}de ine Ci=CUi(Ui+1/Ci+1). No e
ha Ciis well-de ined, since using ha o e e y ℓ he subg oup Uℓis no mal in
G1· · · Gℓ, one can easily show by induc ion ha Cℓ⊴G1· · · Gℓ.
I j≥2, le
Y={[x1u1, . . . , xj−1uj−1]|ui∈Ui, i = 1, . . . , j −1}.
Then, we can ew i e φJ∪{j}(xiUi;Ui) = 1 as
[Y, xjUj]⊆CGj(Uj+1/Cj+1).
Fo e e y i∈ {1, . . . , j}, ix ui∈Uiand sho en y= [x1u1, . . . , xj−1uj−1]. Then
we ha e [y, xjuj] = [y, uj][y, xj]uj, and since CGj(Uj+1/Cj+1)is a no mal subg oup
o Gjcon aining [y, xjuj]and [y, xj], i ollows ha [y, uj]∈CGj(Uj+1/Cj+1). This
shows ha φ(x1U1, . . . , xj−1Uj−1, Uj, Uj+1, . . . , U ) = 1, as we wan ed.
Fo he case j= 1, no e ha bo h x1and x1U1lay in CG1(U2/C2), so ha
U1≤CG1(U2/C2).
(ii) Fo e e y i∈ {j+ 1, . . . , }we de ine Cias in (i). Fo i∈ {1, . . . , }, le
ui∈Uiand sho en y= [x1u1, . . . , xj−1uj−1]. Then,
[y, xjuj] = [y, u′xj] = [y, xj][y, u′]xj= [y, u′]xi[xj,y][y, xj]
o some u′∈Uj, and no e ha z:= [y, u′]xj[xj,y]∈CGj(Uj+1/Cj+1). Then
[z, u′
j+1, . . . , u′
] = 1 o e e y u′
i∈Ui,i∈ {j+ 1, . . . , }, so ha
[y, xjuj, uj+1, . . . , u ] = [z[y, xj], uj+1, . . . , u ] = [y, xj, uj+1, . . . , u ],
whe e he las equali y ollows om Lemma 4.21. The lemma ollows.
86
De ini ion 4.27 (Subg oup Nσ).Le G1, . . . , G be p onilpo en subg oups o a
p o ini e g oup Gsuch ha Gj≤NG(Gi) o all j≤i. Le σbe a ini e se o
p imes. We de ine he no mal subg oup
Nσ=hφ∗
{j}(Hi;Gi)|jis such ha |π(Gj)|=∞iG,
whe e Hiis he Hall σ-subg oup o Gi o e e y i. I |π(Gi)|<∞ o all i, hen
Nσ=h∅iG= 1 o e e y σ.
The subg oups G1, . . . , G o G o which he de ini ion o Nσapplies will be
clea om he con ex . No ice ha o any ini e se s o p imes σ1and σ2such
ha σ1⊆σ2we ha e
Nσ1≤Nσ2.(4.4)
Lemma 4.28. Le G1, . . . , G be p onilpo en subg oups o a p o ini e g oup G
such ha Gj≤NG(Gi) o all j≤i. Fix ℓ∈ {1, . . . , }and xℓ∈Gℓ. Fo
i∈ {1, . . . , ℓ −1}, le Xi⊆Gi, and o i∈ {ℓ, . . . , }le Ui⊴oGibe such
ha Gj≤NG(Ui) o j≤i. Suppose ha (|xℓ|,|xℓ−1|)=(|xℓ|,|Uℓ+1|)=1 o
e e y xℓ−1∈Xℓ−1. I φ∗(X1, . . . , Xℓ−1, xℓUℓ, Uℓ+1, . . . , U ) = 1, hen we ha e
φ∗(X1, . . . , Xℓ−1, Uℓ, . . . , U ) = 1.
P oo . Fi s o all, obse e ha since φ∗(X1, . . . , Xℓ−1, xℓUℓ, Uℓ+1, . . . , U )6=∅,
he e a e y1, . . . , yℓ−1such ha yi∈Xiand
(|yj|,|yj+1|) = 1 (4.5)
o all j∈ {1, . . . , ℓ −2}. No e ha he uple (y1, . . . , yℓ−1,1, . . . , 1) is in Cand
hen φ∗(X1, . . . , Xℓ−1, Uℓ, . . . , U )6=∅.
Fix hen a uple (x1, . . . , xℓ−1, uℓ, . . . , u )∈ C wi h xj∈Xjand uj∈Uj. In
o de o conclude we wan o p o e ha φ∗(x1, . . . , xℓ−1, uℓ, . . . , u ) = 1. Fo
i∈ {ℓ, . . . , }, le Hibe he minimal Hall subg oup o Uicon aining ui, and no ice
ha we ha e
(|xℓ−1|,|Hℓ|) = (|Hj|,|Hj+1|) = 1 (4.6)
o all j∈ {ℓ, . . . , −1}. Since Gℓis p onilpo en , we ha e π(xℓh)⊆π(xℓ)∪
π(h) o all h∈Hℓ, and hence, as (|xℓ|,|xℓ−1|) = (|xℓ|,|Uℓ+1|) = 1, we ha e
φ(x1, . . . , xℓ−1, xℓHℓ, Hℓ+1, . . . , H )⊆φ∗(X1, . . . , Xℓ−1, xℓUℓ, Uℓ+1, . . . , U ), and i
is hen equal o he i ial subg oup.
Lemma 4.26(i) now gi es φ(x1, . . . , xℓ−1, Hℓ, . . . , H ) = 1, and he e o e we ha e
φ∗(x1, . . . , xℓ−1, uℓ, . . . , u ) = 1.
Lemma 4.29. Le Gi, ℓ, Xi, Uibe as in Lemma 4.28.
87
(i) Fo i∈ {ℓ, . . . , }, suppose ha ei he |π(Gi)|=∞, in which case we w i e
Yi=Gi, o |π(Gi)|= 1, in which case we w i e Yi=Ui. Assume mo eo e
ha i π(Gi) = {p}consis s o a single p ime, hen p /∈π(Gi−1)∪π(Gi+1).
Suppose we also ha e ha φ∗(X1, . . . , Xℓ−1, xℓUℓ, . . . , x U ) = 1 o some
xℓ∈Gℓsuch ha (|xℓ|,|xℓ−1|)=1 o e e y xℓ−1∈Xℓ−1. Then, he e
exis s a ini e se o p imes σsuch ha φ∗(X1, . . . , Xℓ−1, Yℓ, . . . , Y )⊆Nσ
(c . De ini ion 4.27).
(ii) Suppose ha we ix xi∈Gi,i=ℓ, . . . , , such ha (|xi|,|xi+1|) = 1
o all i∈ {ℓ, . . . , −1}and (|xℓ|,|xℓ−1|) = 1 o all xℓ−1∈Xℓ−1. I
he se φ∗(X1, . . . , Xℓ−1, xℓUℓ, . . . , x U )is emp y, hen we also ha e ha
φ∗(X1, . . . , Xℓ−1, Gℓ, . . . , G ) = ∅.
P oo . (i) W i e L={ℓ, . . . , }, and o i∈L, de ine
σi=
π(Gi/Ui)i |π(Gi)|=∞,
π(Gi)i |π(Gi)|= 1.
Le σ=σℓ∪ · · · ∪ σ . Up o changing he ep esen a i e, we can assume
ha e e y xiis a good ep esen a i e o xiUi, and in pa icula ha hey a e
all σ-elemen s by Lemma 4.24. Fu he mo e, since φ∗
L(xiUi;Xi)6=∅and
π(xj)⊆π(xjuj) o e e y uj∈Uj, i ollows ha (|xi|,|xi+1|) = 1 o all
i∈ {ℓ, . . . , −1}.
Fo i∈Lwi h |π(Gi)|=∞, le Vibe he Hall σ′-subg oup o Gi, and o
i∈Lwi h |π(G)|= 1, se Vi=Ui(no ice ha Vi≤Uii |π(Gi)|=∞).
We wan o apply Lemma 4.28 −ℓ+ 1 imes, i s o he index , hen
dec easing un il we each he index ℓ, wi h he Vi aking he ole o he Ui.
Say we a e applying i o he index ℓ≤j≤ and le us check ha he
wo cop imali y condi ions o Lemma 4.28 a e sa is ied. We i s check he
hypo hesis (|xj|,|Vj+1|)=1. I |π(Gj+1)|=∞, hen π(Vj)⊆σ′and he
hypo hesis is sa is ied. I π(Gj+1) = {p}, hen p /∈π(Gj)and in pa icula
p /∈π(xj). As o he o he condi ion, i j=ℓ, i is simply one o he
hypo heses o he lemma. I ℓ+ 1 ≤j≤ , we ha e ha (|xj|,|xj−1|) = 1
and (|xj|,| j−1|) = 1 o all j−1∈Vj−1, ei he because Vj−1is a σ′-subg oup
i |π(Gj−1)|=∞o by hypo hesis i |π(Gj−1)|= 1.
A he end o his p ocess we ob ain φ∗
L(Vi;Xi) = 1. Now, i |π(Gi)|= 1,
hen Yi=Ui=Vi. I |π(Gi)|=∞, w i ing Hj o he Hall σ-subg oup o
88
Gj, hen φ∗(X1, . . . , Xℓ, Gℓ+1, . . . , Gj−1, Hj, Gj+1, . . . , G )⊆Nσby de ini ion
and by Lemma 4.22(i) we ob ain ha φ∗
L(Yi;Xi)⊆Nσ.
(ii) I φ∗(X1, . . . , Xℓ−1, xℓUℓ, . . . , x U ) = ∅ hen in pa icula we ha e ha
φ∗(X1, . . . , Xℓ−1, xℓ, . . . , x ) = ∅. The only way o his o happen is ha
he e exis s an index j∈ {1, . . . , l −2}such ha (|xj|,|xj+1|)6= 1 o all
xj∈Xj,xj+1 ∈Xj+1, and he lemma ollows.
