Profinite R-analytic groups Talde R-analitiko profinituak

PhD Thesis / Dok o ego Tesia
P o ini e R-analy ic g oups
Talde R-anali iko p o ini uak
Andoni Zozaya U suegui
Supe iso s / Zuzenda iak
Gus a o A. Fe nández-Alcobe
Jon González-Sánchez
FEBRUARY 2023 OTSAILA
(cc)2023 ANDONI ZOZAYA URSUEGUI (cc by-nc 4.0)
© Copy igh by Andoni Zozaya U suegui, 2023.
Abs ac
While he heo y o analy ic g oups is ex ensi ely de eloped o e he p-adic
numbe s, ela i ely li le is known abou g oups ha a e analy ic o e al e na i e
coefficien ings such as Zp[[ 1, . . . , m]] o Fp[[ 1, . . . , m]]. This hesis is a con i-
bu ion o he heo y o analy ic g oups o e gene al p o-pdomains by means o a
sys ema ic in es iga ion o s uc u al concep s o ha heo y, including associa ed
Lie algeb as, submani olds and analy ic quo ien s.
Mo eo e , we s udy se e al g oup- heo e ical p ope ies in he se ing o ana-
ly ic g oups, namely linea i y, wo d p oblems and ac al dimensions. Wi h ega d
o he i s wo p ope ies –linea ep esen a ions and wo d-conciseness o some
analy ic g oups– we ex end esul s ha a e well-es ablished in he p-adic case. In
con as , he s udy o he Hausdo dimension –whe e we mainly ocus on g oups
ha a e analy ic o e Fp[[ ]]– shows signi ican di e ences be ween g oups ha a e
analy ic o e he p-adic numbe s and hose ha a e analy ic o e o he coefficien
ings.
Labu pena
Talde anali ikoen eo ia aski ga a u a dago zenbaki p-adikoen gainean, baina alde-
a u a eze gu xi ezagu zen da bes e koe izien e e az un ba zuen, Zp[[ 1, . . . , m]]
edo Fp[[ 1, . . . , m]] kasu, gainean anali ikoak di en aldeei bu uz. Tesi hau p o-p
domeinu o oko en gainean anali ikoak di en aldeen ingu uko hainba eka penek
osa zen du e. Alde ba e ik, eo ia ho e ako egi u azko kon zep u ani z xeheki
az e zen di a, esa e ba e ako, elka u iko Lie en aljeb ak, azpiba ie a eak e a
za idu a anali ikoak.
Bes e alde ba e ik, alde anali ikoen es uingu uan alde eo iako zenbai p o-
pie a e az e zen di a, lineal asuna, hi z-p oblemak e a dimen sio ak alak hain
zuzen e e. Lehenengo bi gaiei dagokienez – alde anali iko ba zuen adie azpen li-
nealak e a hi z-labu asuna–, e die si iko emai zek kasu p-adikoan ezagunak di en
eo emak o oko zen di uz e. Alabaina, Hausdo en dimen sioa en az e ke an
–eskua ki Fp[[ ]]- en gainean anali ikoak di en aldee a a muga uko ga a– alde
naba mena dago zenbaki p-adikoen gainean e a bes e koe izien e e az un ba zuen
gainean anali ikoak di en aldeen a ean.
iii

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 Spanish Minis y o Science,
Inno a ion and Uni e si ies’ g an FPU17/04822. In addi ion, he au ho was
suppo ed by he Basque Go e nmen , 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.
Tesi hau Euskal He iko Unibe si a ean (UPV/EHU) ida zi da Espainiako
Zien zia, Be ikun za e a Unibe si a e Minis e ioa en FPU17/04822 lagun za-
ekin. E a be ean, au o eak Eusko Jau la i za en, IT974-16 e a IT483-22 p oiek-
uak, e a Espainiako Gobe nua en, MTM2017-86802-P e a PID2020-117281GB-
I00 p oiek uak (pa zialki EGEFk inan za u a) lagun za jaso du.
i
Con en s
I P o ini e R-analy ic g oups 1
Index o no a ion 3
In oduc ion 7
1R-analy ic g oups 11
1.1 P o-pdomains............................. 11
1.2 R-analy icg oups........................... 15
1.3 R-s anda dg oups .......................... 19
1.4 Liealgeb as.............................. 26
1.5 Cons uc ion o mani olds . . . . . . . . . . . . . . . . . . . . . . 35
1.6 No es ................................. 45
2 Linea i y o compac R-analy ic g oups 47
2.1 Linea i y o compac p-adic analy ic g oups . . . . . . . . . . . . 49
2.2 Change o p o-pdomains....................... 51
2.3 Disc imina ion ............................ 57
2.4 Model heo y and he linea i y o compac R-analy ic g oups . . . 60
2.5 No es ................................. 64
3 Hausdo dimension in compac R-analy ic g oups 65
3.1 Hausdo and box dimension . . . . . . . . . . . . . . . . . . . . 67
3.2 Hausdo dimension o submani olds . . . . . . . . . . . . . . . . 78
3.3 Abelian compac R-analy ic g oups . . . . . . . . . . . . . . . . . 82
3.4 Compac Fp[[ ]]-analy ic g oups . . . . . . . . . . . . . . . . . . . 83
3.5 Classical Che alley g oups . . . . . . . . . . . . . . . . . . . . . . 88
3.6 No es ................................. 94
ii
4 Wo ds in compac R-analy ic g oups 95
4.1 Conciseness in R-s anda d g oups . . . . . . . . . . . . . . . . . . 100
4.2 Conciseness in compac Fp[[ ]]-analy ic g oups . . . . . . . . . . . 101
4.3 Conciseness in compac R-analy ic g oups . . . . . . . . . . . . . 105
4.4 S ong conciseness in R-analy ic g oups . . . . . . . . . . . . . . . 107
4.5 No es ................................. 108
Appendix A Ado’s Theo em o e p incipal ideal domains 111
A.1 In oduc ion.............................. 111
A.2 Adjoin and egula ep esen a ions . . . . . . . . . . . . . . . . . 114
A.3 Ado’sTheo em ............................ 116
A.4 No es ................................. 127
Bibliog aphy 129
Index 135
II Talde R-anali iko p o ini uak 139
No azio indizea 141
Sa e a 145
1 Talde R-anali ikoak 149
1.1 P o-pdomeinuak ........................... 149
1.2 Talde R-anali ikoak.......................... 154
1.3 Talde R-es anda ak ......................... 157
1.4 Lie enaljeb a............................. 164
1.5 Ba ie a een e aikun za . . . . . . . . . . . . . . . . . . . . . . . . 174
1.6 Oha ak................................ 184
2 Lineal asuna alde R-anali iko inkoe an 187
2.1 Talde p-adiko anali iko inkoen lineal asuna . . . . . . . . . . . . 189
2.2 P o-pdomeinualdake a ....................... 191
2.3 Disk iminazioa ............................ 198
2.4 E edu eo ia e a alde R-anali iko inkoen lineal asuna . . . . . . 201
2.5 Oha ak................................ 205
3 Hausdo en dimen sioa alde R-anali iko inkoe an 207
iii
Pa I
P o ini e R-analy ic g oups
1

No azio indizea
Con en ions. We assume ha all ings a e commu a i e and wi h iden i y.
