p-Basilica Groups

Medi e . J. Ma h. (2022) 19:275
h ps://doi.o g/10.1007/s00009-022-02187-z
1660-5446/22/060001-28
published online Oc obe 31, 2022
c
The Au ho (s) 2022
p-Basilica G oups
Elena Di Domenico, Gus a o A. Fe n´andez-Alcobe ,
Ma ialau a Noce and Ani ha Thillaisunda am
Abs ac . We conside a gene alisa ion o he Basilica g oup o all odd
p imes: he p-Basilica g oups ac ing on he p-adic ee. We show ha
he p-Basilica g oups ha e he p-cong uence subg oup p ope y bu no
he cong uence subg oup p ope y no he weak cong uence subg oup
p ope y. This p o ides he fi s examples o weakly b anch g oups wi h
such p ope ies. In addi ion, he p-Basilica g oups gi e he fi s exam-
ples o weakly b anch, bu no b anch, g oups which a e supe s ongly
ac al. We compu e he o de s o he cong uence quo ien s o hese
g oups, which enable us o de e mine he Hausdo ff dimensions o he
p-Basilica g oups. Las ly, we show ha he p-Basilica g oups do no pos-
sess maximal subg oups o infini e index and ha hey ha e infini ely
many non-no mal maximal subg oups.
Ma hema ics Subjec Classifica ion. 20E08, 20E18, 28A78, 20E28.
Keywo ds. G oups ac ing on oo ed ees, weakly b anch g oups, con-
g uence subg oup p ope ies, Hausdo ff dimension, maximal subg oups.
1. In oduc ion
Le pbe a p ime and le Tbe he p-adic ee. G oups ac ing on p-adic ees
ha e been well s udied o e he pas decades, owing o hei nice s uc u e,
hei impo ance in he heo y o jus infini e g oups, and he ac ha many
such g oups ha e exo ic p ope ies; see [6] o a good in oduc ion. Lo s o
he in e es ing examples o such g oups sha e one common p ope y: ha o
being b anch o weakly b anch, whe e b anchness is a measu e o how close
The i s h ee au ho s a e suppo ed by he Spanish Go e nmen g an MTM2017-86802-
P, pa ly wi h FEDER unds, and by he Basque Go e nmen g an IT974-16. The i s
and hi d au ho s a e also pa ially suppo ed by he Na ional G oup o Algeb aic and
Geome ic S uc u es, and hei Applica ions (GNSAGA—INdAM). The i s au ho ac-
knowledges suppo om he Depa men o Ma hema ics o he Uni e si y o T en o. The
hi d au ho acknowledges inancial suppo om a London Ma hema ical Socie y Join
Resea ch G oups in he UK (Scheme 3) g an . The ou h au ho acknowledges suppo
om EPSRC, g an EP/T005068/1.
275 Page 2 o 28 E. Di Domenico e al. MJOM
he s uc u e o he g oup esembles he s uc u e o he ull au omo phism
g oup o he ee T; see Sec . 2 o p ecise defini ions.
An example o such a g oup is he Basilica g oup. This g oup ac s on he
bina y ee, is weakly b anch bu no b anch, is o sion- ee, is o exponen-
ial g ow h, is gene a ed by a fini e au oma on, and is no subexponen ially
amenable [22]. The Basilica g oup is gene a ed by wo elemen s, aand b,
which a e ecu si ely defined as ollows:
a=(1,b)andb=(1,a)σ,
whe e σis he cyclic pe mu a ion (1 2), which swaps he wo maximal sub-
ees, and he no a ion (x, y) indica es he independen ac ions on he e-
spec i e maximal sub ees, o xand yau omo phisms o he bina y ee.
The Basilica g oup is also he i e a ed monod omy g oup o he complex
polynomial z2−1, and is a no able example in Nek ashe ych’s heo y which
links au oma a g oups o complex dynamics; see [27, Sec ion 6.12.1]. Fu he -
mo e, he Julia se o z2−1, which is he se o accumula ion poin s o he
backwa d i e a ions o an a bi a y poin in he complex plane unde z2−1,
can be app oxima ed by a sequence o fini e Sch eie g aphs ob ained by he
ac ion o he Basilica g oup a each le el o he bina y ee; see [12], whe e
all limi s o fini e Sch eie g aphs o he Basilica g oup we e classified up o
isomo phism. The Basilica g oup has also been s udied in o he con ex s in
g oup heo y, o example in [13] i has been p o ed ha he Basilica g oup
has no non i ial Engel elemen s.
In his pape , we a e in e es ed in a na u al gene alisa ion o he Basilica
g oup, which we call he p-Basilica g oup, ha ac s on he p-adic ee, o p
any p ime. Such a g oup Gis gene a ed by he ollowing 2 elemen s:
a=(1,p−1
...,1,b)andb=(1,p−1
...,1,a)σ,
whe e σis he cyclic pe mu a ion (1 2 ··· p). Clea ly he 2-Basilica g oup
coincides wi h he Basilica g oup. This gene alisa ion o he Basilica g oup
mi o s Sidki and Sil a’s gene alisa ion o he B unne –Sidki–Viei a g oup;
see [32] and [11]. A diffe en gene alisa ion o he Basilica g oup o he p-adic
ee, wi h pgene a o s, was fi s in es iga ed by Sasse in he Mas e hesis
[30], and Sasse’s wo k has been ecen ly de eloped u he by Pe schick and
Rajee [29]. As seen below, ou 2-gene a o p-Basilica g oups, also known
in [29] as Basilica g oups o le el 2, a e mo e simila o he Basilica g oup.
The gene alisa ions o he Basilica g oups conside ed by Sasse, Pe schick and
Rajee include he Basilica g oups o le els s ic ly g ea e han 2, and hey
diffe mo e significan ly om he Basilica g oup.
We p o e he ollowing in Sec s. 3and 4; see Theo em 3.3, Theo em 3.5
and Lemma 4.10.
Theo em A. Le Gbe a p-Basilica g oup, o pa p ime. Then Gis no
b anch, bu i is weakly egula b anch o e G. Fu he mo e:
(i) G/G∼
=Z×Z.
(ii) G/γ3(G)∼
=Z.
(iii) G/G ∼
=Z2p−1.
(i ) γ3(G)/G ∼
=Z2p−2.
MJOM p-Basilica G oups Page 3 o 28 275
Addi ionally in Sec s. 3and 4we es ablish o he basic p ope ies o he
p-Basilica g oups G, such as being o sion- ee (Theo em 3.6), con ac ing
(Theo em 3.8), jus non-sol able (Co olla y 4.3), and ha ing i s au omo -
phism g oup Au (G) equal he no malise o Gin Au (T) (Co olla y 4.6). We
also show ha he g oups a e supe s ongly ac al (Theo em 4.5), which
means o any n∈N, he p ojec ion o he n h le el s abilise S G(n)a any
n h le el e ex is he whole o G; see Sec ion 2 o he p ecise defini ion.
This yields he fi s examples o fini ely gene a ed weakly b anch, bu no
b anch, g oups ha a e supe s ongly ac al.