The ollowing lemma is he ocal poin o he p oo o P oposi ion 4.33, as i will
allow us o unnel some alues o ce ain cop ime commu a o s in o an accu a ely
chosen subg oup.
Lemma 4.30. Le G1, . . . , G be p onilpo en subg oups o a p o ini e g oup G
such ha Gj≤NG(Gi) o all j≤i, and |δ∗
−1{G}| <2ℵ0. Then, he e exis a
ini e se W⊆φ∗(Gi)and a ini e se σo p imes such ha φ∗(Gi)⊆NσhWiG.
As his is he mos echnical p oo , we will i s gi e an example o he p ocedu e
o a speci ic case o cla i y he main ideas.
Example 4.31. We es ic o he case = 2, so we a e s udying φ∗(G1, G2), in
he speci ic case when |π(G1)|= 1,|π(G2)|=∞and π(G1)∩π(G2) = ∅. No ice
ha o = 2 some easie easoning could lead o an analogous esul , bu we will
ollow he algo i hm benea h he p oo o Lemma 4.30 in o de o illus a e i .
By Lemma 4.25, o i∈ {1,2}, we ob ain Ui⊴oGiand xi∈Gisuch ha
φ∗(x1U1, x2U2) = {w}consis s o a single alue. Se W={w}, we will wo k in
G/hWiGand assume w= 1. We ecall ha by Rema k 4.16, we can always e ine
an open no mal subg oup U2⊴G2wi h ano he no mal open subg oup which is
no malized by G1 oo, so we will always assume ha G1≤NG(U2).
Lemma 4.29 (wi h ℓ= 1) gi es a se σ(∅)o p imes such ha φ∗(U1, G2)⊆
Nσ(∅). We can ac o ou his subg oup and assume φ∗(U1, G2) = 1. Fix now
a se S={s1= 1, . . . , sm}o cose ep esen a i es o U1in G1. As 1∈Sand
π(G1) = 1, e e y elemen o Sis a good ep esen a i e o U1.
Se now V0=G2. Fo e e y ℓ∈ {1, . . . , m}, i φ∗(sℓ, Vℓ−1) = ∅, hen se
Vℓ=Vℓ−1, o he wise Lemma 4.25 gi es a cose Vℓ⊆Vℓ−1, such ha φ∗(sℓ, Vℓ) = 1.
No ice ha each Vℓis a cose o an open subg oup o G2. Repea ing his p ocedu e
m imes we ge Vm=gV o V⊴oG2,g∈G2such ha φ∗(sℓ, gV )is ei he emp y
o consis s o he i ial elemen o e e y ℓ= 1, . . . , m. No ice ha , being 1∈S,
he se φ∗(S, gV )is non-emp y.
89
Applying now Lemma 4.29, his ime wi h ℓ= 2, we can ob ain a ini e se
o p imes σsa is ying φ∗(S, G2)⊆Nσ. Now, i we wo k in G/Nσ, we can apply
Lemma 4.22 and ob ain ha φ∗(sℓU1, G2)is ei he emp y o i ial o e e y
ℓ∈ {1, . . . , m}. As Swas a se o cose ep esen a i es o U1in G1, we ha e ha
φ∗(G1, G2) = 1. Since he beginning o he p oo , we ha e ac o ed ou he no mal
subg oups hWiGand Nσ(∅)∪σ, se ling Lemma 4.30 in ou case.
O e all wi h se e al subg oups G1, . . . G some addi ional s eps migh be neces-
sa y, bu his case exempli ies he main ideas o he p oo .
P oo o Lemma 4.30. Le
I={i∈ {1, . . . , } | |π(Gi)|=∞}.
I suffices o p o e he heo em in he case when |π(Gi)|= 1 o all Giwi h i /∈ I.
The gene al case, whe e each Gi,i /∈ I, is he p oduc o i s Sylow subg oups
ollows by applying Lemma 4.22.
Fo i /∈ I, le pibe a p ime such ha π(Gi) = {pi}. Then we ha e
φ∗(Gi) = φ∗(G1, . . . , Gi−2, Hi−1, Gi, Hi+1, Gi+2, . . . , G ),
whe e Hi−1and Hi+1 a e he Hall p′
i-subg oups o Gi−1and Gi+1, espec i ely.
We can he e o e assume, again by Lemma 4.22(i), ha o all i /∈ I we ha e
pi/∈π(Gi−1)∪π(Gi+1).(4.7)
We claim ha ha o e e y J⊆ {1, . . . , } I he e exis a ini e se WJ⊆
φ∗(Gi), a ini e se o p imes σ(J)and subg oups UJ
i⊴oGiwi h i /∈ I ∪ Jsuch
ha φ∗
I∪J(Gi;UJ
i)⊆Nσ(J)hWJiG.
We p oceed by induc ion on |J|. Assume i s J=∅. By Lemma 4.25, o
e e y i∈ {1, . . . , } he e exis elemen s xi∈Giand subg oups U∅
i⊴oGisuch
ha φ∗(xiU∅
i) = {w∅} o a sui able w∅∈G. Mo eo e , by Rema k 4.16, we may
assume ha Gj≤NG(U∅
i) o e e y j≤i. Hence, Lemma 4.29 p oduces a ini e
se σ(∅)o p imes such ha φ∗
I(Gi;U∅
i)⊆Nσ(∅)hw∅iG, so he claim ollows o
|J|= 0.
Assume now ha |J| ≥ 1and ha o e e y J−(J he e exis a ini e se
WJ−⊆φ∗(Gi), a ini e se o p imes σ(J−)and subg oups UJ−
i⊴oGi,i /∈
I ∪ J−, such ha φ∗
I∪J−(Gi;UJ−
i)⊆Nσ(J−)hWJ−iG. Fo con enience, we also se
UJ−
i=Gii i∈J−, so ha UJ−
iis de ined o all i /∈ I. Le WJ=SJ−WJ−,
ρ=SJ−σ(J−)and Vi=TJ−UJ−
i o all i /∈ I, so ha , by (4.4), we ha e
90
φ∗
I∪J−(Gi;Vi)⊆NρhWJiG o e e y J−(J. Fu he mo e, by ac o ing ou
NρhWJiG, we may assume ha
φ∗
I∪J−(Gi;Vi) = 1 (4.8)
o e e y J−(J. Mo eo e , aking in o accoun Rema k 4.16 we may u he
assume ha Viis in a ian unde he conjugacy ac ion o Gj o e e y j≤i.
W i e J={j1, . . . , jn}wi h j1<· · · < jn, and o e e y i∈J, ix a se Sio
cose ep esen a i es o Viin Gicon aining he iden i y. W i e
Sj1× · · · × Sjn={s1, . . . , sm}
wi h sℓ= (sℓ,j1, . . . , sℓ,jn) o ℓ∈ {1, . . . , m}. Deno e Vi=Gi o i∈ I. Since
1∈Si o e e y i, we ha e φ∗
J(Si;Vi)6=∅, so applying Lemma 4.25 we ob ain
elemen s xi∈Viand subg oups Ui⊴oVisuch ha φ∗
J(Si;xiUi) akes a single
alue. Ac ually, since 1 = φ∗
J(1; xiUi)⊆φ∗
J(Si;xiUi), we ha e φ∗
J(Si;xiUi)=1.
Thus, o e e y ℓ∈ {1, . . . , m}, we ei he ha e
φ∗
J(sℓ,i;xiUi) = ∅o φ∗
J(sℓ,i;xiUi) = 1.(4.9)
We may assume xi o be a good ep esen a i e o he cose xiUiand he e o e, i
Jdoes no con ain nei he ino i+ 1, hen (|xi|,|xi+1|)=1. Also, by Rema k
4.16 we may u he assume ha Uiis in a ian unde he conjugacy ac ion o Gj
o e e y j≤i.
Le J0=∅, and o ∈ {1, . . . , n}, le J ={j1, . . . , j }. We also w i e j0= 0
o con enience. We will show ha o e e y ∈ {0, . . . , n}, he e exis s a ini e
se o p imes τ( )such ha φ∗
J (sℓ,i;Y( )
i)⊆Nτ( ) o e e y ℓ∈ {1, . . . , m}, whe e
Y( )
i=
Gii i≥j , i ∈ I ∪ J,
Uii i > j , i 6∈ I ∪ J,
xiUii i < j .
No ice ha igh now we a e no using Y( )
j , bu i will be con enien o ha e
i de ined o la e . We a gue by e e se induc ion on ∈ {0, . . . , n}; assume i s
=n. Since j 6∈ I, we deduce om (4.7) ha (|sℓ,j |,|Gj −1|) = (|sℓ,j |,|xj +1|) =
1. Thus, o all ℓ∈ {1, . . . , m}, we ob ain om (4.9) and Lemma 4.29 a ini e se
91
o p imes τ( , ℓ)such ha φ∗
J (sℓ,i;Y( )
i)⊆Nτ( ,ℓ). De ining τ( ) = Sm
ℓ=1 τ( , ℓ),
we ob ain φ∗
J (sℓ,i;Y( )
i)⊆Nτ( ) o e e y ℓ∈ {1, . . . , m}.
Hence, we assume ≤n−1. By induc ion, we know ha he e exis s a ini e
se o p imes τ( + 1) such ha
φ∗
J +1 (sℓ,i;Y( +1)
i)⊆Nτ( +1) (4.10)
o e e y ℓ∈ {1, . . . , m}.