Mo eo e , h oughou all he hesis ps ands o a p ime numbe , and q o a
powe o p.
No a ion. Mos o he no a ion is s anda d, excep o A(n),which s ands
o he n h Ca ensian powe o he se A(we shall use his no a ion, since i
will be common o w i e exp essions o he o m (an)(m) o an ideal a, so i is
con enien o dis inguish he n h Ca esian powe a(n) om he n h powe ideal
an). Mo eo e , i :A→Bis a map, we deno e by (n) he map A(n)→B(n),
(a1, . . . , an)7→ ( (a1), . . . , (an)) .
The emaining e minology is lis ed below:
N he na u al numbe s
N0 he na u al numbe s oge he wi h 0
Z he in ege s
Zp he p-adic in ege s
Q he a ional numbe s
Qp he p-adic numbe s
R he eal numbe s
R≥0 he non-nega i e eal numbe s
C he complex numbe s
Fq he ini e ield o size q
loga he loga i hm o basis a
3
P(A) he pa s o A
A×B he Ca esian p oduc o Aand B
Qi∈IAi he Ca esian p oduc o he di ec ed amily {Ai}i∈I
A(k) he k h Ca esian powe o A
A⊕B he di ec sum o Aand B
A⋉B he semidi ec p oduc o Aand B
Li∈IAi he di ec sum o he di ec ed amily {Ai}i∈I
Qi∈IAi/U he ul ap oduc o he di ec ed amily {Ai}i∈I
AU he ul apowe o A
A⊆B A is a subse o B
A⊆oB A is an open subse o B
A⊆cB A is a closed subse o B
A≤B A is a subg oup o B
A≤oB A is an open subg oup o B
A≤cB A is a closed subg oup o B
A⊴B A is a no mal subg oup o B
A⊴oB A is a no mal open subg oup o B
A⊴cB A is a no mal closed subg oup o B
Acha B A is a cha ac e is ic subg oup o B
Le Gbe a g oup and le g, x, y ∈G:
xyy−1xy
[x, y]x−1y−1xy
[x1, . . . , xn] [[x1, . . . , xn−1], xn]
Z(G) he cen e o G
CG(g) he cen alise o g∈G
G′= [G, G] he de i ed subg oup o G:h[x, y]|x, y ∈Gi.
Gn he n h powe subg oup: hgn|g∈Gi
[H1, . . . , Hn]h[h1, . . . , hn]|hi∈Hii
cy he conjuga ion isomo phism G→G, x 7→ xy
Ly he le mul iplica ion map G→G, x 7→ yx
Ry he igh mul iplica ion map G→G, x 7→ xy
4
ke he ke nel o he g oup ( esp. ing) homomo phism
im he image o he g oup ( esp. ing) homomo phism
Hom(A, B)g oup ( esp. ing) homomo phisms :A→B.
Le Qbe a ing:
U(Q) he uni s o Q
cha Q he cha ac e is ic o Q
dimK ull Q he K ull dimension o Q
F ac(Q) he ac ion ield o he in eg al domain Q
Q[[ 1, . . . , m]] he ing o o mal powe se ies in m a iables and coe -
icien s in Q
Mn×m(Q)n×mma ices wi h coefficien s in Q
Mn(Q)n×nma ices wi h coefficien s in Q
GLn(Q)gene al linea g oup wi h coefficien s in Q
SLn(Q)special linea g oup wi h coefficien s in Q
SOn(Q)special o hogonal g oup wi h coefficien s in Q
Spn(Q)symplec ic g oup wi h coefficien s in Q
Un(Q)uppe iangula ma ices wi h coefficien s in Q
Le Kbe a ield:
Kalg he algeb aic closu e o K
dimKK- ec o space dimension
Le Mbe an R-analy ic mani old:
dimxM he analy ic dimension o he mani old Ma x
dim Ganaly ic dimension o he analy ic g oup G
Le Gbe a linea algeb aic g oup:
Ru(G)unipo en adical
k ank o a ma ix
es. k esidual ank o a ma ix
de de e minan o a ma ix
ace o a ma ix
5
DUdi e en ial o a uple o powe se ies
JxF he Jacobian ma ix o Fa x
k M he ank o a ee module M
IsoM(N)isola o o Nin M
EndR(M)endomo phisms o he R-module M
hdim Hausdo dimension
hspec Hausdo spec um
hdims s anda d Hausdo dimension
hspecs s anda d Hausdo spec um
bdim box dimension
bdims s anda d box dimension
lbdim lowe box dimension
lbdims s anda d lowe box dimension
ubdim uppe box dimension
ubdims s anda d uppe box dimension
w{G} he se o w- alues o G
w(G) he e bal subg oup o w
w∗(G) he ma ginal subg oup o w
X∗ℓ he se o p oduc s o `elemen s o X∪X−1∪{1}
deg Ldeg ee o he Lie algeb a L
deg φdeg ee o he ep esen a ion φ
Z(L)cen e o he Lie algeb a L
Rn(L)nilpo en adical o L
Rs(L)soluble adical o L
TR(L) o sion algeb a o L
UR(L)uni e sal en eloping algeb a o L
De R(L)de i a ions o L
Cen (L)cen oid o L
6

“I end o hink oo much.
My g ea es successes came om decisions I made when I
s opped hinking and simply did wha el igh .
E en i he e was no good explana ion o wha I did. [...]
E en i he e we e e y good easons o me no o do wha
I did.”