Now, one o he main p ope ies conce ning he p-Basilica g oups ha
we in es iga e is he cong uence subg oup p ope y, whe e we say ha G≤
Au (T) has he cong uence subg oup p ope y i e e y fini e-index subg oup
o Gcon ains a le el s abilise S G(n) o some n∈N. Equi alen ly, he
g oup Ghas he cong uence subg oup p ope y i he p ofini e comple ion
o Gequals i s closu e in Au (T).
Ga ido and U ia-Albizu i [19] in oduced a weake e sion o he con-
g uence subg oup p ope y: a g oup G≤Au (T) is said o ha e he p-
cong uence subg oup p ope y i e e y no mal subg oup o p-powe index
con ains some le el s abilise . In [19], examples o weakly b anch, bu no
b anch, g oups wi h he p-cong uence subg oup p ope y and no he con-
g uence subg oup p ope y we e p o ided. Fo podd, hei examples we e
he G igo chuk–Gup a–Sidki (GGS-)g oups defined by he cons an ec o ,
and o p= 2, hei example was he Basilica g oup. In Sec . 5, we ex end
his esul o p-Basilica g oups, o all odd p imes p:
Theo em B. Le Gbe a p-Basilica g oup, o pa p ime. Then Ghas he p-
cong uence subg oup p ope y bu no he cong uence subg oup p ope y no
he weak cong uence subg oup p ope y.
We ecall ha a g oup G≤Au (T) has he weak cong uence subg oup p ope y
i e e y fini e-index subg oup con ains he de i ed subg oup o some le el
s abilise ; c . [31]. The p-Basilica g oups a e he fi s examples o weakly
egula b anch g oups wi h he p-cong uence subg oup p ope y bu no he
weak cong uence subg oup p ope y.
In Subsec . 5.2, we compu e he o de s o he cong uence quo ien s
G/ S G(n) o all n∈N, o ap-Basilica g oup G. This enables us o com-
pu e he Hausdo ff dimension o he closu e o he p-Basilica g oup Gin he
g oup Γ o p-adic au omo phisms o T. We ecall ha
Γ∼
=lim
←−
n∈N
Cpn
···Cp
is a Sylow p o-psubg oup o Au (T) co esponding o he p-cycle (1 2 ··· p).
Fo a subg oup Go Γ, he Hausdo ff dimension o he closu e o Gin Γ is
gi en by
hdimΓ(G) = lim
n→∞
log |G :S
G(n)|
log |Γ:S
Γ(n)|∈[0,1],(1.1)
275 Page 4 o 28 E. Di Domenico e al. MJOM
whe e lim ep esen s he lowe limi . The Hausdo ff dimension o Gis a
measu e o how dense Gis in Γ. This concep was fi s applied by Abe c om-
bie [1] and by Ba nea and Shale [2] in he mo e gene al se ing o p ofini e
g oups. We no e ha he Hausdo ff dimension o he closu es o se e al p omi-
nen weakly b anch g oups, such as he fi s [20] and second [28] G igo chuk
g oups, he siblings o he fi s G igo chuk g oup [35], he GGS-g oups [15],
he b anch pa h g oups [14], and gene alisa ions o he Hanoi owe g oups
[33], ha e been compu ed.
Theo em C. Le Gbe a p-Basilica g oup, o pa p ime. Then
(i) The o de s o he cong uence quo ien s o Ga e gi en by
logp|G:S
G(n)|=pn−1+pn−3+···+p3+p+n
2 o ne en,
pn−1+pn−3+···+p4+p2+n+1
2 o nodd.
(ii) The Hausdo ff dimension o he closu e o Gin Γis
hdimΓ(G)= p
p+1.
In Sec . 6, we gi e a ecu si e p esen a ion, a so-called L-p esen a ion,
o he p-Basilica g oups (P oposi ion 6.3), plus we show ha he p-Basilica
g oups a e amenable bu no elemen a y subexponen ially amenable
(Lemma 6.2), and ha e exponen ial g ow h (Theo em 6.1); we e e o Sec . 6
o he defini ions. To he bes o ou knowledge, he only o he infini e amily
o weakly b anch g oups ha a e amenable bu no elemen a y subexponen-
ially amenable is he amily o p-gene a o Basilica g oups ac ing on he
p-adic ee; see [30].
F ancoeu [17, Thm. 4.28] p o ed ha he Basilica g oup does no pos-
sess maximal subg oups o infini e index, hus p o iding he fi s example o
a weakly b anch bu no b anch g oup wi hou maximal subg oups o infi-
ni e index. Also, he Basilica g oup has non-no mal maximal subg oups [16,
Co . 8.3.2]. In Subsec . 6.3, we ex end hese esul s o p-Basilica g oups o
all p imes p, likewise gi ing ano he infini e amily o weakly b anch g oups
wi h such p ope ies. No e ha he fi s infini e amily o weakly b anch, bu
no b anch, g oups wi hou maximal subg oups o infini e index was gi en
by F ancoeu and Thillaisunda am in [18], namely he GGS-g oups defined
by he cons an ec o .
Theo em D. Le Gbe a p-Basilica g oup, o pa p ime. Then all maximal
subg oups o Gha e fini e index, and Ghas infini ely many non-no mal max-
imal subg oups.
No a ion. Th oughou , we use le -no med commu a o s, o example,
[x, y, z]=[[x, y],z]. Fo a g oup G, a subg oup H≤Gand g∈G, we w i e
[H,g]=[h, g]|h∈H.Alsoi NG hen we w i e g≡Nh o mean ha
he images o gand hin G/N coincide. Fo Ga g oup and pa p ime, we w i e
Wp(G) o he w ea h p oduc o Gwi h a cyclic g oup o o de p.
MJOM p-Basilica G oups Page 5 o 28 275
2. P elimina ies
2.1. The G oup Au (T)
Le pbe a p ime and le Tbe he p-adic ee, i.e. he oo ed ee ha ing
pdescendan s a e e y e ex. I we choose an alphabe Xwi h ple e s,
Tcan be ep esen ed as he g aph whose e ices a e he elemen s o he
ee monoid X∗, he oo is he emp y wo d ∅,andwis a descendan o u
p o ided ha w=ux wi h x∈X.
Fo a gi en e ex u, he se o e ices u wi h ∈X∗a e said o
succeed u.They o ma eeTu oo ed a u, which is isomo phic o T.We
deno e by |u| he leng h o uas a wo d. Fo e e y n∈N∪{0}, he se Lno
all wo ds o leng h nis called he n h laye o he ee.
Au omo phisms o Tasag aph o mag oupAu (T) unde composi-
ion. Le ube a e ex o Tand le ∈Au (T). We use exponen ial no a ion
o images o au omo phisms and, mo e gene ally, pe mu a ions. Thus he
image o uunde ∈Au (T)isu . No e ha au omo phisms lea e each
laye in a ian , so u and ubelong o he same laye . The label o a uis
he pe mu a ion (u) o he alphabe Xdefined by he ule
(ux) =u x (u), o e e y x∈X.