The induc i e s ep will be di ided in wo phases. We will i s show ha
φ∗
J (sℓ,i;Y( +1)
i)⊆Nτ( +1) (meaning ha he only di e ence om (4.10) is posi ion
j +1). In o de o ob ain his, we ha e o subs i u e in he j +1- h posi ion i s
sℓ,j +1 , and hen sℓ,j +1 Uj +1 o all ℓ∈ {1, . . . , m}. We will hen conclude he
induc i e s ep by p o ing ha he e exis s a ini e se τ( )o p imes such ha
φ∗
J (sℓ,i;Y( )
i)⊆Nτ( ) o e e y ℓ∈ {1, . . . , m}.
We begin by no ing ha Y( +1)
i≤Vi o e e y i6∈ I ∪ Jand ha Uj +1 ≤Vj +1 ,
so (4.8) yields
φ∗
J (sℓ,i;e
Yi)⊆Nτ( +1),(4.11)
whe e e
Yi=Y( +1)
ii i6=j +1 and e
Yj +1 =Uj +1 . As we chose he se s o ep-
esen a i es Sjin such a way ha he iden i y is con ained in hem, o e e y
ℓ∈ {1, . . . , m}, ei he sℓ,j +1 is i ial o |π(sℓ,j +1 )|= 1, so in pa icula sℓ,j +1 is
a good ep esen a i e. Thus, by (4.10) and (4.11), we deduce om Lemma 4.22
ha φ∗
J (sℓ,i;Yi)⊆Nτ( +1), whe e Yi=Y( +1)
ii i6=j +1 and Yj +1 =sℓ,j +1 Uj +1 .
Since his holds o e e y ℓ∈ {1, . . . , m}, and since Gj +1 =Ss∈Sj +1 sUj +1 , we
ob ain φ∗
J (sℓ,i;Y( +1)
i)⊆Nτ( +1), as we wan ed.
Now using (4.7) and Lemma 4.29, we conclude exac ly as in he case =n ha
he e exis s a ini e se τ( )o p imes such ha φ∗
J (sℓ,i;Y( )
i)⊆Nτ( ) o e e y
ℓ∈ {1, . . . , m}.
This comple es he e e se induc ion on . In pa icula , o = 0, i ollows
ha φ∗
J(Gi;Ui)⊆Nτ(0), so his, in u n, concludes he induc i e s ep on |J|, and
he claim is p o ed.
Finally, aking Jin such a way ha I ∪ J={1, . . . , }, we ob ain a ini e se o
p imes σ(J)and a ini e se W⊆φ∗(Gi)such ha φ∗(G1, . . . , G )⊆Nσ(J)hWiG,
as desi ed.
Recall ha i Gis a p osol able g oup o Fi ing heigh k+ 1, he e exis some
p onilpo en subg oups U0, . . . , Uksa is ying P oposi ion 4.15.
We ema k ha φand φ∗we e de ined wi h a iables {xi|i= 1, . . . , } o
a gene ic posi i e in ege . Since we now wan o apply he p e ious esul s o
92
he subg oups U0, . . . , Uk, we will se =k+ 1 and we will w i e φ(Ui−1) o
φ(U0, . . . , Uk)and φ∗(Ui−1) o φ((U0× · · · × Uk)∩ C), whe e Cis de ined as in
(4.3).
Lemma 4.32. Le G=U0· · · Ukbe as in P oposi ion 4.15 wi h Uk=δ∗
k(G)
abelian, and assume |δ∗
k{G}| <2ℵ0. Le g∈φ∗(Ui−1). Then, he e exis s a ini e
no mal subg oup N⊴Gsuch ha g∈N.
P oo . W i e g= [x0, . . . , xk], whe e xj∈Uj o all jand (|xℓ|,|xℓ+1|)=1 o all
ℓ∈ {0, . . . , k−1}. By Lemma 4.9, [x0, . . . , xj]is a δ∗
j- alue o e e y j∈ {0, . . . , k}.
In pa icula x:= [x0, . . . , xk−1]is a δ∗
k−1- alue. Le Hbe he minimal Hall
subg oup o δ∗
k(G)con aining xk, so ha (|x|,|H|)=1, again by Lemma 4.9.
Since, again, [x, h]is a δ∗
k- alue o e e y h∈H, he se K:= {[x, h]|h∈H}
has less han 2ℵ0 alues, and, since His abelian and no mal in G, i ollows
ha Kis ac ually a closed subg oup o G. In pa icula , Kis ini e, so e e y
elemen o Khas ini e o de . Thus, we deduce om Lemma 3.17 ha he se
S:= S{kG|k∈K}is ini e, and he e o e N=hSiis ini e by Die zmann’s
Lemma (see Lemma 14.5.7 o [73]).
We a e now eady o p o e he s ong conciseness o δ∗
kin p osoluble g oups o
Fi ing heigh k+ 1.
P oposi ion 4.33. Le Gbe a p osoluble g oup o Fi ing heigh k+ 1. Assume
ha |δ∗
k{G}| <2ℵ0. Then δ∗
k(G)is ini e.
P oo . In iew o Lemma 4.12, we may assume ha δ∗
k(G)is abelian. Thus, we can
ake U0, . . . , Uk≤Gas in P oposi ion 4.15, so ha G=U0· · · Ukwi h Uk=δ∗
k(G)
abelian.
We claim ha o e e y amily o subg oups Gi−1≤Ui−1wi h i∈ {1, . . . , k+1}
such ha Gj≤NG(Gi) o j≤i, we ha e |φ∗(Gi−1)|<∞. We a gue by induc ion
on |I|, whe e
I={i∈ {1, . . . , k + 1} | |π(Gi−1)|=∞}.
I |I| = 0, hen Lemma 4.30 gi es he esul since o e e y ini e se W⊆
φ∗(Gi−1), he no mal subg oup hWiGis ini e by Lemma 4.32, and since, by
de ini ion, Nσ= 1 o e e y ini e se o p imes σ. Suppose hus |I| ≥ 1. Then,
Lemma 4.30 p oduces a ini e se o p imes σand a ini e se W⊆φ∗(Gi−1)such
ha φ∗(Gi−1)⊆NσhWiG. Obse e ha by induc ion, o e e y j∈ I, we ha e
|φ∗
{j}(Hi−1;Gi−1)|<∞, whe e Hi−1is he Hall σ-subg oup o Gi−1, and he e o e
Nσis ini e by Lemma 4.32. Again by Lemma 4.32, hWiGis also ini e, and he
claim ollows.
93
When s udying ac ions o g oups on ees, we o en need o es ic o minimal
in a ian sub ees, whose exis ence is gua an eed by he ollowing lemma, which
is P oposi ion 2.4.12 o [70].
Lemma 5.4. I Gis a p o-Cg oup ac ing on a p o-C ee Γ, hen he e exis s a
minimal G-in a ian p o-Csub ee ∆o Γ. I ∆con ains mo e han one e ex,
hen i is unique.
The ac ion o Gon a p o-C ee Γis i educible i Γhas no p ope G-in a ian
sub ees. F om now on, o e e y subse So a g oup Gac ing on a p o-C ee
T, we deno e by TS he minimal p o-Csub ee on which hSi ≤ Gac s. Simila ly
o he abs ac case, when elemen s commu e we can ob ain some addi ional
in o ma ion on hei ac ion.
Lemma 5.5. Le Gbe a p o-Cg oup ac ing ai h ully on a p o-C ee T.
1. Le g, h ∈Gbe such ha hno malises hgi, hen hlea es Tgin a ian and
in pa icula , i [g, h] = 1 hen Tg=Th.
2. Le S={g1, . . . , gk}be a se o elemen s such ha he ac ion o each gi,
i∈ {1, . . . , k}, is ellip ic. I [gi, gj] = 1 o e e y i, j ∈ {1, . . . , k}, hen
he e exis s a e ex o T ixed by he whole se S.
P oo . Pa (1) ollows immedia ely by obse ing ha h·(Tg) = Thgh−1⊆Tg.
We i s p o e pa (2) o wo elemen s g1, g2∈G. I bo h g1and g2a e ellip ic,
conside he sub ees Tg1and Tg2 ixed by g1and g2 espec i ely; by (1) we
ha e ha Tg1is a non-emp y p o-Csub ee in a ian unde he ac ion o g2. By
Co olla y 4.1.9 o [70], g1 ixes a e ex o Tg2, hence Tg2∩Tg1is no i ial.
Applying he case k= 2 o each pai , we ha e ha Tgi∩Tgj6=∅and giand gj
ix poin wise he in e sec ion o e e y i, j ∈ {1, . . . , k}so we can apply Lemma
5.3 o he se {Tg1, . . . , Tgk}and conclude ha Ti∈ITi6=∅and each gi ixes his
in e sec ion, hus (2) ollows.
Le ∆ = (V(∆), E(∆)) be a g aph. We se m∈∆i m∈V(∆) o m∈E(∆).
A ini e g aph o p o-Cg oups (G,∆) o e a ini e abs ac g aph ∆is a collec ion
o p o-Cg oups G(m) o each m∈∆, and con inuous monomo phisms ∂i:
G(e)−→ G(di(e)) o each edge e∈E(∆),i∈ {0,1}. We only wo k wi h ini e
g aphs o p o-Cg oups, in he sense ha he g aph ∆is ini e, bu i is possible
o de ine an analogous concep o g aphs o p o-Cg oups o e p o ini e g aphs ∆
(see Chap e 6 o [70]). A g aph o g oups is educed i edge g oups co esponding
o edges ha a e no loops a e p ope ly con ained in adjacen e ex g oups.