K o he, (Pa ick Ro h uss, The Name o he Wind)
In oduc ion
This disse a ion is a monog aph on analy ic g oups. These comp ise an abs ac
g oup oge he wi h an analy ic mani old s uc u e o e a con enien opological
ing in such a way ha bo h s uc u es a e compa ible, in he sense ha he
mul iplica ion map and he in e sion map a e analy ic unc ions.
The heo y o analy ic Lie g oups o e opological ields is a sou ce o examples
o p o ini e g oups. O cou se, o e he classical ields Rand C,analy ic Lie g oups
canno be p o ini e unless hey a e ini e, as hey should be bo h compac and
locally homeomo phic o a o ally disconnec ed subse o C(n). Howe e , p o ini e
analy ic g oups migh a ise i he unde lying g oup o he base ing is a p o ini e
g oup in i s own igh . Fo ins ance, in he ea ise G oupes analy iques p-adiques
[50], Laza d ex ensi ely s udied he p-adic analy ic g oups, ha is, analy ic Lie
g oups o e he ield o p-adic numbe s Qp– equi alen ly, o e he alua ion ing
o p-adic in ege s Zp–; and he showed ha compac p-adic analy ic g oups a e
ac ually p o ini e g oups.
In addi ion o he o iginal pu ely analy ic poin o iew, he e a e se e al al-
e na i e cha ac e isa ions o p-adic analy ic g oups (we e e o [24, In e lude A]
o a comp ehensi e lis ). Among hese cha ac e isa ions, in o de o p o e wha
could be ega ded as Hilbe ’s 5 h p oblem o p-adic analy ic g oups, Laza d him-
sel p o ed ha compac p-adic analy ic g oups a e p ecisely he p o ini e g oups
ha a e i ually p o-pg oups o ini e ank– hese a e he p o ini e g oups which
con ain a p o-pg oup o ini e index such ha all he subg oups o ha p o-p
g oup a e ini ely gene a ed, and such ha he necessa y numbe o gene a o s is
bounded. The heo y o p-adic analy ic g oups has since e ol ed in o a ich a ea,
7
and a numbe o exci ing p ope ies ha e been es ablished: all o hem, wi h he
excep ion o he g oup Zp, sa is y Golod-Sha a e ich inequali y (Lubo zky [52]),
hey ha e polynomial subg oup g ow h (Lubo zky and Mann [53]), hey a e e -
bally ellip ic (Jaikin-Zapi ain [44]), e c.
Fu he mo e, i one s a s wi h a gene al opological ing Rand de ine analy ic
g oups by analogy, he concep o p-adic analy ic g oup is gene alised o ha o
analy ic g oup o e R, which he eina e will be e e ed o as R-analy ic g oup.
The eby, Bou baki (o be e said, he bou baquis s) [11] and Se e [68] s udied
analy ic g oups o e he local ield Fp(( )),i.e., he posi i e cha ac e is ic coun-
e pa o p-adic analy ic g oups. Mo eo e , he second edi ion o he celeb a ed
book Analy ic p o-p g oups [24] was p o ided wi h a u he chap e conce ning
analy ic g oups o e gene al p o-pdomains and hus ook he i s s eps o his
b oade heo y. We ecall ha a p o-pdomain is a local Noe he ian in eg al
domain Rwhich is comple e wi h espec o he me ic de ined by he maximal
ideal and whose esidue ield is ini e o cha ac e is ic p(Sec ion 1.1 is de o ed o
explo ing hese ings and he concep s equi ed in hei de ini ion). These mo e
gene al g oups possess in e es ing algeb aic p ope ies (see [14], [42], [43], [45] and
[54]), albei no as hose enjoyed by he p-adic analy ic g oups. Wo se, he e is
no cha ac e isa ion, e en a a conjec u al le el, o R-analy ic g oups pu ely in
g oup- heo e ic e ms.
This hesis aims o de elop u he he heo y men ioned abo e. I s objec i es
a e wo old: on he one hand, o ad ance in he sys ema ic s udy o analy ic
g oups o e gene al p o-pdomains, which cons i u es a somewha bela ed sequel
o [24, Chap e 13]; and, on he o he hand, o p o ide his heo y wi h new e-
sea ch esul s. Those mainly gene alise known p ope ies o p-adic o he b oade
con ex o R-analy ic g oups. We poin ou ha , mo eo e , he p-adic case is usu-
ally a undamen al ing edien o ou p oo s.
We now ou line he con en s in g ea e de ail: Chap e 1 is an in oduc ion o
R-analy ic g oups aking [24, Chap e 13] as a s a ing poin . We will pay special
a en ion o s anda d g oups, which pe haps cons i u e he main example o R-
analy ic g oups, as well as o he Lie algeb a associa ed wi h hem. In iew o
he ac ha many elemen a y concep s conce ning analy ic g oups had s ill o be
de eloped, we shall es ablish he machine y we will use h oughou . Fo ins ance,
in [24, p. 349], he au ho s highligh ed ha “ o mo e gene al analy ic g oups o
he p esen chap e , such concep s [o submani old and quo ien mani old] would
8
need o be de eloped”, which is p ecisely wha we y o do in Sec ion 1.5.
Chap e 2 is abou linea i y. Speci ically we show ha when Ris a p o-p
domain o cha ac e is ic ze o, e e y compac R-analy ic g oup is linea , i.e., i
can be embedded in he ene al linea g oup GLn(K) o a sui able ield K. This
pa ially answe s a ques ion om Lubo zky and Shale (see Ques ion 2 in page
311 o [54]). The p oo we shall gi e is nea and based on he linea i y o p-adic
analy ic g oups. Besides, i has a model- heo e ic la ou , as we embed he g oup
in ques ion in a con enien ul apowe o GLn(Zp).Since model heo y is no a
cen al opic o his hesis, he p oo is w i en, as a as possible, in such a way
ha p io knowledge is no equi ed.
Chap e 3 is de o ed o he Hausdo dimension in compac R-analy ic g oups.
We shall show ha in a compac R-analy ic g oup G, he e exis s a me ic ha
encodes i s analy ic s uc u e, and we will ecall how o de ine he Hausdo di-
mension co esponding o ha me ic, namely a ac al dimension hdim: P(G)→
[0,1]. The chap e consis s o wo main pa s. Fi s ly, we shall s udy he ela ion-
ship be ween he analy ic and he Hausdo dimensions o a closed submani old.