The po ai o is he se o all labels o , and he e is a one- o-one
co espondence be ween au omo phisms o Tand po ai s. The suppo o
is he se o e ices wi h non- i ial label. We say ha is oo ed i he
suppo is con ained in he oo , and is di ec ed i he suppo is infini e
and consis s only o descendan s o a gi en infini e pa h s a ing a he oo .
In a simila way, he sec ion uo a uis he au omo phism o T
defined by
(u ) =u u, o e e y ∈X∗.
Fo all ,g ∈Au (T)andu, ∈X∗,weha e( u) = u ,( g)u= ugu ,
( g)ug=(gu)−1 ugu .(2.1)
2.2. Subg oups o Au (T)
Fo a e ex uo T, he e ex s abilise S (u) is he subg oup consis ing o
all au omo phisms o Tfixing u. The map ψu: → uis a homomo phism
om S (u) on o Au (T). Fo e e y n∈N, hen h le el s abilise is
S (n)= 
u∈Ln
S (u).
Then S (n) is a no mal subg oup o Au (T)andAu (T) is isomo phic o he
in e se limi o he fini e g oups Au (T)/S (n). Hence Au (T) is a p ofini e
g oup wi h {S (n)}n∈Nas a basis o neighbou hoods o he iden i y. Fo e e y
n∈N, we ha e an isomo phism
ψn:S (n)−→ Au (T)×pn
···×Au (T)
−→ ( u)u∈Ln.

275 Page 6 o 28 E. Di Domenico e al. MJOM
The map ψ1can be ex ended o an isomo phism
ψ: Au (T)−→ Au (T)Sym(X)
−→ ψ1( )τ,
whe e τis he label o a he oo . Thus we can define au omo phisms o
Au (T) by gi ing hei image unde ψ.
Le Sym(X) be he symme ic g oup o e he alphabe X.I σ∈
Sym(X) is a fixed p-cycle, we can ob ain a Sylow p o-psubg oup Γ(σ)o
Au (T) by conside ing all au omo phisms whose po ai only con ains la-
bels om σ; ha is,
Γ(σ)= ∈Au (T)| (u)∈σ o all u∈X∗∼
=lim
←−
n∈N
Cpn
···Cp.
As men ioned in he in oduc ion, we also w i e Γ = Γ(1 2 ··· p).
Now le Gbe a subg oup o Au (T). We w i e S G(u) = S (u)∩G
and S G(n) = S (n)∩G. The la e is a no mal subg oup o Gand we se
Gn=G/ S G(n), which is called he n h cong uence quo ien o G.Wesay
ha Gis
•Le el- ansi i e i i ac s ansi i ely on Ln o e e y n∈N.
•Sel -simila i all sec ions o elemen s o Ga all e ices belong o G.
•F ac al i i is sel -simila and ψu(S G(u)) = G o e e y e ex uo he
ee.
•Supe s ongly ac al i i is sel -simila and ψu(S G(n)) = G o e e y
u∈Lnand e e y n∈N.
No e ha i Gis sel -simila hen ψn(S G(n)) ⊆G×pn
···×G o all n,and
i u he mo e Gis con ained in a Sylow p o-psubg oup Γ(σ) hen ψ(G)⊆
Gσ=Wp(G). The n h cong uence quo ien o Γ(σ) is isomo phic o he
i e a ed w ea h p oduc o ncopies o Cpand Γ(σ) is he in e se limi o
hese fini e p-g oups.
The igid e ex s abilise Ris G(u)o a e exuin Gis he subg oup
consis ing o all au omo phisms in G ha fix all e ices ou side Tu. Then
o e e y n∈Nwe define he igid n h le el s abilise as
Ris G(n)=Ris G(u)|u∈Ln=u∈Ln
Ris G(u)G.
I Gis le el- ansi i e we say ha Gis a b anch g oup i |G:Ris
G(n)|<∞
o all n∈N, and ha i is weakly b anch i Ris G(n)= 1 o all n.I Gis
le el- ansi i e and sel -simila , and K×···×K⊆ψ(K) o some K≤G,
we say ha Gis egula b anch o e Ki |G:K|<∞and ha Gis weakly
egula b anch o e Ki K= 1. Obse e ha being (weakly) egula b anch
implies being (weakly) b anch.
2.3. p-Basilica G oups
Le X={x1,...,x
p}and le Γ be he Sylow p o-psubg oup o Au (T)
co esponding o he p-cycle σ=(x1x2··· xp). The p-Basilica g oup is he
subg oup Go Γ gene a ed by he au omo phisms aand bgi en by
ψ(a)=(1,...,1,b)andψ(b)=(1,...,1,a)σ.
MJOM p-Basilica G oups Page 7 o 28 275
(a) (b)
Figu e 1. The po ai o he gene a o s o a p-Basilica g oup
Figu e 2. The 3-Basilica au oma on
No e ha he 2-Basilica g oup is he well-known Basilica g oup men ioned
in he in oduc ion. The po ai s o aand ba e desc ibed in Fig. 1.
We ecall ha an au omo phism ∈Au (T) is called bounded i he
se s {w∈Xn| w=1}ha e uni o mly bounded ca dinali ies o e all n.A
g oup G≤Au (T) is said o be a bounded au oma a g oup i Gis fini ely
gene a ed, sel -simila and e e y elemen g∈Gis bounded and fini e s a e
(i.e. he se {g | ∈X∗}is fini e). Fo e e y p ime p he p-Basilica g oup is
a g oup gene a ed by a fini e bounded au oma on wi h se o s a es {Id; a;b}.
In Fig. 2, he e is he gene a ing au oma on in he case p=3.
We ema k ha being an au oma a g oup, o any p ime p, hep-Basilica
g oup has sol able wo d p oblem, [27, P op. 2.13.8].
3. Fi s P ope ies
In his sec ion, we p o e some basic p ope ies o he p-Basilica g oups. We
s a wi h he ollowing elemen a y bu essen ial esul .
Lemma 3.1. Le Gbe a p-Basilica g oup, o a p ime p.ThenGis ac al
and le el- ansi i e.
P oo . By [36, Lem. 2.7], i suffices o show ha Gac s ansi i ely on he
fi s laye and ha ψx(S G(x)) = G o some x∈X(see also [21, Sec. 3]).
275 Page 8 o 28 E. Di Domenico e al. MJOM
This is s aigh o wa d since bac s ansi i ely on he fi s laye and since
ψ(a)=(1,...,1,b)andψ(bp)=(a,...,a). 
Nex we conside he s abilise s in Go he fi s wo laye s. Recall ha
Gn=G/ S G(n), and o con enience, we se A=aGand B=bG.
Lemma 3.2. Le Gbe a p-Basilica g oup, o a p ime p.Then:
(i) S G(1) = Abp=a, ab,...,a
bp−1,b
pand G1=bS G(1)∼
=Cp.
(ii) G2∼
=CpCpis a p-g oup o maximal class o o de pp+1.