100
De ini ion 5.6 (P o-C undamen al g oup).Gi en a ini e g aph o p o-Cg oups
(G,∆), we de ine i s p o-C undamen al g oup G= Π1(G,∆) as ollows. Fix a
maximal sub ee Do ∆; hen Gis a p o-Cg oup, oge he wi h a collec ion o
con inuous homomo phisms
νm:G(m)−→ G(m∈∆)
and a con inuous map E(∆) −→ G, deno ed e7→ e(e∈E(∆)), such ha e= 1
i e∈E(D), and such ha
(νd0(e)∂0)(x) = e(νd1(e)∂1)(x) −1
e∀x∈ G(e), e ∈E(∆);
ha sa is ies he ollowing uni e sal p ope y:
whene e we ha e
• a p o-Cg oup H,
• a collec ion o con inuous homomo phisms βm:G(m)−→ H,(m∈∆),
• a map e7→ se(e∈E(∆)) wi h se= 1 i e∈E(D), and
•(βd0(e)∂0)(x) = se(βd1(e)∂1)(x)s−1
e∀x∈ G(e), e ∈E(∆),
hen he e exis s a unique con inuous homomo phism δ:G−→ Hwi h δ( e) = se
(e∈E(∆)) such ha o each m∈∆ he diag am
G
δ
G(m)
νm
<<
y
y
y
y
y
y
y
y
βm""
E
E
E
E
E
E
E
E
H
commu es.
I was p o en in [84] ha his de ini ion does no depend on he choice o he
maximal sub ee D, mo eo e he exis ence and uniqueness o his g oup is p o en
in P oposi ion 6.2.1 and Theo em 6.2.4 o [70].
One can cons uc he undamen al g oup o a g aph o p o-Cg oups by i e a ing
wo ope a ions, namely p o-Camalgama ed p oduc s and p o-CHNN ex ensions,
deno ed by G1qHG2and HNN(G1, H, ) espec i ely, and whe e G1and G2a e
p o-Cg oups, H≤G1, and :H→H′≤G1is an isomo phism. Bo h o hese
101
cons uc ions a e de ined by means o a uni e sal p ope y and can be ob ained
as a ce ain p o-Ccomple ion o he abs ac amalgama ed p oduc and HNN
ex ension o he co esponding g oups. We e e o Sec ions 9.2 and 9.4 o [72] o
he p ecise de ini ions and basic p ope ies.
I is impo an o ema k ha , con a y o he abs ac case, he ac o s G1and
G2( esp. he base g oup G1) do no necessa ily embed in o G1qHG2( esp.
HNN(G1, H, )). Whene e hey embed, he amalgama ed p oduc ( esp. HNN
ex ension) is said o be p ope . Some necessa y and sufficien condi ions o p o-
Camalgama ed p oduc s and HNN ex ensions o be p ope we e desc ibed in
Theo em 9.2.4 and P oposi ion 9.4.3 o [72]. We ema k ha p ope ness is assu ed
i i he amalgama ed subg oup His a i ual e ac o G1and G2(as G1and
G2would induce he ull p o-C opology on Hand he hypo hesis o Thm 9.2.4
in [72] hold in his case).
Abs ac Bass-Se e heo y ela es undamen al g oups o g aphs o g oups wi h
g oups ac ing on ees. Such a ela ion is ue o he p o-Ccase assuming ha
he ac ion on a p o-C ee is co ini e and no ue in gene al. Namely gi en a
undamen al g oup o a g aph o p o-Cg oups (G,∆), he e is a na u al p o-C ee
Ton which i ac s. The cons uc ion o his ee, called he s anda d p o-C ee,
is desc ibed in Chap e 6 o [70]. The con e se is ue only o he co ini e ac ion.
I he undamen al g oup o he g aph o p o-Cg oups is a p o-Camalgama ed
p oduc G=G1qHG2o a p o-CHNN ex ension G=HNN(G1, H, ), hen
each e ex s abilise G o a e ex is a conjuga e o G1o G2(o o G1i
G=HNN(G1, H, )) and each edge s abilise Geis a conjuga e o H.
Abs ac Bass-Se e heo y is ex emely use ul o s udying he s uc u e o
subg oups o undamen al g oups o g aphs o g oups. The same is ue o he
p o-C e sion o Bass-Se e heo y, and he main ool is Theo em 7.1.7 o [70].
We s a e he applica ions o hese esul s o he case when he g oup ac ing on
he p o-C ee is a p o-Camalgama ed p oduc o HNN ex ension. As usual, we
deno e by b
ZC=Qp∈π(C)Zp he p o-Ccomple ion o Z o any se o p imes π(C).
Theo em 5.7. Le Kbe a subg oup o a p ope ee amalgama ed p o-Cp oduc
G=G1qHG2o p o-Cg oups. Then one o he ollowing holds:
1. K≤gGig−1 o g∈Gand i∈ {1,2};
2. Khas a non-abelian ee p o-psubg oup P o a ce ain p∈π(C)such ha
P∩gGig−1= 1 o all g∈Gand i∈ {1,2};
3. he e exis s a subg oup H0⊴K(which is he ke nel o he ac ion o Kon
TK) ha is con ained in a conjuga e o Hand such ha K/H0is sol able
102
and isomo phic o a p ojec i e g oup Zσo Zρ(σ, ρ ⊆π(C)wi h σ∩ρ=∅)
o ZσoCn(wi h σ⊆π(C)and Cna ini e cyclic g oup). In he las case,
i can be a p o ini e F obenius g oup o , i Cn=C2and 2∈σ, an in ini e
dihed al p o-σg oup.
Theo em 5.8. Le Kbe a subg oup o a p ope p o-CHNN ex ension G=
HNN(G1, H, ). Then one o he ollowing holds:
1. K≤gG1g−1 o g∈G;
2. Khas a non-abelian ee p o-psubg oup P o p∈π(C)such ha P∩
gG1g−1= 1 o all g∈G;
3. he e exis s a subg oup H0⊴K(which is he ke nel o he ac ion o Kon
TK) ha is con ained in a conjuga e o Hand such ha K/H0is sol able
and isomo phic o a p ojec i e g oup Zσo Zρ(σ, ρ ⊆π(C)wi h σ∩ρ=∅)
o ZσoCn(wi h σ⊆π(C)and Cna ini e cyclic g oup). In he las case,
i can be a p o ini e F obenius g oup o , i Cn=C2and 2∈σ, an in ini e
dihed al p o-σg oup.
A use ul ema k is ha , in he hi d case o he p e ious heo ems, H/H0is
o sion ee i and only i i is isomo phic o Zσo Zρ. In his case, as his is a
p ojec i e g oup, we ha e ha H∼
=H0o(Zσo Zρ).
Finally, we eco d he ollowing obse a ion.
Lemma 5.9. Le G=G1qHG2be a p ope amalgama ed p o-Cp oduc o wo
p o-Cg oups G1and G2and le Tbe he s anda d p o-C ee associa ed wi h his
spli ing. Le g1, . . . , gkbe a sequence o ellip ic elemen s such ha [gi, gi+1]=1
o all i∈ {1, . . . , k −1}. Then he e a e some e ices 1, . . . , k∈V(T)(no
necessa ily dis inc ) such ha g1∈G 1and gi∈G i o each i∈[ i−1, i].
P oo . By Lemma 5.5 he e exis s a e ex is abilized by e e y pai o commu ing
elemen s gi, gi+1 o e e y i∈ {1, . . . , k −1}. De ine k o be any e ex s abilized
by gk. In his se ing, gis abilizes bo h i−1and i, hence i s abilizes he whole
sub ee [ i−1, i]by Co olla y 4.1.6 o [70].
5.2 Basics on p o-CRAAGs
The aim o his sec ion is o desc ibe basic p ope ies p o-CRAAGs. The abs ac
e sion o he de ini ions and esul s ha we discuss can be ound, o example,
in [18].
103
Le Γ=(V(Γ), E(Γ)) be an undi ec ed ini e g aph wi hou double edges o
loops, whe e V(Γ) and E(Γ) a e he se o e ices and edges espec i ely. A
subg aph ∆<Γis called ull i o all e∈Γwi h d0(e), d1(e)∈∆we ha e
ha e∈∆. No ice ha ull subg aphs a e uniquely de e mined by he subse o
e ices V(∆) o V(Γ).
De ini ion 5.10 (Righ -angled A in p o-Cg oups).The igh -angled A in p o-C
g oupp o-CRAAG (p o-CRAAG o sho ) GΓis he p o-Cg oup gi en by he
p o-Cp esen a ion
GΓ=hV(Γ)|[u, ] = 1 i and only i uand a e adjacen in Γi.
We ecall some s anda d e minology.
De ini ion 5.11 (Canonical Gene a o s).The gene a o s associa ed wi h he
e ices o Γa e called canonical gene a o s and, abusing he no a ion, we deno e
hem wi h he same le e as he co esponding e ex.
De ini ion 5.12 (S anda d subg oups).A subg oup o GΓis called a s anda d
subg oup i i is he subg oup gene a ed by a subse V′⊆V(Γ). I Γ = ∅, by
con en ion we se GΓ o be he i ial subg oup.
Abusing he no a ion, i S⊆V(Γ), we deno e by GS he s anda d subg oup
gene a ed by he ull subg aph gene a ed by S. We begin by s a ing some p op-
e ies o s anda d subg oups.
Lemma 5.13. Le GΓbe a p o-CRAAG. Then:
1. GΓis he p o-Ccomple ion o he abs ac RAAG G(Γ);
2. he s anda d subg oup gene a ed by a subse o e ices V′⊆V(Γ) is he
p o-CRAAG G∆gene a ed by he ull subg aph ∆⊆Γde e mined by V′;
3. he s anda d subg oups o GΓa e e ac s;
4. he in e sec ion o s anda d subg oups is a (possibly i ial) s anda d sub-
g oup.
P oo . Fo e e y g oup G, we deno e by b
Gi s p o-Ccomple ion.
1. Follows om he p o-Cp esen a ion (see De ini ion 5.10).
2. In he abs ac case, he subg oup o G(Γ) gene a ed by V′is exac ly G(∆),
see o example Co olla y 2.11 o [50]. As his subg oup is a e ac o G(Γ),
he p o-C opology o G(Γ) induces on i he ull p o-C opology, so he
p o-Csubg oup hV′i ≤ GΓis
G(∆), ha by (1) coincides wi h G∆.