This s udy is based on he a icle [27] by Fe nández-Alcobe , Giannelli and Gon-
zález-Sánchez. Secondly, we will ocus mainly on he case R=Fp[[ ]],and desc ibe
he Hausdo spec um o compac Fp[[ ]]-analy ic g oups, namely he se
hspec(G) = {hdim(H)|H≤Gis closed}.
Chap e 4 is conce ned wi h wo ds. A wo d is no hing bu an elemen w=
w(x1, . . . , xk)o he ee g oup F(x1, . . . , xk)in k-gene a o s; and gi en a g oup G,
i na u ally de ines a map w:G(k)→G, which sends (g1, . . . , gk) o he elemen
o Gob ained by subs i u ing gi o xiin w. In he se ing o analy ic g oups,
we will s udy some p oblems ega ding wo ds ha we e o iginally p oposed by
P. Hall [33]. In keeping wi h his e minology, we will p o e ha in a compac
R-analy ic g oup e e y wo d is concise, i.e., whene e im w, he image o he map
win G, is ini e, he e bal subg oup w(G) = him wiis also ini e.
Finally, Appendix A con ains a p oo o Ado’s Theo em o Lie algeb as o e
p incipal ideal domains ha a e addi ionally ee modules, since we will need his
e sion o he heo em in Chap e 2.
A he end o each chap e , he e is a sec ion o No es, whe e we de ail he
au ho ’s o iginal con ibu ions, o we make a ious commen s.
This manusc ip in ends o be s and-alone, and acco dingly, mos concep s a e
ho oughly in oduced. Howe e , amilia i y wi h p o ini e and p o-pg oups is
assumed, and i necessa y he eade is e e ed o [24, Chap e 1].
9
10
(U2, φ2, n2)a e wo R-cha s such ha U1∩U26=∅, hen n1=n2.The e o e, he
dimension o Ma xis well-de ined as he common dimension o he R-cha s o
x, and i will be deno ed by dimxM.
I is wo h ema king ha o e p o-pdomains, unlike in eal o complex man-
i olds, he dimension is no a opological p ope y, bu an analy ic p ope y de-
e mined by he cha . Fo example, Zpand Z(2)
pa e isomo phic o each o he as
opological g oups.
(i ) An R-analy ic mani old is said o be pu e when dimxMis cons an o all
x∈M.
Mimicking De ini ion 1.9, we can de ine R-analy ic maps be ween R-analy ic
mani olds:
De ini ion 1.12. Le Mand Nbe R-analy ic mani olds. A unc ion F:N→M
is R-analy ic a x∈Ni he e exis s an R-cha (U, φ, n)o xin Nand an R-cha
(V, ψ, m)o F(x)in Msuch ha F−1(V)is open in Nand
ψ◦F◦φ−1|ϕ(U∩F−1(V)) :φU∩F−1(V)→R(m)(1.2)
is an R-analy ic map (acco ding o De ini ion 1.9 (i)). Simila ly, Fis an R-analy ic
map when i is R-analy ic a all he poin s o N.
Mo eo e , i is habi ual o call he map (1.2) by Fin coo dina es, and on
occasions we will use his e m in o mally wi hou speci ying he R-cha s we
wo k wi h. Fu he mo e, we duly e e o Fas s ic ly analy ic a S⊆N, i in
De ini ion 1.12 we can ake he R-cha s such ha S⊆U∩F−1(V)and he e
exis s H∈R[[X1, . . . , Xn]](m)such ha
ψ◦F◦φ−1(φ(x)) = H(x)∀x∈S.
Adop ing a e m used by Se e [68] we will also de ine he ollowing:
De ini ion 1.13. Le Mbe an R-analy ic mani old, U⊆oMand x∈U. A amily
o R-analy ic unc ions F={ i:U→R}n
i=1 is a coo dina e sys em o Ma x, i
he e exi s U′⊆oUsuch ha x∈U′and (U′, F|U′),whe e F= ( 1, . . . , n),is an
R-cha .
Obse e om he de ini ion ha whene e Fis a coo dina e sys em a x, hen
i is so locally a ound x.
17

Examples 1.14. (i) The canonical example o an R-analy ic mani old is M=
mN(n)whe e N, n ∈N.The canonical coo dina e sys em is {πi}n
i=1 whe e πi
is he i h p ojec ion map πi:mN(n)→mN.Besides, o e e y x∈M, he
componen s o he ansla ion map x:M→M, y 7→ y+x, i.e. {πi◦ x}n
i=1 ,
a e also a coo dina e sys em.
(ii) Le K= F ac(R),and endow K(n)wi h he opology gi en by he neigh-
bou hood basis nk+mN(n)oN∈No k∈K(n).Then K(n)is an R-analy ic
mani old wi h espec o he a las {(Uk, ϕk, n)}k∈K(n)whe e Uk=k+R(n)
and ϕk:Uk→R(n), x 7→ x−k( ecall ha he opology in K(n)we a e im-
posing does no in gene al coincide wi h he na u al opology on he ac ion
ield; in u h i ne e does unless Ris a PID).
(iii) The se o ma ices Mn×m(m), which can be na u ally iden i ied wi h m(nm),
is clea ly an R-analy ic mani old, and so is he gene al linea g oup GLn(R)
wi h espec o he a las {(UA, φA, n2)}A∈GLn(R),whe e UA=A+ Mn(m)
and φA:UA→Mn(m), A +M7→ M.
(i ) Le Mand Nbe R-analy ic mani olds, wi h a lases {(Ui, φi, ni)}i∈Iand
{(Vj, ψj, mj)}j∈J.The di ec p oduc M×Nis an R-analy ic mani old wi h
espec o he a las {(Ui×Vj, φi×ψj, ni+mj)}i∈I, j∈J.
The ollowing elemen a y p ope ies can be easily deduced om Lemma a 1.5
and 1.7.
Lemma 1.15. (i) E e y R-analy ic map is con inuous.
(ii) (c . [24, Lemma 13.4]) The composi ion o wo R-analy ic maps is an R-
analy ic map.
(iii) The composi ion o wo s ic ly R-analy ic maps is a s ic ly R-analy ic map.
We inish wi h he main de ini ion:
De ini ion 1.16. An R-analy ic g oup is a opological g oup G ha is an R-
analy ic mani old such ha
(i) he mul iplica ion map m:G×G→G, (g, h)7→ g·hand
(ii) he in e sion map ι:G→G, g 7→ g−1
a e R-analy ic maps.