P oo . (i) Obse e ha a∈S G(1), and ha bn∈S G(1) i and only i p|n.
Since S G(1) Gwe ge Abp≤S G(1). No e ha Abpis no mal in
Gsince G/A is cyclic. As a consequence, he inclusion Abp≤S G(1) is
an equali y, since G/Abphas o de pand S G(1) is a p ope subg oup
o G.
(ii) Since ψ(bp)=(a,...,a)anda∈S G(1), we ha e bp∈S G(2). By (i), i
we conside G2=G/ S G(2), and we deno e an elemen in he quo ien
using he ba no a ion in G2,weha e
S G2(1) = a, ab,...,abp−1.
Since ahas o de pin G2and he uples
ψ(a)=(1,...,1,b)andψ(abi)=(1,i−1
...,1,a
−1ba, 1,...,1), o 1 ≤i≤p−1,
commu e wi h each o he , S G2(1) is elemen a y abelian o o de pp.
Thus |G2|=pp+1. To comple e he p oo , obse e ha since G≤Γ, he
quo ien G2=G/ S G(2) embeds in Γ2=Γ/S Γ(2) ∼
=CpCp, and ha
he la e is a p-g oup o maximal class o o de pp+1.
Ou nex goal is o s udy he abelianisa ion o G. In he emainde , le
A=aGand B=bGas be o e. Since G=a, b,weha eA=aG,
B=bG,andG=AB. Obse e ha ψ(a)=(1,...,1,b)andGbeing
sel -simila imply
ψ(A)⊆B×···×B. (3.1)
On he o he hand, he map π:Wp(G)→G/Gsending (g1,...,g
p)σi o
g1···gpGis clea ly a g oup homomo phism. Since ψ(b)=(1,...,1,a)σ,i
ollows ha
(π◦ψ)(B)⊆A/G.(3.2)
Theo em 3.3. Le Gbe a p-Basilica g oup, o a p ime p.Then:
(i) G/A =bAand G/B =aBa e infini e cyclic. In pa icula , he
elemen s aand bha e infini e o de in G.
(ii) G/G=aG×bG∼
=Z×Z.
(iii) A∩B=G.
P oo . Since G/G=A/G·B/G, wi h A/G=aGand B/G=bG,
bo h (ii) and (iii) ollow immedia ely om (i).
We p o e ha G/A and G/B a e infini e simul aneously. Assume o a
con adic ion ha , o some n∈N,weha eei he an∈Bo bn∈A, and le
MJOM p-Basilica G oups Page 9 o 28 275
us choose nas small as possible. I bn∈A⊆S G(1) hen n=pm o some
m, and consequen ly,
ψ(bn)=(am,...,a
m)∈ψ(A)⊆B×···×B,
by (3.1). Hence, am∈B, which is impossible since m<n. On he o he
hand, i an∈B hen
bnG=(π◦ψ)(an)∈(π◦ψ)(B)⊆A/G
by (3.2). Thus, bn∈A, which we jus p o ed is no he case. This comple es
he p oo . 
Nex we s udy igid s abilise s and he b anch s uc u e o G. To his
pu pose, he ollowing esul is e y use ul. I is gi en in [15, P op. 2.18] o
GGS-g oups, bu he same p oo wo ks mo e gene ally o le el- ansi i e
ac al g oups. We s a e his gene al e sion he e o he con enience o he
eade .
Lemma 3.4. Le Gbe a le el- ansi i e ac al subg oup o Au (T),andle
Land Nbe wo no mal subg oups o G. Suppose ha L=SGand ha
(1,...,1,s,1,...,1) ∈ψ(N) o e e y s∈S,whe esappea s always a he
same posi ion in he uple. Then L×···×L⊆ψ(N).
Theo em 3.5. Le Gbe a p-Basilica g oup, o a p ime p.Then
(i) Ris G(1) = Awi h ψ(A)=B×···×B. In pa icula , he g oup Gis
no b anch.
(ii) ψ(S G(1))=G×···×G. As a consequence, he g oup Gis weakly
egula b anch o e G.
P oo . (i) We al eady know om (3.1) ha ψ(A)⊆B×···×B, and he
e e se inclusion ollows om Lemma 3.4, since ψ(a)=(1,...,1,b).
Hence A≤Ris G(1) ≤S G(1) = Abpand so Ris G(1) = Abpn o
some n. Since
ψ(Abpn)=(B×···×B)(an,...,a
n)
and ahas infini e o de modulo Bby Theo em 3.3(i), i ollows om
he defini ion o he igid s abilise ha n= 0. Hence Ris G(1) = A,
which has infini e index in G,soGis no b anch.
(ii) The inclusion ⊆is clea . Fo he e e se inclusion, obse e ha
ψ([bp,a]) = (1,...,1,[a, b]). Since G=[a, b]G, he esul ollows om
Lemma 3.4.

Theo em 3.6. Le Gbe a sel -simila subg oup o Au (T)and suppose ha
he e exis s a o sion- ee quo ien G/N wi h N≤S G(1).ThenGis o sion-
ee. In pa icula , he p-Basilica g oups a e o sion- ee.
P oo . Fo e e y n∈N∪{0},le Pns and o he se o o sion elemen s in
S G(n) S G(n+ 1). Then ou goal is o p o e ha hese se s a e all emp y.
By way o con adic ion, suppose ha Pn=∅ o some n, which we choose
as small as possible.
275 Page 16 o 28 E. Di Domenico e al. MJOM
which is con a y o (4.8). This p o es he case n= 2. The gene al case
ollows in a simila ashion by induc ion on n.
We close his sec ion by de e mining he s uc u e o he quo ien s
G/γ3(G), G/G,andγ3(G)/G, which is key o Sec . 5. We need a couple
o lemmas.
Lemma 4.8. Le Gbe a p-Basilica g oup, o a p ime p. Fo e e y n∈N,we
ha e
ψ(S G(n)) = S G(n−1) ×···×S G(n−1)Cpβ(n−1) .
P oo . The inclusion ⊇is ob ious om Theo em 4.1(i) and Theo em 4.4(i).
Fo he o he di ec ion, le g∈S G(n) and w i e ψ(g)=(w1,...,w
p)
(bi1,...,b
ip), whe e he fi s ac o is in G×···×Gand he second is in C.Fix
an index j∈{1,...,p}. Since wjbij∈S G(n−1) we ha e bij∈GS G(n−1),
and hen pβ(n−1) di ides ijby Theo em 4.4.Thus,bij∈S G(n−1) and i
ollows ha also wj∈S G(n−1). This p o es he esul . 
Lemma 4.9. Le Gbe a p-Basilica g oup, o pa p ime. Then he o de o
[a, b]modulo γ3(G)S
G(n)is a leas pβ(n−1) o e e y n∈N.