104
3. The map p ∆:GΓ→G∆whose es ic ion o G∆is he iden i y and such
ha p ∆( )=1 o e e y ∈V(Γ) V′is su jec i e. Since by (2) G∆is a
subg oup o GΓ, we ha e ha p ∆is a e ac ion on o G∆.
4. Conside wo s anda d subg oups G∆, GΛo GΓ. By (3), a non- i ial ele-
men go GΓis in G∆∩GΛi and only i p ∆(p Λ(g)) = g, bu his com-
posi ion o maps co esponds exac ly o p ∆∩Λ(g), and he e o e G∆∩GΛ=
G∆∩Λ.
I ollows om he p o-C e sion o Theo em 9.2.4 o [72] ha , i His a e ac
o wo g oups G1and G2, hen a p o-CG1qHG2is a p ope p o-Camalgam-
a ed p oduc . Simila ly, i ollows om Theo em 9.4.3 ha p o-CHNN-ex ension
HNN(G1, H, )is p ope i His a e ac o G1. As s anda d subg oups o a
RAAG a e e ac s, we deduce he ollowing.
Co olla y 5.14. Le GΓbe a p o-CRAAG. I GΓis a p o-Camalgama ed p oduc
G1qHG2o a p o-CHNN ex ension HNN(G1, H, )wi h G1, G2, H, (H)s and-
a d subg oups o GΓ, hen he ee p oduc wi h amalgama ion o HNN ex ension
is p ope .
We now wan o de ine he no ion o suppo o an elemen , bu we i s begin
by p o ing ha his concep is well-de ined.
Lemma 5.15. Le GΓbe a p o-CRAAG and le g∈GΓ. Then he e exis s a
unique minimal s anda d subg oup con aining g. Mo eo e he e exis s an elemen
hin he conjugacy class o gwhose co esponding minimal s anda d subg oup is
con ained in each s anda d subg oup con aining conjuga es o g.
P oo . The unique minimal s anda d subg oup con aining gis he in e sec ion o
all he s anda d subg oups con aining i , and his in e sec ion is s ill a s anda d
subg oup by Lemma 5.13. Suppose now ha ∆1,∆2a e ull subg oups o Γsuch
ha g∈G∆1and g ∈G∆2 o ∈GΓ. We claim ha he e exis s s∈GΓsuch
ha gs∈G∆1∩∆2. Indeed le p ∆1be he e ac ion o GΓ o G∆1and de ine
s=p ∆1( ). Then gs=p ∆1(g )∈G∆1∩∆2. In o de o p o e he lemma i
suffices o apply his obse a ion o he la ice o ull subg aphs o Γcon aining a
conjuga e o g. No ice ha i g= 1 we ha e ha g∈G∅and by con en ion, he
s anda d subg oup gene a ed by he emp y se is he i ial g oup.
105
De ini ion 5.16 (Suppo o an elemen ).Le gbe an elemen o a (p o-C) RAAG
GΓ. The suppo α(g)o gis he se o canonical gene a o s o he unique minimal
s anda d subg oup o GΓcon aining g.
In iew o Lemma 5.15, in any conjugacy class he e exis s an elemen gsuch
ha α(g)⊆α(g ) o e e y ∈G, in his case we say ha g is an elemen o
minimal suppo among i s conjuga es.
De ini ion 5.17 (Links and s a s).Le gbe an elemen o a (p o-C) RAAG GΓ.
The link Link(g)o gis he se o e ices o Γ α(g) ha a e adjacen o each
o he e ices in α(g).
I is a canonical gene a o , we deno e by S a ( ) he ull subg aph gene a ed by
Link( )∪ .
Rema k 5.18. I ∈V(Γ), we can spli GΓas a p o-CHNN ex ension as
GΓ=HNN(GΓ { }, GLink( ), id)(5.2)
wi h s able le e , and by Co olla y 5.14 his is a p ope p o-CHNN ex ension.
I ollows ha i gis an elemen wi h minimal suppo among i s conjuga es and
∈α(g), hen Theo em 5.8 gua an ees ha i s ac ion on he s anda d p o-C ee
Tassocia ed wi h his spli ing is hype bolic.
Abs ac igh -angled A in g oups a e o sion- ee, bu he p o-Ccomple ion
o o sion- ee g oups is no always o sion- ee (e en he p o ini e comple ion as
shown in [54],[19]). Howe e , in he case o p o-CRAAGs his is ue.
Theo em 5.19. P o-CRAAGs a e o sion- ee p o ini e g oups.
P oo . A p o-CRAAG is he p o-Ccomple ion o he co esponding (abs ac )
RAAG. In [30], he au ho s p o ed ha abs ac RAAGs a e esidually ( ini ely
gene a ed o sion- ee nilpo en ), and hence he p o-Ccomple ion o a RAAG
embeds in a di ec p oduc o he p o-Ccomple ions o ini ely gene a ed o sion-
ee nilpo en g oups. By Theo em 4.7.10 o [72] he p o ini e comple ion b
No a
ini ely gene a ed o sion- ee nilpo en g oup Nis o sion- ee. Bu b
N=Qpb
Np
is he di ec p oduc o he p o-pcomple ions and he p o-Ccomple ion o Nis he
di ec p oduc Qp∈π(C)Np. Hence he p o-Ccomple ion o Nis o sion- ee.
5.3 Di ec p oduc decomposi ion o p o-CRAAGs
Ou goal is o show ha he di ec p oduc decomposi ion o a p o-CRAAG is
de e mined by he de ining g aph. Mo e p ecisely GΓ≃A1×A2, whe e A1and
A2a e non- i ial p o-Cg oups, i and only i Γis a join, see Theo em 5.22.
106
Lemma 5.20. Le GΓbe a p o-CRAAG and le g∈GΓbe an elemen wi h
minimal suppo among i s conjuga es. Then, he cen alise o gis con ained in
he s anda d subg oup gene a ed by Link(g)∪α(g). In pa icula , i g= is a
s anda d gene a o , hen CG( ) = GS a ( )=h i × GLink( ).
P oo . Suppose owa ds con adic ion ha he e is an elemen hcommu ing wi h
gwhose suppo is no con ained in Link(g)∪α(g). Then he e exis s ∈α(h)
such ha /∈Link(g)∪α(g). Deno ing by G0=GΓ { }and by A=GLink( )and
using Rema k 5.18, we ha e ha he g oup GΓspli s as a p ope HNN ex ension
o he o m
GΓ=HNN(G0, A, id)
whe e he ac ion by conjuga ion o on Ais i ial. No ice ha om he as-
sump ion on h, we ha e ha h /∈G0. We nex s udy he ac ion o gand hon he
s anda d p o-C ee Tassocia ed wi h his spli ing.
No ice ha g∈G0and so gis ellip ic. Howe e , gcanno belong o any edge
s abilise . Indeed, o he wise, he e would exis an elemen ∈GΓsuch ha g ∈A
and in his case, since ghas by assump ion minimal suppo , i would ollow om
Lemma 5.15 ha α(g)⊆α(g )⊆Link( )and so ∈Link(g)con adic ing he
choice o . Since gcanno be in any edge s abilise , we conclude ha gonly
ixes he e ex s abilised by G0, i.e. Tg={ }. F om Lemma 5.5 (1), hhas o
lea e Tg={ }in a ian and, in pa icula , h ixes . Then hbelongs o G0, a
con adic ion.
Lemma 5.21. Suppose a p o-CRAAG G=GΓdecomposes as a di ec p oduc
GΓ=A1×A2o non- i ial g oups. Then o each canonical gene a o ∈Γ, a
leas one ac o Aiis con ained in GS a ( ).
P oo . Le be a canonical gene a o . Since α( ) = { }, by Lemma 5.20 we ha e
ha CG( ) = GS a ( )=h i × GLink( ).
Suppose ha =a1·a2whe e ai∈Ai,i= 1,2. Since CG( ) = CA1(a1)×CA2(a2)
and ai∈CAi( ), om he desc ip ion o he cen alise CG( ), we deduce ha
ai= eia′
i o ei∈Zπ(C)and a′
i∈GLink( ). Since =a1·a2, we ha e ha ei6= 0
o ei he i= 1 o i= 2; wi hou loss o gene ali y assume e16= 0. Le be
an elemen such ha −1a1 has minimal suppo among i s conjuga es, we can
assume ∈A1because A2⊆CG(a1). Applying Lemma 5.20 we ha e
A2 −1=A2⊆CG(a1) = CG( −1a1 ) −1⊆ Gα( −1a1 )×GLink( −1a1 ) −1.
No ice ha by Lemma 5.15 α( −1a1 )⊆α(a1)⊆S a ( )and, since ∈α( −1a1 ),
he de ini ion o link implies ha Link( −1a1 )⊆S a ( ). O e all, we conclude
ha A2⊆GS a ( ).
107
We a e now eady o ully cha ac e ize when a p o-CRAAG spli s as a di ec
p oduc . We ecall ha a g aph is a join i and only i he e is a non-emp y
subg aph ∆Γsuch ha o each ∈∆and each w∈Γ ∆, , w a e adjacen .
Theo em 5.22. Le GΓbe a p o-CRAAG. Then GΓhas a non- i ial di ec
p oduc decomposi ion i and only i Γis a join. In pa icula , each ac o in a
di ec p oduc decomposi ion o GΓis a s anda d subg oup.
P oo . The analogous esul o abs ac RAAGs is classical (see o example
Co olla y 2.15 in [50]). F om he abs ac esul and Lemma 5.13, i is s aigh -
o wa d ha whene e Γis a join, hen GΓspli s as a di ec p oduc .
We now wan o p o e he con e se implica ion. By Lemma 5.21 o each
canonical gene a o , a leas one among A1o A2is con ained in GS a ( ).