18
Pa icula ly, Zp-analy ic g oups a e he o emos example o hese g oups, as
well as he ge m o he abo e de ini ion. In he li e a u e hese g oups ha e been
e e ed o as p-adic analy ic g oups.
1.3 R-s anda d g oups
The R-s anda d g oups a e a no ewo hy amily o R-analy ic g oups.
De ini ion 1.17. An R-s anda d g oup o le el Nand dimension dis an R-
analy ic g oup Swi h a global cha {(S, φ, d)}such ha
(i) φ(S) = mN(d),
(ii) φ(1) = 0and
(iii) o all j∈ {1, . . . , d} he e exis s a o mal powe se ies Fj∈R[[X1, . . . , X2d]]
such ha
φ(xy) = (F1(φ(x), φ(y)), . . . , Fd(φ(x), φ(y))) ∀x, y ∈S.
Any uple o powe se ies F= (F1, . . . , Fd)sa is ying condi ion (iii) o he abo e
de ini ion mus ce ainly also sa is y
(F1) F(X,0) = Xand F(0,Y) = Y(in pa icula , each Fihas cons an e m
equal o ze o), and
(F2) F(X,F(Y,Z)) = F(F(X,Y),Z),
(he e X,Yand Za e d- uples o a iables) as hey a e s aigh o wa d conse-
quences o he ac ha φ(1) = 0and he associa i i y o he g oup law. Con-
e sely, any uple F∈R[[X1, . . . , X2d]](d) ha sa is ies he p eceding wo con-
di ions endows mN(d), o any N∈N,wi h an R-s anda d g oup s uc u e.
Acco dingly, a uple o powe se ies sa is ying (F1) and (F2) is said o be a d-
dimensional o mal g oup law, and he e exis s a o mal in e se o i , namely a
uple o powe se ies I= (I1, . . . , Id)∈R[[X1, . . . , Xd]](d)such ha
F(I(X),X) = F(X,I(X)) = 0
(see [24, P oposi ion 13.16 (ii)]). Some imes we will deno e an R-s anda d g oup
by (S, φ)o (S, F) o emphasise he ôle o he homeomo phism o he o mal
19
g oup law.
F om (F1) we can u he conclude ha
F(X,Y) = X+Y+B(X,Y) + G(X,Y),(1.3)
whe e Bis bilinea and whe e all he monomials in ol ed in Gha e o al deg ee
a leas 3.Mo eo e , e e y monomial in ol ed in Band Gcon ains a non-ze o
powe o Xiand Yj o some i, j ∈ {1, . . . , d}.
Rema k 1.18 (c . [68, Pa II, Chap e IV, §7]).S a ing om (1.3) we can
ob ain simila exp essions o he o mal in e se and he conjuga ion maps. In
e ec , i is a ou ine exe cise o e i y ha i Iis he o mal in e se o Fin (1.3),
hen
I(X) = −X+B(X,X) + ˜
G(X),(1.4)
whe e e e y monomial in ol ed in ˜
G(X)has o al deg ee a leas 3; and conse-
quen ly, he conjuga ion map has he nex o m in coo dina es:
F(I(Y),F(X,Y)) = X+B(X,Y)−B(Y,X) + ˆ
G(X,Y),(1.5)
whe e e e y monomial in ol ed in ˆ
Ghas o al deg ee a leas 3and i con ains a
non-ze o powe o Xiand Yj o some i, j ∈ {1, . . . , d}.
Examples 1.19. (i) The addi i e g oup mN(d)is an R-s anda d g oup wi h
he addi i e o mal g oup law F(X,Y) = X+Y.
(ii) The mul iplica i e R-s anda d g oup G= 1 + mN,which is so wi h espec
o he global cha φ:G→mN,1 + m7→ mand he mul iplica i e o mal
g oup law F(X, Y ) = X+Y+XY.
(iii) We can gene alise he p e ious wo o mal g oup laws: i is easy o e i y ha
any 1-dimensional polynomial o mal g oup law has he o m Fc(X, Y ) =
X+Y+cXY o some c∈R(c . [9, Co olla y 2.2.4]).
(i ) Le GL1
n(R)be he ke nel o he modulo m educ ion map GLn(R)→
GLn(R/m), ha is, GL1
n(R) = In+ Mn(m).This g oup is R-s anda d wi h
he n2-dimensional R-cha gi en by he assigna ion In+A7→ Aand he
o mal g oup law F(X,Y) = X+Y+X·Y,whe e Xand Ys and o
n-by-nma ices o inde e mina es.
20
( ) In posi i e cha ac e is ic p, we ha e he 2-dimensional o mal g oup law
F(X1, X2, Y1, Y2) = (X1+Y1, X2+Y2+Xp
1Y2),
which is a ibu ed o Che alley (c . [18, Chap e II, §10, Example V]).
Gi en an R-s anda d g oup (S, φ)we can de ine he R-s anda d il a ion se ies
by
Sn:= φ−1mN+n(d)∀n∈N0,(1.6)
whe e Nand ds and o he le el and he dimension o S. I eadily ollows om
(1.5) ha Snis an open no mal subg oup o S, o e e y n∈N.Mo eo e , since
Ris compac , Sis a compac opological g oup, so Snhas ini e index in S. We
can speci y be e :
Lemma 1.20 (c . [27, Lemma 2.3]).Le (S, φ, d)be an R-s anda d g oup o le el
N. Then,
|S:Sn|=mN(d):mN+n(d),
whe e he la e s ands o he index as addi i e g oups.
P oo . F om (1.3),
φ(x) = φxy−1y=Fφxy−1, φ(y)=φ(xy−1) + φ(y) + Hφxy−1, φ(y),
whe e all he monomials in ol ed in H(X,Y)ha e o al deg ee a leas 2and
con ain a non-ze o powe o Xiand Yj o some i, j ∈ {1, . . . , d}.
Thus, i φ(x)−φ(y)∈mK(d) mK+1(d), hen
φxy−1+Hφxy−1, φ(y)=φ(x)−φ(y)∈mK(d) mK+1(d),
and so φ(xy−1)∈mK(d) mK+1(d).Con e sely, i φ(xy−1)∈mK(d)
mK+1(d), hen
φ(x)−φ(y)≡φxy−1mod mK+1(d),
and so φ(x)−φ(y)∈mK(d) mK+1(d).