P oo . Since [a, b] and [a, b−1] a e in e se conjuga e, we p o e he esul o
he o de o [a, b−1]. We use induc ion on n. The esul is ob ious i n=1,so
we suppose ha n≥2. I [a, b−1]pm∈γ3(G)S
G(n), we wan o show ha
m≥β(n−1). By way o con adic ion, we assume ha m<β(n−1). By
applying ψ o [a, b−1]pmand using Theo em 4.1(iii) and Lemma 4.8,wege
c−pm
0=ydgc, (4.9)
whe e y=yk0
0···ykp−2
p−2 o some k0,...,k
p−2∈Z,d∈D,g∈S G(n−1) ×
···×S G(n−1), and c∈Cpβ(n−1) . I we educe (4.9) modulo G×···×G
and use ha y0 educes o c−p
0,wege
cpm−pk0+k1
0c−k1+k2
1···c−kp−3+kp−2
p−3c−kp−2
p−2∈Cpβ(n−1) .(4.10)
Since c0,...,c
p−2 o m a basis o he ee abelian g oup C, i ollows ha
all exponen s in (4.10) a e di isible by pβ(n−1) and, as a consequence, so is
pm−pk0. Since m<β(n−1), i ollows ha he p-pa o pk0is pm.Now
since B=bGis abelian modulo γ3(G), he map
τ:B×···×B−→ B/γ3(G)
(g1,...,g
p)−→ g1···gpγ3(G)
is a g oup homomo phism. Obse e ha bo h Cand Dlie in he ke nel o τ,
and ha τ(y0)=[a, b−1]γ3(G). Hence, by applying τ o (4.9), we ge
[a, b−1]k0∈γ3(G)S
G(n−1).
Since he p-pa o k0is pm−1, by he induc ion hypo hesis we ha e m−1≥
β(n−2), and so m≥β(n−2) + 1 ≥β(n−1), con a y o ou assump ion.
This comple es he p oo . 
Theo em 4.10. Le Gbe a p-Basilica g oup, o pa p ime. Then

MJOM p-Basilica G oups Page 17 o 28 275
(i) G/γ3(G)∼
=Z.
(ii) G/G ∼
=Z2p−1.
(iii) γ3(G)/G ∼
=Z2p−2.
P oo . (i) We obse e ha G/γ3(G) is cyclic and gene a ed by he image
o [a, b]. F om Lemma 4.9, he o de o [a, b] ends o infini y modulo
γ3(G)S
G(n)asngoes o infini y. Hence he s a emen immedia ely
ollows.
(ii) The esul ollows om (i) and om Theo em 4.1 since we ha e
G
G ∼
=ψ(G)
ψ(G)∼
=G
γ3(G)×p
···× G
γ3(G)×C
∼
=Z×p
···×Z×Zp−1.
This comple es he p oo since (iii) is a s aigh o wa d consequence o
(i) and (ii). 
We ecall he in eg al Heisenbe g g oup H3(Z), which is he 2-gene a ed
g oup o 3×3 uppe uni iangula ma ices wi h in eg al en ies, and has he
p esen a ion x, y, z |z=[x, y],xz=zx, yz =zy.
Co olla y 4.11. Le Gbe a p-Basilica g oup, o pa p ime. Then G/γ3(G)is
isomo phic o he in eg al Heisenbe g g oup H3(Z).
P oo . The p oo is analogous o [19, P op. 23] o o [17, P op. 4.8], whe e i
was p o ed o he case p= 2, using diffe en app oaches. 
As no ed in [17, Co . 4.9], he p e ious esul yields an al e na i e p oo
ha he p-Basilica g oups a e no b anch. As e e y p ope quo ien o a
b anch g oup is i ually abelian and he in eg al Heisenbe g g oup is no
i ually abelian, he esul ollows.
5. Cong uence Subg oup P ope ies and Hausdo ff Dimension
5.1. Cong uence Subg oup P ope ies
Le Gbe a p-Basilica g oup, o pa p ime. Since G/G∼
=Z×Z, he g oup G
does no ha e he cong uence subg oup p ope y as all quo ien s o Gby
le el s abilise s a e p-g oups. In his subsec ion we show ha Ghas he p-
cong uence subg oup p ope y (p-CSP o sho ) bu no he weak cong uence
subg oup p ope y.
We ecall ha i Gis a subg oup o Au (T)andNG, we say ha
G/N has he p-cong uence subg oup p ope y (o ha Ghas he p-cong uence
subg oup p ope y modulo N)i e e yJG, such ha G/J is a fini e p-g oup
and N≤J, con ains some le el s abilise o G. Acco ding o [19, Lem. 6], i
N≤Ma e wo no mal subg oups o Gsuch ha bo h G/M and M/N ha e
he p-CSP hen also G/N has he p-CSP.
We need a couple o lemmas be o e p o ing ha he p-Basilica g oups
ha e he p-CSP.
275 Page 18 o 28 E. Di Domenico e al. MJOM
Lemma 5.1. Le Nand Gbe subg oups o Au (T)wi h NGand G/N ee
abelian o ank , o some ∈N. Suppose ha , o la ge enough n∈N,we
ha e
G/N S G(n)∼
=Cpλ1(n)×···×Cpλ (n),(5.1)
wi h limn→∞ λi(n)=∞ o 1≤i≤ .ThenG/N has he p-CSP.
P oo . Le N≤JG, whe e |G :J|=pm, o some m∈N. Then Gpm≤J.
Now choose an in ege nsuch ha λi(n)≥m o 1 ≤i≤ .By(5.1), o
la ge enough nwe ha e
|G/N S G(n):(G/N S G(n))pm|=p m =|G/N :(G/N )pm|,
which implies NS G(n)Gpm=NGpm, since by he hi d isomo phism heo-
em,
G/N S G(n)
(G/N S G(n))pm=G/N S G(n)
NS G(n)Gpm/N S G(n)∼
=G
NS G(n)Gpm
and
G/N
(G/N )pm=G/N
NGpm/N ∼
=G
NGpm.
Hence, S G(n)≤NGpm≤J,andG/N has he p-CSP. 
The ollowing lemma is a sligh gene alisa ion o [19, Thm. 1], which
co esponds o he case when Nis chosen so ha L≤K, and i s p oo is
e y simila . Fo con enience, we include he p oo below.
Lemma 5.2. Le Gbeasubg oupo Au (T) ha is weakly egula b anch o e
a no mal subg oup K.Le Nbe a no mal subg oup o Gsuch ha :
(i) K≤N≤K.
(ii) I L=ψ−1(N×···×N) hen G/N ,N/L,andN/Kha e he p-CSP.
Then Ghas he p-CSP.
P oo . Se Lm=ψ−1
m(N×pm
···×N) o e e y m∈N. No e ha LmG, since
Gis sel -simila and NG. We will p o e by induc ion on m ha G/Lmhas
he p-CSP. Since L1=Land bo h G/N and N/L ha e he p-CSP, he esul
is ue o m= 1. Now we suppose ha he esul holds o mand we will
show i o m+ 1. Obse e ha i suffices o p o e ha Lm/Lm+1 has he
p-CSP. Recall ha
ψm(Lm)=N×pm
···×N
and
ψm(Lm+1)=L×pm
···×L,
so ψminduces an isomo phism be ween Lm/Lm+1 and N/L ×pm
···×N/L.