Le Γ1⊆V(Γ) be he se o canonical gene a o s such ha A1<S a ( )and
Γ2= Γ Γ1. Then, o each canonical gene a o ∈Γ2, since by de ini ion o Γ2
we ha e ha A16<S a ( ), by Lemma 5.21 again we conclude ha A2≤GS a ( ).
Fo i= 1,2de ine ∆i⊆Γsuch ha G∆i=T ∈ΓiGS a ( ); by Lemma 5.13
G∆iis a s anda d subg oup and by de ini ion i con ains Aiand each ∈Γiis
connec ed o each w∈∆i. In pa icula ∆ia e non-emp y g aphs. No ice ha
i he e is a canonical gene a o w∈∆1∩∆2, hen wis by de ini ion in he s a
o each e ex in Γiand so Γ1,Γ2<S a (w). Hence such a canonical gene a o
w∈∆1∩∆2would be cen al and Γwould decompose as a join. Fo his eason
we can assume ha G∆1and G∆2a e disjoin and since A1and A2gene a e G,
so do G∆1and G∆2.
Hence, we can decompose V(Γ) as he disjoin union o he (possibly emp y)
se s Γ2∩∆1,Γ1∩∆2and Λ = (Γ1∩∆1)∪(Γ2∩∆2).
Since ∆iis non-emp y o i= 1,2, hen ei he Λ6=∅o Γ2∩∆1and Γ1∩∆2
a e non-emp y. I a leas wo o he se s a e non-emp y, hen hey de ine a join,
because each e ex in a se is connec ed o each e ex in he o he se , because
each elemen in Γiis connec ed o each elemen in ∆i o i= 1,2.
We a e le o conside he case when only Λis non-emp y, so ha Λ = V(Γ).
In his case, each e ex in Γiis in ∆i oo and in pa icula hey a e connec ed
o each o he . I ollows ha Γi∩∆i= Γi= ∆iis a comple e g aph o i= 1,2.
Since Ai≤G∆iand G∆iis abelian, so is Ai. Hence G=A1×A2is abelian and
Γis a comple e g aph and a join.
These esul s a e in line wi h o he p ope ies o p o-CRAAGs ha can be
ecognized om he abs ac g aph. Fo example, abs ac RAAGs spli as a
108
ee p oduc i and only i he unde lying g aph is disconnec ed, and Wilkes and
K opholle p o ed ha he same is ue o p o ini e RAAGs in [51]. Simila ly,
bo h abs ac and p o-pRAAGs a e cohe en i and only i he unde lying g aph
is cho dal, see [76].
5.4 Cen alise s and no malise s o elemen s
In his sec ion, we desc ibe explici ly he s uc u e o cen alise s o elemen s in
p o-CRAAGs, ob aining a desc ip ion simila o he one ha Baudisch p o ed o
abs ac RAAGs in [12]. In a ee p o-pg oup, cen alise s o elemen s a e cyclic.
Howe e , in he p o-Ccase, he si ua ion is subs an ially di e en as he cen alise
o an elemen does no need o be cyclic. Indeed, o example, he p ojec i e g oup
Z3o Z2, wi h he gene a o o Z2, say a, ac ing on Z3by in e sion, embeds in a
ee p o ini e g oup, so he cen alise o a2con ains his sol able p ojec i e g oup.
Theo em 5.23. Le G=GΓbe a p o-CRAAG and le g0∈G. Then he e is an
elemen gin he conjugacy class o g0such ha i s cen alise is o he o m
CG(g) = H1× · · · × Hs× hLink(g)i
whe e:
1. α(Hi), α(Hj),Link(g)a e all disjoin o i6=j;
2. Gα(g)=Gα(H1)× · · · × Gα(Hs);
3. Hia e p ojec i e p o-Cg oups;
4. i Gis p o-p,Hi=hhiiand g=hk1
1· · · hks
s, o some ki∈Zp.
P oo . We begin he p oo wi h some educ ions.
I gis i ial, hen V(Γ) = Link(g)and he esul holds i ially, so we u he
assume g6= 1.
Among he conjuga es o g0, we choose an elemen go minimal suppo among
i s conjuga es, so ha by Lemma 5.20 CG(g)is con ained in he s anda d subg oup
gene a ed by Link(g)∪α(g). Hence, we can assume ha V(Γ) = Link(g)∪α(g).
In his case, we ha e om Theo em 5.22 ha G=Gα(g)×GLink(g). Clea ly
GLink(g)≤CG(g), so i suffices s udying he cen alise in he s anda d subg oup
Gα(g)and hen
CG(g) = CGα(g)(g)×GLink(g).
We u he assume ha α(g) = V(Γ). I Gis decomposable as a di ec p oduc
GΓ=G1× · · · × Gs, hen g=g1× · · · × gs o gi∈Gi,i∈ {1, . . . , s}, and he
109
116
6
Abelian spli ings o RAAGs
In his sec ion we will s udy how a p o-CRAAG can spli as an amalgama ed
p oduc o HNN ex ension o e an abelian subg oup.
In [35], Hull and G o es p o ed ha an abs ac RAAG spli s o e an abelian
subg oup i and only i he unde lying g aph ei he has a sepa a ing comple e
g aph o i is disconnec ed. This esul ex ends a p e ious heo em o Clay [21],
who p o ed i in he case o cyclic spli ings. In he i s sec ion we p o e ha he
same condi ions a e necessa y and sufficien in o de o ha e abelian spli ings o
p o-CRAAGs. We also poin ou ha , i he unde lying g aph is connec ed, a
conjuga e o a s anda d subg oup is always con ained in he abelian amalgama ed
subg oup.
Desc ibing all he abelian spli ings o a g oup is in gene al difficul as some o
hem a e no compa ible wi h each o he . In any case, he e is a cons uc ion,
called he JSJ decomposi ion, ha encodes all he “uni e sal” spli ings o a g oup
o e a chosen class o subg oups. In he second sec ion we gi e a desc ip ion o JSJ
decomposi ions in p o ini e g oups, which is ob ained ollowing he app oach o
Gui a del and Le i in [37]. In pa icula , we can de ine A-JSJ decomposi ions,
meaning ha we desc ibe all spli ings o a g oup when he amalgama ed sub-
g oups a e in he class o g oups A, and hen ela i e (A,H)-JSJ decomposi ions,
in he sense ha e e y subg oup in he class His ellip ic in he decomposi ion.
In he hi d sec ion we ob ain a (A,H)-JSJ decomposi ion o p o-CRAAGs
117
in he case ha Ais he class o abelian g oups and His he class o p ocyc-
lic subg oups gene a ed by a canonical gene a o . The p oo is cons uc i e, in
he sense ha i inhe i ely p o ides an algo i hm o ob ain he a o emen ioned
decomposi ion.
In he las sec ion we e ine he ela i e decomposi ion in o de o ob ain a
gene al A-JSJ decomposi ion. We conclude wi h an explici example showing he
algo i hm benea h he cons uc ion o hese decomposi ions.
6.1 Abelian spli ings o p o ini e RAAGs
The main goal o his sec ion is o desc ibe when and how a p o-CRAAGs spli s
o e a p o-Cabelian g oup. We begin wi h wo auxilia y lemmas.
Lemma 6.1. Le G=GΓbe a p o-CRAAG associa ed wi h a connec ed g aph
Γ. Suppose ha Gac s on a p o-C ee Twi hou a global ixed poin , and ha
all canonical gene a o s a e ellip ic. Then he e exis wo canonical gene a o s
, w ∈V(Γ) such ha ( , w)/∈E(Γ) and h , widoes no s abilize any e ex o T.
P oo . Le T be he sub ee o ixed poin s o a canonical gene a o . I , by
con adic ion, T ∩Tw6=∅ o each couple o canonical gene a o s , w ∈V(Γ),
by Lemma 5.3 he e is a poin con ained in T ∈V(Γ) T ixed by all he gene a o s
and so ixed by G, con adic ing he hypo hesis. This implies ha he e a e a
leas wo e ices , w ∈V(Γ) such ha h , widoes no s abilize any e ex o T.
No ice ha such e ices canno be adjacen by Lemma 5.5 (2).
Lemma 6.2. Le GΓbe a p o-CRAAG o e a connec ed g aph Γac ing on a p o-C
ee Twi h abelian edge s abilise s. Suppose ha a canonical gene a o ∈GΓis
hype bolic, hen:
1. S a ( )is a comple e g aph;
2. ei he V(Γ) = S a ( )o he se
S:= {u∈Link( )|S a (u)is no a comple e g aph}
sepa a es S a ( ) Sand Γ S a ( );
3. he s anda d subg oup gene a ed by Ss abilizes an edge.
P oo . 1. I he e exis s a single e ex adjacen o , hen he esul holds.
Suppose hen ha he e exis wo dis inc e ices w1, w2∈Link( ).
Fo each canonical gene a o wcommu ing wi h we can es ic o he
118
minimal sub ee T⟨ ,w⟩on which he abelian subg oup h , wiac s. By The-
o ems 5.7 and 5.8, he g oup h , wiis a p ocyclic ex ension o he ke nel
o his ac ion and since h , wiis abelian o ank 2, he e exis s an elemen
g=ab wi h a∈ h i ≤ Gand b∈ hwi ≤ Gwi h b6= 1 (as is hype bolic)
in he ke nel o he ac ion, i.e. g ixes poin wise he minimal sub ee T⟨ ,w⟩.
Pick now wo elemen s gi=aibiwi h i∈ {1,2} o a1, a2∈ h i,b1∈ hw1i,
b2∈ hw2isuch ha b1, b2a e no i ial and such ha g1, g2s abilize poin -
wise T . By hypo hesis g1, g2a e con ained in he abelian s abilise s o he
edges o T . Le K=hg1, g2iand le be he e ac ion o Gon o he
s anda d subg oup gene a ed by w1, w2. The image (K)≤G{w1,w2}is an
abelian subg oup ha con ains b1and b2. The elemen b1is in he cen al-
ise o b2and hey a e bo h wi h minimal suppo among hei conjuga es,
so applying Lemma 5.20 his can happen only i w1∈Link(b2) = Link(w2),
so w1, w2a e adjacen and S a ( )is a comple e g aph.