In o he wo ds,
xy−1∈Sn⇐⇒ φ(x)−φ(y)∈mN+n(d).
21
The e o e, since log|R/m||mi:mi+1|= dimR/m(mi/mi+1),we ha e ha
|S:Sn|=pcd Pn−1
i=0 dimR/m(mN+i/mN+i+1),(1.7)
whe e pcis he size o he esidue ield R/m,and pa icula ly, S/Snis a ini e
p-g oup. Fo asmuch as Sis a compac opological g oup wi h an open neigh-
bou hood sys em o he iden i y {Sn}n∈Nsuch ha S/Snis a ini e p-g oup o
all n∈N, we conclude ha any R-s anda d g oup is ac ually a coun ably based
p o-pg oup.
By he nex esul he s udy o R-analy ic g oups can be educed, o some
ex en , o R-s anda d g oups.
Lemma 1.21 (c . [24, Theo em 13.20]).Le Gbe an R-analy ic g oup and (U, φ, d)
an R-cha o he iden i y. Then, Ucon ains an open R-s anda d subg oup o
dimension dim1G. In pa icula , e e y R-analy ic g oup con ains an open R-
s anda d subg oup.
P oo . We can assume ha φ(1) = 0by composing wi h a con enien ansla-
ion. Since he mul iplica ion map mand he in e sion map ιa e R-analy ic
a 1, he e exis s N∈Nsuch ha mN(d)⊆φ(U)and some powe se ies
Fj∈Λ0[[X1, . . . , X2d]] and Ij∈Λ0[[X1, . . . , Xd]], j ∈ {1, . . . , d},such ha
φ◦m◦(φ, φ)−1(x, y) = (F1(x, y), . . . , Fd(x, y)) ∀x, y ∈mN(d)
and
φ◦ι◦φ−1(x) = (I1(x), . . . , Id(x)) ∀x∈mN(d)
(ac ually, i is sufficien o conside he mul iplica ion, since in mN(d) he uple
o powe se ies Iis no hing bu he o mal in e se o F).
In pa icula , i F= (F1, . . . , Fd)and I= (I1, . . . , Id), hen
0=φ(1) = F(φ(1), φ(1)) = F(0,0),
and
0=φ(1) = I(φ(1)) = I(0),
so each powe se ies Fjand Ijhas cons an e m equal o ze o. The e o e, mN(d)
is closed wi h espec o bo h Fand I.Thus,
H:= φ−1mN(d)⊆U
22

is an open subg oup o Gsuch ha
φ(xy) = (F1(φ(x), φ(y)), . . . , Fd(φ(x), φ(y))) ∀x, y ∈H.
When Ris no a PID, Λ0[[X]] = R[[X]] so (H, φ|H)is an R-s anda d g oup o
le el N, dimension dand o mal g oup law F.Suppose now ha Ris a PID wi h
uni o mise πand ac ion ield K, acco ding o (1.3),
Fj(X,Y) = X+Y+X
α,β∈N(d)
0 {0}
|α|+|β|≥2
aj,α,βXαYβ∈K[[X,Y]].
Since Fis con e gen in mN(d), he e exis s L∈N0such ha aj,α,βπL(|α|+|β|)∈R.
De ine he powe se ies
Fj(X,Y) := π−LFjπLX, πLY=X
α,β∈N(d)
0 {0}
π−Laj,α,βπL(|α|+|β|)XαYβ,
which is a powe se ies wi h coefficien s in R. Consequen ly,
¯
H:= φ−1mN+L(d)
is an open subg oup o G, which is an R-s anda d g oup wi h he R-cha ψ:¯
H→
mN(d), h 7→ π−Lφ(h)and he o mal g oup law F= (F1, . . . , Fd).Indeed,
ψj(xy) = π−Lφj(xy) = π−LFj(φ(x), φ(y)) (1.8)
=Fjπ−Lφ(x), π−Lφ(y)=Fj(ψ(x), ψ(y)),∀x, y ∈¯
H.
An open R-s anda d g oup (S, φ, d)can be used o ob ain a na u al a las o G.
Indeed, conside {(xS, φx, d)}x∈G,whe e φx:xS →mN(d)is de ined by φx(y) =
φ(x−1y).Those R-cha s a e compa ible, as
φx◦φ−1
y=φ◦Lx−1◦Ly◦φ−1=φ◦Lx−1y◦φ−1
and Lx−1yis R-analy ic. Mo eo e , his a las is compa ible wi h he ini ial R-
analy ic s uc u e o G.
As a by-p oduc we obse e ha
Co olla y 1.22. Le Gbe an R-analy ic g oup. Then dimxGis cons an o all
x∈G.
23
P oo . We ha e indica ed in Rema k 1.11, al hough i will be p o ed in Co olla y
1.30, ha dimxGis independen o he R-cha s. By Lemma 1.21, he e exis s
an open R-s anda d subg oup S≤Go dimension d:= dim1G, and he R-a las
{(xS, φx, d)}x∈Gshows ha dimx(G) = d o all x∈G.
This common alue is e e ed o as he (analy ic) dimension o an R-analy ic
g oup, and on accoun o i , we will w i e he R-cha s simply as he pai (U, φ).
1.3.1 R-s anda d g oups and g oup ope a ions
By i ue o Lemma 1.21 e e y R-analy ic g oup con ains an open p o-psubg oup,
so compac R-analy ic g oups a e p o ini e g oups. Fu he mo e, assuming com-
pac ness Lemma 1.21 can be s eng hened:
Lemma 1.23. Le Rbe a p o-pdomain ha is no a PID. A compac R-analy ic
g oup Gcon ains an open no mal R-s anda d subg oup S such ha o all g∈G
he conjuga ion map cg:S→Sis s ic ly R-analy ic.
P oo . Le Gbe a compac R-analy ic g oup o dimension d. By Lemma 1.21,
he e exis s a ini e index R-s anda d subg oup (H, φ)o dimension d, le el N,
o mal g oup law Fand o mal in e se I.Le Tbe a le ans e sal o Hin G.
Since he conjuga ion maps a e R-analy ic a 1, o each ∈T he e exis s an
in ege N ≥Nand some powe se ies C
j∈R[[X1, . . . , Xd]], j ∈ {1, . . . , d}, such
ha
φx =C
1(φ(x)), . . . , C
d(φ(x))∀x∈φ−1mN (d).