Since
ψm(S Lm(n)) = S N(n−m)×pm
···×S N(n−m) o e e y n≥m,
MJOM p-Basilica G oups Page 19 o 28 275
and N/L has he p-CSP, i ollows ha
N/L ×pm
···×N/L ∼
=(N×pm
···×N)/(L×pm
···×L)∼
=Lm/Lm+1
has he p-CSP.
To conclude ha Ghas he p-CSP, le JGbe such ha G/J is a fini e
p-g oup. Since Gis le el- ansi i e, we ha e Ris G(m)≤J o some m∈N
by [19, Lem. 4]. Define he subg oup Km≤Ris G(m) by he condi ion
ψm(Km)=K×pm
···×K.
Since K≤Ni ollows ha K
m≤Lm. Now aking in o accoun ha N/K
has he p-CSP, he same a gumen as in he fi s pa ag aph o he p oo yields
ha Lm/K
mhas he p-CSP as well. Thus G/K
mhas he p-CSP o e e y
m∈N. Since K
m≤Ris G(m)≤J, his p o es ha Jcon ains some le el
s abilise in G, and consequen ly Ghas he p-CSP. 
Theo em 5.3. Le Gbe a p-Basilica g oup, o a p ime p.ThenGhas he
p-CSP.
P oo . We apply Lemma 5.2 wi h K=N=G.Thusi L=ψ−1(G×···×G)
hen i suffices o p o e ha G/G,G/L and L/G ha e he p-CSP. No e
ha hen also G/G has he p-CSP.
Fi s o all, he ac o g oup G/Ghas he p-CSP by Lemma 5.1, since
G/GS G(n)∼
=Cpβ(n−1) ×Cpβ(n)wi h β(n)=n/2, acco ding o Theo-
em 4.4(ii).
Nex we deal wi h G/L, which is ee abelian o ank p−1 by Theo-
em 4.1(i). By Lemma 4.8,weha eψ(LS G(n)) = (G×···×G)Cpβ(n−1)
and consequen ly G/L S G(n)∼
=C/Cpβ(n−1) . Hence, his case also ollows
om Lemma 5.1.
Le us finally conside he case o L/G.Weha e
L/G ∼
=(G×···×G)/(γ3(G)×···×γ3(G)) ∼
=Zp.
Since
ψ(G S L(n)) = γ3(G)S
G(n−1) ×···×γ3(G)S
G(n−1),
i ollows ha
L/G S L(n)∼
=G/(γ3(G)S
G(n−1)) ×···×G/(γ3(G)S
G(n−1)),
and we can once again apply Lemma 5.1,by akingin oaccoun Lemma4.9.

We no e ha in [29, Thm. 1.10] i was shown ha s-gene a o Basilica
g oups, o s>2, ha e he p-CSP. I is wo h men ioning ha he e a e
key s uc u al diffe ences be ween hese g oups and he p-Basilica g oups;
compa e [29, Thm. 1.9].
Now we comple e he p oo o Theo em B. Recall ha a g oup G≤
Au (T) has he weak cong uence subg oup p ope y i e e y fini e-index sub-
g oup con ains he de i ed subg oup o some le el s abilise .
275 Page 20 o 28 E. Di Domenico e al. MJOM
Theo em 5.4. Le Gbe a p-Basilica g oup, o a p ime p.ThenGdoes no
ha e he weak cong uence subg oup p ope y.
P oo . Le q=pbe a p ime, and le N=aq,b
q,[a, b]qγ3(G), which is
no mal and o fini e index in G. By Co olla y 4.11,weha eG/N ∼
=H3(q).
We claim ha S G(n)≤ N o e e y odd n, which is enough o p o e he
heo em. A guing by way o con adic ion, since by Theo em 4.7 we ha e
ψn(S G(n))=G×pn
···×G,andacco ding o(4.7)
ψn([ap(n−1)/2,b
p(n+1)/2]) = (1,pn−1
... ,1,[b, a]) ∈G×pn
···×G
o odd n, i ollows ha [ap(n−1)/2,b
p(n+1)/2]∈N.Asγ3(G)≤N,wege
[a, b]pn∈N. Since also [a, b]q∈N, we conclude ha [a, b]∈N. This con a-
dic s he ac ha G/N ∼
=H3(q). 
5.2. Hausdo ff Dimension
In his subsec ion, we de e mine he o de s o he cong uence quo ien s o
he p-Basilica g oups, and we compu e hei Hausdo ff dimensions. Tha is,
we p o e Theo em C, which o con enience we ecall he e.
Theo em C.Le Gbe a p-Basilica g oup, o pa p ime. Then:
(i) The o de s o he cong uence quo ien s o Ga e gi en by
logp|G:S
G(n)|=pn−1+pn−3+···+p3+p+n
2 o ne en,
pn−1+pn−3+···+p4+p2+n+1
2 o nodd.
(ii) The Hausdo ff dimension o he closu e o Gin Γ is
hdimΓ(G)= p
p+1.
P oo . (i) We a gue by induc ion on n. The case n= 1 is clea , so we
assume n≥2. W i e n=2m+e, wi h e= 0 o 1. We need o es ablish
ha
logp|G:S
G(n)|=pn+1 −p1+e
p2−1+m+e.
No e ha , by Theo em 4.4,
|G:S
G(n)|=|G:GS G(n)||GS G(n):S
G(n)|
=pn|G:S
G(n)|
and ha |G:S
G(n)|coincides wi h
|ψ(G):ψ(S G(n))|
=|(G×p
···×G)C:(S
G(n−1) ×p
···×S G(n−1))Cpβ(n−1) |
=p(p−1)β(n−1) |G:S
G(n−1)|p
=p(p−1)β(n−1)−p(n−1) |G:S
G(n−1)|p,
MJOM p-Basilica G oups Page 21 o 28 275
whe eweha eusedLemma4.8 and he ac ha Cis ee abelian o
ank p−1. He e β(n−1) = (n−1)/2as be o e. Thus,
logp|G:S
G(n)|
=plogp|G:S
G(n−1)|+(p−1)(β(n−1) −n+1)+1
=plogp|G:S
G(n−1)|−(p−1)(m+e−1) + 1,
since β(n−1) = m. Now he esul ollows om he induc ion hypo h-
esis.
(ii) To ge he Hausdo ff dimension o Gin Γ, we jus need o ake in o
accoun o mula (1.1) and he ac ha
logp|Γ:S
Γ(n)|=1+p+···+pn−1=pn−1
p−1.

We ema k ha he Hausdo ff dimension o he Basilica g oup was gi en
by Ba holdi in [4]. Also he Hausdo ff dimensions o he gene alised Basilica
g oups we e compu ed in [29, Thm. 1.7] using a e y diffe en app oach.
6. Fu he P ope ies
6.1. G ow h and Amenabili y
Be o e p o ing he main esul s o his subsec ion, we need some p elimina y
defini ions, namely he no ions o g ow h o g oups and amenabili y.