2. Suppose V(Γ) 6= S a ( ), as Γis connec ed we ha e ha
S={u∈Link( )|S a (u)is no a comple e g aph}
is non-emp y. I is immedia e o see ha Ssepa a es he subg aphs gene -
a ed by S a ( ) Sand Γ S a ( )because, as Link( )is a comple e g aph
by (1), each e ex in S a ( ) Sis connec ed only o e ices in S a ( ).
3. By Lemma 5.5(1), each e ex o S ixes he sub ee T , which con ains a
leas an edge because is hype bolic. By (1), he ac ion on To any elemen
o Sis ellip ic, and hence T is ixed poin wise by S.
We a e now eady o p o e he main heo em o his sec ion.
Theo em 6.3. Le G=GΓbe a p o-CRAAG associa ed wi h a connec ed g aph
Γ. Then Gac s on a p o-C ee wi h abelian edge s abilise s wi hou a global ixed
poin i and only i ei he Γis a comple e g aph o Γhas a disconnec ing comple e
g aph.
In he second case, he e exis s a disconnec ing comple e g aph whose s anda d
subg oup is con ained in one edge s abilise o T.
P oo . The case when Γis a comple e g aph is clea : indeed, deno ing by π=
π(C), he p o-CRAAG GΓis isomo phic o Zn
π, ha spli s as an HNN-ex ension
119
HNN(Zn−1
π,Zn−1
π, id). Simila ly, i he e is a comple e g aph K ha disconnec s
Γ, i.e. Γ K= Γ1∪Γ2wi h Γ1,Γ2disjoin subg aphs, hen Gspli s as
GΓ=GΓ1∪KqGKGΓ2∪K
and so Gac s on he s anda d p o-C ee associa ed wi h his spli ing.
Suppose now ha Gac s on a p o-C ee Twi h abelian edge s abilise s. I he e
exis s a hype bolic canonical gene a o o G, by Lemma 6.2 we know ha ei he
Γis comple e o he se S:= {u∈Link( )|S a (u)is no a comple e g aph}is a
disconnec ing comple e g aph con ained in an edge s abilise . We a e le o he
case when each canonical gene a o o Gis ellip ic.
By Lemma 6.1 he e exis wo e ices , w ∈V(Γ) such ha no e ex o Tis
s abilized by bo h and w. As each canonical gene a o ac s ellip ically, le , w
be wo e ices o Ts abilized by and w espec i ely. Le S= [ , w]be he
geodesic be ween hese wo e ices in T.
By Lemma 6.1 Scon ains a leas one edge, and mo eo e he e exis s a leas one
edge o S ha is no s abilised ei he by o by w, as by collapsing he sub ees
S∩T and S∩Tw o a poin (no icing ha we chose , w such ha T ∩Tw=∅),
we would o he wise ha e S o be disconnec ed.De ine Kas a maximal (by he
numbe o e ices con ained) comple e subg aph o Γcon ained in an edge g oup
o S ha is no s abilized by ei he o w, say e∈E(T). I Kis emp y, le ebe
any edge o S, no s abilized by o w. I is impo an o no ice ha e en i S
migh con ain in ini ely many edges sa is ying he p ope ies, Γis ini e hence K
is well de ined. We claim ha Kis a comple e g aph o Γ ha disconnec s he
e ices and w.
Suppose by con adic ion ha i is no , hen we could ind a ini e pa h p=
( , u1, . . . , uk, w)in Γ, such ha no e ex o pis con ained in K. By Lemma
5.9 he e exis some e ices 1, . . . , k+2 such ha s abilizes 1,uis abilizes
he geodesic Si= [ i, i+1] o i= 1, . . . , k, and ws abilizes [ k+1, k+2]. Se
0= and Tk+3 = w. In his se ing, s abilizes S0= [ 0, 1]and ws abilizes
Sk+1 = [ k+1, w]. Fu he mo e, he union S′=Si∈{0,...,k+1}Sio he Siis a p o-p
ee ha con ains and w, hence i con ains he whole [ , w]. In pa icula ,
S′con ains e, so e∈[ j, j+1] o some j∈ {0, . . . , k + 3}. By he choice o e,
i canno be s abilized by o w, so he e exis s a e ex ujsuch ha uj∈Ge.
Now Geis an abelian p o-Csubg oup o G ha con ains ujand each e ex o K,
bu by Theo em 5.23 his is only possible i ujis adjacen o e e y e ex o K.
By maximali y o K,uj∈K, bu his con adic s he ac ha no elemen o he
pa h pis con ained in K. I we assumed K o be emp y, we ha e anyway p o ed
ha he e is a e ex ujcon ained in an edge s abilise o Tcon adic ing ha
120
K=∅.
This p o es ha he g aph Kis a disconnec ing comple e g aph con ained in an
edge s abilise , as equi ed.
6.2 JSJ decomposi ions
P e equisi es
In he p e ious sec ion, we ha e cha ac e ised when a p o-CRAAGs admi s a
spli ing o e an abelian subg oup. Ou nex goal is o desc ibe all he spli -
ings o hese g oups o e abelian subg oups. In he abs ac case, he abelian
spli ings o a ini ely gene a ed g oup a e encoded in a cons uc ion called he
JSJ decomposi ion o a g oup. We de elop his heo y ollowing he app oach o
Gui a del and Le i in [37]. We show ha i can be na u ally ex ended o he
p o-Cwo ld; o addi ional esul s and al e na i e de ini ions on he heo y o JSJ
decomposi ions see he e e ences in [37].
De ini ion 6.4 (A- ees).Fo each class o p o-Cg oups Aclosed o subg oups
and conjuga ion, we de ine an A- ee (T, G)as a p o-C ee Twi h an ac ion o a
p o-Cg oup Gsuch ha each edge s abilise is a g oup in he class A.
We o en deno e he A- ee as T a he han (T, G)whene e he p o-Cg oup
Gac ing on i is clea by he con ex and we will say ha an A- ee (T, G)is
i ial i Tconsis s o a single e ex s abilized by he whole G.
We say ha a subg oup Ho a p o-Cg oup Gis uni e sally ellip ic ( o ac ions
o e A- ees) i he ac ion o His ellip ic o e any A- ee (T, G)on which Gac s.
De ini ion 6.5 (JSJ decomposi ions).
• An A- ee (T, G)is uni e sally ellip ic i i s edge s abilise s Ge≤Ga e
uni e sally ellip ic o ac ions on A- ees.
• An A- ee (T, G)domina es ano he A- ee (T′, G)i he same g oup Gac s
on bo h o hem and he ac ion o e ex s abilise s G , ∈Tis ellip ic on
T′ oo.
• Two A- ees (T, G)and (T′, G)a e equi alen i he same p o-Cg oup G
ac s on bo h o hem and hey domina e each o he . An equi alence class
o A- ees o his ela ion is said o be a de o ma ion space.
• The de o ma ion space o he A- ees ha a e uni e sally ellip ic and ha
domina e any o he uni e sally ellip ic A- ee on which Gac s is he JSJ
de o ma ion space and i s elemen s a e called he JSJ ee decomposi ions.
121
No ice ha he de o ma ion space is unique, bu he e migh be many non-
isomo phic ee decomposi ions o a p o-Cg oup G.
De ini ion 6.6 (Rigid and lexible e ices).A e ex o a JSJ- ee is said o
be igid i i is uni e sally ellip ic o he ac ion on any A- ee (e en i he ee is
no uni e sally ellip ic) and lexible o he wise.
No ice ha i all e ex g oups o an A- ee a e igid, hen he A- ee is a JSJ
ee, bu he con e se is no ue, as he ollowing example shows.
Example 6.7. I G∼
=Zn
ρ o n≥2,ρan a bi a y se o p imes, he abelian JSJ
decomposi ion is i ial.
We claim ha o each elemen g∈Gwe can p oduce an A- ee (T, G)such
ha he ac ion o gon Tis hype bolic. Conside a maximal p ocyclic subg oup
Ccon aining g∈G. Any gene a o o a maximal p ocyclic g oup can be pa
o a basis o Zn
ρ, so we can pick a complemen B∼
=Zn−1
ρo Cin Gand w i e
G=HNN(B, B, id)wi h a gene a o o Cas he s able le e . The s anda d
p o-C ee associa ed wi h his p o-CHNN ex ension is a e ex wi h a single
loop and gis hype bolic by cons uc ion. This p o es ha no edge g oup can be
uni e sally ellip ic, hence he e exis s a single uni e sally ellip ic A- ee (T, G)on
which Gac s, which is a ee Twi h a single poin . This is he JSJ decomposi ion
o G, which has a single lexible e ex.
Some imes is con enien o s udy ela i e JSJ decomposi ions, which a e de ined
as ollows.
De ini ion 6.8 (Rela i e JSJ Decomposi ions).Le Hbe an a bi a y amily o
subg oups o a p o-Cg oup G. An A- ee (T, G)is an (A,H)- ee i all he sub-
g oups in he class Ha e ellip ic. An (A,H)- ee is an (A,H)-JSJ decomposi ion
i i is uni e sally ellip ic o ac ions on (A,H)- ees and i domina es e e y o he
uni e sally ellip ic (A,H)- ee.
We now u n ou a en ion o he s udy o he JSJ-decomposi ion o a p o-C
RAAG o e abelian g oups. Le G=GΓbe a p o-CRAAG o e a ini e connec ed
g aph Γ. F om he e on, we assume A o be he class o abelian p o-Csubg oups
o Gand H o be he class o p ocyclic g oups gene a ed by canonical gene a o s
o G.