Le L= max ∈TN , since mL(d)is closed wi h espec o he uples o powe
se ies Fand I, hen S:= φ−1mL(d)is an open R-s anda d subg oup. Mo e-
o e ,
C
j(0) = C
j(φ(1)) = 0,
so each C
jhas cons an e m equal o ze o, and hus, Sis closed wi h espec o
he conjuga ion by e e y ∈Tand c :S→Sis s ic ly analy ic in S o e e y
∈T. In addi ion, whene e h∈Hand x∈S hen
φxh=F(I(φ(h)),F(φ(x), φ(h))),
so Sis closed wi h espec o conjuga ing by hand ch:S→Sis s ic ly analy ic
in S. Since e e y elemen g∈Gcan be w i en as h whe e ∈Tand h∈H,
hen Sis closed wi h espec o he conjuga ion wi h g–i.e. Sis no mal in G–
and cg=ch◦c is s ic ly analy ic in S.
24
Fo he emainde o he sec ion, we keep he no a ion used h oughou he
p e ious p oo and we eco e he a las associa ed o (S, φ),acco dingly cgis
gi en in coo dina es by he uple o powe se ies Cg= (Cg
1, . . . , Cg
d)o he p oo ,
and we can gi e an explici desc ip ion o he g oup ope a ions in G:
Lemma 1.24. Le Gbe an R-analy ic g oup and le Sbe a open no mal R-
s anda d subg oup such ha o all g∈G he conjuga ion map cg:S→Sis
s ic ly analy ic. Suppose ha wi h espec o he R-a las induced by S he map cg
is gi en in coo dina es by he uple o powe se ies Cg.Le , ∈G.
(i) The in e se in S is gi en in coo dina es by he uple o powe se ies C −1◦I.
Tha is,
φ −1x−1= (C −1◦I) (φ (x)) ∀x∈ S.
(ii) The mul iplica ion in S × S is gi en in coo dina es by he uple o powe
se ies F(C (X),Y).Tha is,
φ (xy) = F(C (φ (x)), φ (y)) ∀x∈ S, y ∈ S.
P oo . (i) Take x= x ∈ S, hen
C −1(I(φ (x))) = C −1φx−1=φx−1 −1=φ −1x−1.
(ii) Take x= x ∈ S and y= y ∈ S, hen
φ (xy) = φ(x y) = F(C (φ(x)), φ(y)) = F(C (φ (x)), φ (y)).
Finally, i we ix a le ans e sal T o Sin G, we can wo k solely wi h he a las
{( S, φ )} ∈T.In ac , o x, y ∈Gsuch ha xS =yS, le Ay
x:mN(d)→mN(d)
be he R-analy ic homeomo phism φy◦φ−1
x.Since φy=Ay
x◦φx,Lemma 1.24(i)
is es a ed as
φ x−1= (A
−1◦C −1◦I) (φ (x)) ∀x∈ S,
whene e S = −1S. Di o mul iplica ion, i.e.
φp(xy) = Ap
(F(C (φ (x)), φ (y))) ∀x∈ S, y ∈ S,
whene e S =pS.
25
1.4 Lie algeb as
Any R-s anda d g oup is associa ed wi h a so-called Lie algeb a. The objec i e o
his sec ion is o desc ibe his cons uc ion by ollowing [24, Sec ion 13.3], as well
as o ep oduce o gene al p o-pdomains he esul s in [68, Pa II, Chap e III,
§ 10].
Le (S, F)be an R-s anda d g oup o le el Nand dimension dand o mal
in e se I.In acco dance wi h he no a ion in (1.3) we can associa e o F he Lie
b acke
[X,Y]F:= B(X,Y)−B(Y,X),
which will be simply deno ed by [·,·]when he e is no isk o con usion. Le us
e i y ha [·,·]Fis an ac ual o mal Lie b acke (see Appendix A o he p ecise
de ini ion o Lie b acke ). Ob iously, i is bilinea and [X,X]=0.Fu he i
sa is ies he Jacobi iden i y:
Lemma 1.25 (c . [24, Lemma 13.24] and [68, Pa II, Chap e IV, § 7.6]).Le
X,Yand Zbe d- uples o a iables. Then,
[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0.
P oo . The esul ollows om he so-called Hall-Wi iden i y (c . [68, Pa I,
P oposi ion 1.1]), he non-commu a i e e sion o Jacobi’s iden i y. Acco dingly,
e e y g oup Gsa is ies he iden i y:
[xy,[y, z]] [yz,[z, x]] [zx,[x, y]] = 1.(1.9)
He eina e , O(n)s ands o o mal powe se ies in wo d- uples o a iables
Xand Y,all whose monomials ha e deg ee a leas nand con ain a non-ze o
powe o Xiand Yj o some i, j ∈ {1, . . . , d}.Mo eo e , he o mal conjuga ion
F(I(Y),F(X,Y)) will be abb e ia ed by XY,and i
C(X,Y) := F(I(X),F(I(Y),F(X,Y)))
is he o mal commu a o , acco ding o (1.3) - (1.5), we ha e ha XY=X+O(2)
and [X,Y] = C(X,Y) + O(4).The e o e,
[X,[Y,Z]] = CXY,C(Y,Z)+O(4).
Thus,
0=FCXY,C(Y,Z),FCYZ,C(Z,X),CZX,C(X,Y)
= [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] + O(4),
26
assume, by wo king in coo dina es, ha x=0∈U=mL(n)⊆oN, ha
F(x) = 0∈V=mL(m)⊆oMand ha F∈R[[X1, . . . , Xn]](m).
Secondly, o simplici y, we can assume ha he esidue ank o he i s n
columns o J0Fis n, ha is, i ˜
F= (F1, . . . , Fn) hen es. k J0˜
F=n.
Le W=mL(m−n),and de ine he map Φ: N×W→M, (x, w)7→ F(x) +
(0, w).Then,
J0Φ = J0F0
Im−n∈Mm(R),
and so es. k J0Φ = m. By Theo em 1.33, he e exis s a local in e se o Φ, ha
is, he e exis some open subse s U′, V ′and W′o , espec i ely, R(n), R(m)and
R(m−n)such ha Ψ: V′→U′×W′is he in e se o Φ|U′×W′.Tha is, he ollowing
diag am is commu a i e
U′V′
U′×{0}(m−n)U′×W′,
F
ιΨ
Ψ◦F◦ι−1
whe e ι(x) = (x, 0),and
Ψ◦F:U′→U′×W′,(x1, . . . , xn)7→ x1, . . . , xn,0,(m−n)
. . . , 0,
as we we e equi ed.