Le Gbe a g oup gene a ed by a fini e symme ic subse S. The leng h
unc ion on Gis a me ic on Gand he e o e one can define he ball o adius
n:
B(n)={g∈G:|g|≤n}.
We say ha he map γ:N0−→ [0,∞) whe e γ(n)=|B(n)|,is heg ow h
unc ion o G.
I we conside wo g ow h unc ions γ1,γ
2, we say ha γ2domina es
γ1and we w i e γ1γ2i he e exis C, α > 0 such ha γ1(n)≤Cγ2(αn)
o e e y n∈N.I γ1γ2and γ2γ1, we w i e γ1∼γ2.I iseasy osee
ha his is an equi alence ela ion. No ice also ha all g ow h unc ions o
a fini ely gene a ed g oup a e equi alen .
I γ(n)na o some a∈N, we say ha Ghas polynomial g ow h.
Ins ead Gis said o ha e exponen ial g ow h i limn→∞ γ(n)1/n >1 (no ice
ha such a limi always exis s). Finally γ(n)hasin e media e g ow h i γ(n)
is equi alen o nei he o he abo e. No ice ha i is also common o say
ha a g oup Ghas subexponen ial g ow h i limn→∞ γ(n)1/n =1,o equi a-
len ly, i γ(n)=e (n) o some (inc easing) unc ion :N−→ R+sa is ying
limn→∞ (n)/n =0.
Nex , we say ha a g oup Gis amenable i he e is a fini ely addi i e
le -in a ian measu e μon he subse s o Gsuch ha μ(G) = 1. We deno e
he class o amenable g oups by AG. The class EG o elemen a y amenable
g oups is he smalles class o g oups con aining all abelian g oups and fini e

275 Page 22 o 28 E. Di Domenico e al. MJOM
g oups and closed unde quo ien s, subg oups, ex ensions and di ec unions.
We ha e EG ⊆AG, and his inclusion is s ic . Fu he mo e he class SG o
elemen a y subexponen ially amenable g oups is he smalles class o g oups
which con ains all g oups o subexponen ial g ow h and is closed unde aking
subg oups, quo ien s, ex ensions, and di ec unions. O cou se, he class SG
con ains he class EG.
In he ollowing we de e mine he g ow h o a p-Basilica g oup G,
o pan odd p ime, and we p o e ha Gis amenable bu no elemen a y
subexponen ially amenable. The co esponding e sions o Theo em 6.1 and
Lemma 6.2 o p= 2 we e p o ed in [22, Lem. 4 and P op. 4] and in [24,
Co . 9], espec i ely.
Theo em 6.1. Le G=a, bbe a p-Basilica g oup, o pan odd p ime. Then
he semig oup gene a ed by aand bis ee. Consequen ly, he g oup Gis o
exponen ial g ow h.
P oo . Le uand be wo diffe en wo ds ep esen ing he same elemen in
he semig oup gene a ed by aand b, and wi h ρ= max(|u|,| |) minimal. We
no e ha |u|b≡p| |b, whe e |u|bdeno es he b-leng h in u, which is equi alen
o he numbe o occu ences o bin u. A di ec check shows ha ρ≥4.
Suppose fi s ha ucon ains no b’s. The e o e, we ha e u=ai=
ψ−1((1,...,1,b
i)), o some i∈N. Since | |bis a non-ze o mul iple o p,
one deduces ha 1is a non-emp y wo d, whe e ψ( )=( 1,...,
p). Indeed,
e e y componen o ψ( ) con ains an a. Ce ainly | 1|<ρ, and his con a-
dic s he minimali y o ρ. So he numbe o occu ences o bin uis a leas
one.
Suppose ha bo h |u|b≡p| |b≡p1. Apa om he possibili ies amb,
o m∈N, all sec ions o uand will ha e leng h s ic ly less han |u|
and | |, espec i ely. To no con adic he minimali y o ρ,wemus ha e
u=am1b=ψ−1((1,...,1,b
m1a)σ)and =am2b=ψ−1((1,...,1,b
m2a)σ)
o some m1,m
2∈N. Howe e , as no ed abo e, all sec ions o bm1aand bm2a
dec ease in leng h. Hence, |u|b≡p| |b≡pk o k>1. Now in his case, all
sec ions o uand ha e leng h s ic ly smalle han uand espec i ely.
This again con adic s he minimali y o ρ, and he p oo is comple e. 
Lemma 6.2. Fo pa p ime, he p-Basilica g oup Gis amenable bu no ele-
men a y subexponen ially amenable. In pa icula i is no elemen a y amenable.
P oo . Fo p= 2, he esul ollows om [22, P op. 13] and [9]. Hence we
assume ha pis odd. Since he p-Basilica g oup Gis a bounded au oma a
g oup, om [7] i ollows ha Gis amenable. So i suffices o show ha
Gis no elemen a y subexponen ially amenable. Since Gis weakly egula
b anch o e G, om[24, Co . 3] he esul ollows p o ided ha ψu(S G(u))
con ains G o some e ex u. We obse e ha
ψ([b−1,a]p)=(1,p−2
...,1,b
−p,b
p)andψ([a, bp]) = (1,p−1
...,1,[b, a]),
hus ψu([b−1,a]p)=aand ψu([a, bp]) = b, whe e u=xpxp. This comple es
he p oo . 
MJOM p-Basilica G oups Page 23 o 28 275
6.2. An L-P esen a ion
Fo an alphabe S, we deno e by FS he ee g oup on S. A g oup Ghas
an L-p esen a ion, also called endomo phic p esen a ion, i he e exis s an
alphabe S, se s Qand Ro educed wo ds in FS,andase Φo g oup
homomo phisms φ:FS→FSsuch ha Gis isomo phic o a g oup wi h he
ollowing p esen a ion:
S|Q∪
φ∈Φ∗
φ(R),
whe e Φ∗is he monoid gene a ed by Φ; ha is, he closu e o {1}∪Φ unde
composi ion.
An L-p esen a ion is fini e i S,Q,Ra e fini e and Φ = {φ}consis s
o jus one elemen . Fini e L-p esen a ions ha e been compu ed o he fi s
G igo chuk g oup [26], he B unne –Sidki–Viei a g oup [11], he G igo chuk
supe g oup [5], he Fab ykowski–Gup a g oup [3], he Gup a–Sidki 3-g oup
[3], and he wis ed win o he G igo chuk g oup [8].
An L-p esen a ion o he Basilica g oup is gi en in [22], and his L-
p esen a ion is no fini e, since he co esponding se Φ consis s o mo e han
one elemen . This is also he case o he L-p esen a ions o he p-gene a o
Basilica g oups ac ing on he p-adic ee and gene alised Basilica g oups; see
[30] and [29] espec i ely.
Following he s a egy in [22, Sec. 4], one can easily check ha he
p-Basilica g oups ha e he ollowing L-p esen a ion.