We i s cons uc by induc ion a decomposi ion o Go e abelian subg oups
ela i e o H, and p o e ha i is ac ually an (A,H)-JSJ decomposi ion. We hen
e ine his decomposi ion in o de o ob ain he A-JSJ decomposi ion o G.
As we a e in e es ed in spli ings o e s anda d subg oups o disconnec ing
comple e g aphs, we i s need some basic p ope ies o spli ings o his ype.
122
Lemma 6.9. Le GΓbe a p o-CRAAG o e a ini e connec ed g aph Γand K≤Γ
be a comple e subg aph o Γ.
1. I all cyclic subg oups gene a ed by canonical gene a o s in Ka e uni e sally
ellip ic o hei ac ion on A- ees, hen he whole s anda d subg oup GKis
uni e sally ellip ic o i s ac ion on A- ees.
2. I Kis a minimal disconnec ing comple e g aph (in he sense ha no p ope
subse o Kis a disconnec ing comple e g aph), hen he s anda d subg oup
GKis uni e sally ellip ic o i s ac ion on A- ees.
3. I S a ( )is a comple e g aph o ∈V(Γ), hen he e exis s an A- ee on
which he ac ion o is hype bolic.
P oo .
1. Since by assump ion Γis connec ed, his ollows as a consequence o Lemma
5.5 (2).
2. Assume ha Gac s on an A- ee (T, G)and suppose ha he e exis s a
leas one hype bolic canonical gene a o ∈V(K). By Lemma 6.2(1), we
ha e ha S a ( )is a comple e g aph. Since a comple e g aph does no
ha e any disconnec ing subg aphs, i ollows ha Γ6= S a ( ). F om he
minimali y o he disconnec ing comple e g aph K, we ha e ha he ull
subg aph Γ′gene a ed by (V(Γ) V(K)) ∪ { }is connec ed and is a
disconnec ing e ex o Γ′. In pa icula , he e a e wo e ices w1, w2∈
V(Γ′) ha a e adjacen o bu lie in di e en connec ed componen s o
Γ K. This con adic s he ac ha S a ( )is a comple e g aph. Hence
each canonical gene a o o Kmus be ellip ic and by (1) he whole Kis
ellip ic.
3. I suffices o no ice ha he s anda d p o-C ee associa ed wi h he spli ing
(5.2) has abelian edge s abilise s because Link( )is a comple e g aph.
We eco d he ollowing g aph heo e ical obse a ion.
Lemma 6.10 (Disconnec ing g aphs o componen s).Le Γbe a ini e connec ed
simplicial g aph. Le Kbe a disconnec ing comple e subg aph o Γand le {Γi|
i∈ {1, . . . , m}} be he connec ed componen s o Γ K.
I K′is a disconnec ing subg aph o Γj∪K o some j∈ {1, . . . , m}, hen K′
is also a disconnec ing subg aph o Γ.
123
P oo . Suppose on he con a y ha Γ K′is connec ed. Since Kis by assump ion
a disconnec ing subg aph o Γ, i ollows ha Kis no con ained in K′and so
K K′is nonemp y. Since Γ K′is connec ed and Kis disconnec ing, o each
e ex in (Γj∪K) K′ he e is a e ex w( )in K K′such ha and
w( )a e connec ed by a pa h inside (Γj∪K) K′. As Kis comple e, he e
is an edge be ween any wo e ices in K. I ollows ha any pai o e ices
, ′∈(Γj∪K) K′a e connec ed by he pa h which is he composi ion o he
pa hs om o w( ), he edge (w( ), w( ′)) and he pa h om w( ′) o ′. Since
his pa h is in (Γj∪K) K′, we ha e ha (Γj∪K) K′is connec ed, de i ing
a con adic ion.
6.3 (A,H)-JSJ decomposi ion o p o-CRAAGs
We i s cons uc he ( ela i e) abelian JSJ decomposi ion o p o-CRAAGs unde
he assump ion ha all he subg oups in he class H={h i | ∈V(Γ)}o
p ocyclic subg oups gene a ed by canonical gene a o s a e ellip ic.
Theo em 6.11. Le G=GΓbe a p o-CRAAG associa ed wi h a connec ed
abs ac ini e g aph Γ.
The e is a (possibly i ial) decomposi ion o Gas a undamen al p o-Cg oup
o a educed ini e ee o p o-Cg oups (G∆,∆) wi h he ollowing p ope ies:
• e ex g oups o (G∆,∆) a e s anda d subg oups which a e ei he abelian o
hei unde lying g aph does no con ain any disconnec ing comple e subg aph;
•each edge g oup o (G∆,∆) is a s anda d subg oup associa ed wi h a discon-
nec ing comple e subg aph Keo Γand, mo eo e , Keis a minimal (wi h
espec o inclusion) disconnec ing comple e g aph o a subg aph Γ′o Γ.
Fu he mo e, he s anda d p o-C ee associa ed wi h his decomposi ion is an
(A,H)-JSJ ee decomposi ion (T∆, G)o G.
P oo . We p o e he s a emen s by induc ion on he numbe o gene a o s o he
p o-CRAAG.
Assume i s ha Γhas one e ex, i.e. G=Zπ(C). In his case, we conside
he decomposi ion as a undamen al g oup o a g aph o g oups o be i ial, so
∆is a poin and he associa ed g oup is Zπ(C). This decomposi ion sa is ies he
equi ed condi ions. Fu he mo e, since Gis a s anda d subg oup, by assump ion
i is ellip ic and so he (A,H)-JSJ decomposi ion o Gis i ial and ag ees wi h
he decomposi ion as a undamen al g oup o a g aph o g oups.
124
Assume ha we ha e al eady es ablished he decomposi ion o e e y p o-C
RAAG whose unde lying g aph has a mos n−1 e ices as a undamen al g oup
o a g aph o g oups and ha we ha e p o ed ha he (A,H)-JSJ decomposi ion
o Gis de e mined by he g oup decomposi ion as a undamen al g oup o a g aph
o p o-Cg oups sa is ying he p ope ies o he heo em.
Le now Γbe a connec ed g aph wi h n e ices, n≥2. Suppose i s ha Γ
does no ha e any disconnec ing comple e subg aph. In his case, we conside he
decomposi ion as a undamen al g oup o a g aph o g oups o be i ial and so ∆
has one e ex wi h co esponding g oup G. This decomposi ion sa is ies he e-
qui emen s. I Γis a comple e g aph, hen G≃Zn
π(C). Since by assump ion, each
canonical gene a o is ellip ic, hen by Lemma 6.9, he g oup Gs abilizes a poin ,
and hence he (A,H)-JSJ decomposi ion is i ial and coincides wi h he decom-
posi ion o Gas he undamen al g oup o a g aph o g oups. I Γis no comple e
and does no ha e any disconnec ing comple e subg aph, hen by Theo em 6.3 G
canno ac non- i ially on an A- ee, so he (A,H)-JSJ decomposi ion is again
i ial.
Suppose now ha Γhas a disconnec ing comple e g aph. Le Kbe a disconnec -
ing comple e g aph such ha |V(K)|is minimal among disconnec ing comple e
g aphs.
We i s cons uc a spli ing o Gas an amalgama ed ee p oduc o e he
s anda d subg oup GK. Assume ha Γ Khas m≥2non i ial connec ed
componen s Γi, o i∈ {1, . . . , m}. In his case, we conside he spli ing o Gas
a p o-Camalgama ed p oduc o he o m
G=
m
a
i=1
GKGK∪Γi.
By Theo em 6.5.2 o [72], his decomposi ion co esponds o he p o-C unda-
men al g oup o a ee o g oups (G∆,∆) wi h m e ices V(∆) = {x1, . . . , xm},
whose e ex g oups {GK∪Γi|i∈ {1, . . . , m}} espec i ely, and wi h all edges
o E(∆) s abilised by GK. Since Kis a comple e g aph, GKis a p o-Cabelian
subg oup and hence his decomposi ion is an A-decomposi ion o G. No ice ha
i m > 2, he unde lying ee ∆is no unique. Indeed any ee wi h m e ices
p o ides he same undamen al g oup Gsince all edge g oups coincide. Wi hou
loss o gene ali y, we choose he unde lying g aph ∆ o be a pa h consis ing o
mpoin s and m−1edges, e ex g oups GK∪Γiand edge g oups GK(wi h he
na u al embeddings). By cons uc ion, he g aph o p o-Cg oups (G∆,∆) has G
as i s p o-C undamen al g oup.
125
132
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140
Index
canonical gene a o s, 104
cell, 52
decomposi ion, 52
concise
boundedly, 20
on no mal subg oups, 30
s ongly, 22
uni o mly boundedly, 23
wo d, 19
condi ion C′(λ), 55
cop ime commu a o
γ∗
kand δ∗
k, 72
simple, 72
de o ma ion space, 121
diag am, 53
ci cula , 53
o e a g oup, 53
educed, 54
domina ion o A- ees, 121
ellip ic elemen , 99
ex ension o w(N), 44
lexible e ex, 122
ull subg aph, 104
geodesic o a p o-C ee, 99
good ep esen a i e, 84
g aph o p o-Cg oups, 100
undamen al g oup o , 101
educed, 100
hanging e ex, 127
hype bolic elemen , 99
independen se o ela ions, 56
i educible ac ion, 100
isola ed subg oup, 111
join g aph, 108
JSJ decomposi ion, 121
(A,H)-, 122
ela i e, 122
law, 15
link o an elemen , 106
ma ginal subg oup, 15
minimal simple g oup, 94
O-pai s, 63
Ou e commu a o
ex ension, 43
ou e commu a o
heigh , 44
wo ds, 15
pe iod o ank i, 56
piece, 54
p o-CRAAG, 104
141