Theo em 6.3. Le Gbe a p-Basilica g oup, o a p ime p. The g oup Ghas
he p esen a ion
G=a, b |ξkθm([a, abl])=1 o k,m ∈N∪{0}and l∈{1,...,p−1},
whe e
ξ:a→ bpand θ:a→ abp+1
b→ ab→ b
a e endomo phisms o F{a,b}.
6.3. Vi ually Nilpo en Quo ien s and Maximal Subg oups
In his final subsec ion, we s udy nilpo ency and i ual nilpo ency o quo-
ien s o a p-Basilica g oup G, and we p o e Theo em Dabou maximal
subg oups o G. The ollowing lemma will be use ul o bo h pu poses.
Lemma 6.4. Le Gbe a p-Basilica g oup, o a p ime p.ThenGhas a p ope
quo ien isomo phic o Wp(Z).
P oo . Le L=ψ−1(G×···×G). We ha e G=Ab, and on he o he hand
ψ(A)=B×···×Bby Theo em 3.5(i). Hence ψinduces an isomo phism
be ween G/L and he semidi ec p oduc (B/G× ··· × B/G)ψ(b).
Obse e ha ψ(b)=(1,...,1,a)σac s as σon he di ec p oduc o pcopies
o B/G, and ha ψ(bp)=(a,...,a) ac s i ially. I we se N=Lbp
hen i is clea ha NGand ha G/N ∼
=Wp(Z), since B/G∼
=Zby
Theo em 3.3(ii). 
275 Page 24 o 28 E. Di Domenico e al. MJOM
Recall om Co olla y 4.3 ha he p-Basilica g oups a e jus non-sol able.
In [16, Sec. 8.3] i was shown ha he Basilica g oup is no jus non-nilpo en .
On he o he hand, by [16, Lem. 8.3.5 and P op. 8.3.6], all p ope quo ien s
o he Basilica g oup a e i ually nilpo en . We ex end hese esul s o he
p-Basilica g oups o all p imes p.
Theo em 6.5. Le Gbe a p-Basilica g oup, o a p ime p.Then:
(i) The g oup Gis no jus non-nilpo en .
(ii) E e y p ope quo ien o Gis i ually nilpo en , bu Gi sel is no
i ually nilpo en .
P oo . (i) By Lemma 6.4, he g oup Ghas a p ope quo ien isomo phic o
Wp(Z). By he main esul in [10], his w ea h p oduc is no nilpo en .
Hence, Gis no jus non-nilpo en .
(ii) F om Theo em 4.1(ii), he map ψinduces an embedding o G/G in o
he w ea h p oduc Wp(G/γ3(G)). Since he la e is i ually nilpo en ,
he quo ien G/G is also.
Now since Gis weakly egula b anch o e Gand G/G is i u-
ally nilpo en , i ollows ha e e y p ope quo ien o Gis also i u-
ally nilpo en by [17, Thm. 4.10]. On he o he hand, he g oup Gis
no i ually nilpo en by G omo ’s celeb a ed heo em [23], in ligh o
Theo em 6.1.
Le us now conside he maximal subg oups o G. We fi s p o e ha G
does no possess maximal subg oups o infini e index. The p oo is analogous
o ha o [17, Sec. 4.4], howe e wi h a necessa y change o he end o [17,
P op. 4.27]. Due o he p oo being so simila , we e e he eade o [17,
Sec. 4.4], and only eco d he e he pa ha needs o be changed.
Recall ha a subg oup Ho a g oup Gis p odense i HN =G o all
non- i ial no mal subg oups No G. Since a maximal subg oup o infini e
index is a p ope p odense subg oup, i suffices o show ha he e a e no
p ope p odense subg oups in a p-Basilica g oup G.Fo Ha p ope p odense
subg oup o G,by[17, Lem. 3.1 and Thm. 3.2], o all e ices u∈T, he
subg oup ψu(S H(u)) is a p ope p odense subg oup o G. We conside a
p odense subg oup Ho G, and seek a e ex usuch ha ψu(S H(u)) = G,
which hen p o es he heo em.
As in [17, P op. 4.27], he e is a e ex such ha ei he ab, b−1a∈
ψ (S H( )) o ba, b−1a∈ψ (S H( )). In he o me case, we ob ain a2∈
ψ (S H( )). Since pis an odd p ime, i ollows ha b−1ap∈ψ (S H( )). Now
ψ((b−1ap)p)=(a−1bp,...,a
−1bp)andψ(a−1bp)=(a,...,a,b
−1a). The e-
o e, o u= x1x1, we ha e ei he a, ab ∈ψu(S H(u)) o a, ba ∈ψu(S H(u)),
and we a e done. In he la e case, we ha e ba, b−1a∈ψ (S H( )), and so
b2∈ψ (S H( )). As be o e, we ob ain bpa=ψ−1((a,...,a,ab)) ∈ψ (S H( )).
Se ing u= x1, we see ha a, ba ∈ψu(S H(u)) and he esul ollows.
We conclude by showing he exis ence o non-no mal maximal subg oups
in he p-Basilica g oups.
MJOM p-Basilica G oups Page 25 o 28 275
P oposi ion 6.6. Le Gbe a p-Basilica g oup, o pan odd p ime. Then o
e e y p ime qsuch ha pdi ides q−1, he g oup Ghas a non-no mal subg oup
o index q.
P oo . By Lemma 6.4, he g oup Ghas a quo ien isomo phic o Wp(Z),
and so also a quo ien isomo phic o Wp(Z/qZ). Thus i suffices o find a
non-no mal subg oup o index qin he la e g oup.
Le V=Z/qZ×···×Z/qZbe he base g oup o Wp(Z/qZ). The cha -
ac e is ic polynomial co esponding o he ac ion o σon Vis Xp−1, which
by he condi ion ha pdi ides q−1, has pdiffe en oo s in Z/qZ.Le λ=1
be one o hese oo s, and le U=ube he eigenspace o λin V. Then we
can w i e V=U×K o a sui able subg oup K.I wese H=Kσ hen
Hhas index qin Wp(Z/qZ). A he same ime, His no a no mal subg oup
o Wp(Z/qZ), since o he wise [u, σ]=uλ−1= 1 belongs o U∩H=1. 
Obse e ha he e a e ac ually infini ely many non-no mal maximal
subg oups in a p-Basilica g oup, due o Di ichle ’s heo em abou p imes in
a i hme ic p og essions.
We conclude by ema king ha some o he esul s (excep hose ela ed
o he p-CSP) o his pape ca y h ough o he mo e gene al m-Basilica
g oups, o m∈Z≥2. Fo his eason, we es ic ou sel es o conside only
p ime numbe s. As al eady poin ed ou in he in oduc ion, we e e he
eade o mo e gene alisa ions o he Basilica g oup o [29, Sec. 5-8].
Acknowledgemen s
We hank D. F ancoeu , M. Pe schick and K. Rajee o help ul discussions.
Fu he mo e, we a e g a e ul o B. Klopsch o his use ul commen s and o
poin ing ou Sasse’s wo k. We also hank he e e ee o sugges ing aluable
imp o emen s o he exposi ion o he pape .
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