In pa icula , since an R-bianaly ic map is bo h an imme sion and subme sion,
i looks in coo dina es as he iden i y.
De ini ion 1.39. An R-analy ic map F:N→Mis a subimme sion a xwhen
he e exis x∈U⊆oN, F(x)∈V⊆oMand an R-analy ic mani old Wsuch
ha F|Uis he composi ion
Uπ
→Wι
→V,
whe e πis a subme sion and ιis an imme sion.
Lemma 1.40. Le F:N→Mbe an R-analy ic map. The ollowing a e equi a-
len :
(i) Fis subimme sion a x.
33

(ii) Flooks in coo dina es a ound xlike he R-linea homomo phism
F:R(n)→R(m),(x1, . . . , xn)7→ (x1, . . . , x ,0, . . . , 0)
o some ≤min{n, m},whe e n= dimxNand m= dimF(x)M.
P oo . The p oo ollows om Lemma 1.38.
(i) ⇒(ii).By de ini ion Fhas locally he o m Uπ
→Wι
→V. Le = dimπ(x)W.
Acco ding o Lemma 1.38, πand ιlook in coo dina es as
π:R(n)→R( ),(x1, . . . , xn)7→ (x1, . . . , x )
and
ι:R( )→R(m),(x1, . . . , x )7→ (x1, . . . , x ,0,(m− )
. . . , 0),
so Flooks like hei composi ion.
(ii) ⇒(i).The e exis R-cha s (U, φ, n)a xand (V, ψ, m)a F(x)such ha
ψ◦F◦φ−1=F=ι◦πa φ(U),whe e πand ιa e de ined as in he p eceding
implica ion. Hence W=π(φ(U)) is an open subse o R( )and he ollowing
diag am is commu a i e:
U V
φ(U)W ψ(V)
ϕ
F
π ι
ψ−1
The esul ollows since π◦φis a subme sion and ψ−1◦ιis an imme sion.
The ollowing lemma is p o ed ep oducing he a gumen s o [68, Theo em in
pg. 86]. We will ely on he nex basic esul :
Lemma 1.41. Le Qbe a ing o cha ac e is ic ze o and F∈Q[[X1, . . . , Xn]].
Suppose ha he o mal de i a i es on he las m a iables a e 0, ha is, ∂jF= 0
o all j∈ {n−m+ 1, . . . , n}.Then F∈Q[[X1, . . . , Xn−m]].
P oposi ion 1.42. Le Rbe a p o-pdomain o cha ac e is ic ze o and le F:N→
Mbe an R-analy ic map. Suppose ha he e exis s ∈N0such ha k JyF=
es. k JyF= o all yin an open neighbou hood Uo x. Then Fis a subim-
me sion.
34
P oo . Le n= dimxNand m= dimF(x)M. Since he ques ion is local, wo king
in coo dina es, we can assume ha x=0∈U=mL(n)⊆oN, F(x) = 0∈
V=mL(m)⊆oMand ha F∈R[[X1, . . . , Xn]](m).Assume o simplici y ha
i ˜
F= (F1, . . . , F ) hen es. k J0˜
F= . Thus,
{F1, . . . , F , π +1, . . . , πn}
is a coo dina e sys em o Na U. Hence, i we conside Uas mL( )×mL(n− ),
a e a change o coo dina es we can assume ha ˜
F(x1, x2) = x1, ha is, hen
F(x1, x2) = (x1, ψ(x1, x2)) ,
o some ψ∈R[[X1, . . . , Xn]](m− )such ha es. k J0ψ= 0.We shall check up
on ψ, whe he i is independen o he second a iable in a neighbou hood o 0.
Fi s o all, ∂2ψ=0in U, whe e ∂2ψs ands o he ma ix o o mal de i a i es
on he las n− a iables. O he wise, acco ding o (1.10), he e would be y∈U
such ha k JyF > , which is a con adic ion. The e o e, since cha R= 0, om
Lemma 1.41, ψis independen o he las n− coo dina es. Tha is,
F(x1, x2) = (x1, ψ(x1)).
Hence, i π:mL(n)→mL( )is he p ojec ion on o he i s coo dina es,
hen F= (Id ×ψ)◦π, and hus, Fis locally he composi ion o a subme sion and
an imme sion.
1.5 Cons uc ion o mani olds
This sec ion is de o ed o cons uc ing new analy ic s uc u es s a ing om an
ini ial R-analy ic mani old, bo h by changing he coefficien ing o a con enien
sub ing o Ro by de eloping concep s such as submani olds o quo ien mani olds.
1.5.1 Res ic ion o scala s
In his subsec ion we will illus a e how o induce a mani old s uc u e o e a
sui able sub ing. Fo ha we shall ollow he p ocedu e o [24, Example 13.6].
Le (R, m)be a p o-pdomain and Qa sub ing ha is i sel a p o-pdomain
wi h maximal ideal n:= m∩Q. Suppose u he ha Ris a ini ely gene a ed
ee Q-module. Fo ins ance, in iew o Cohen’s S uc u e Theo em whene e
35
dimK ull(R) = 1, hen Ris a ini ely gene a ed ee Zp-module i cha R= 0,o a
ini ely gene a ed ee Fp[[ ]]-module i cha R=pis posi i e.
Le Mbe an R-analy ic mani old and σ:R→Q(e)aQ-module isomo phism,
ha is, i we ix a basis { 1, . . . , e} o Ras a Q-module, hen σis
σ e
X
i=1
qi i!= (q1, . . . , qe).
Fo each R-cha (U, φ, n)o M, we can de ine he iple U, σ(n)◦φ, ne,which
is ac ually a Q-cha . Indeed, we only ha e o check ha σ(n)◦φ(U)is open
in Q(ne).Fi s ly, no e ha i τis he in e se o σ, hen τnN(e)⊆mN o all
N∈N.Secondly, le x∈φ(U),since φ(U)is open in R(n), he e exis s N∈N
such ha
x+mN(n)⊆φ(U),
and he e o e,
σ(n)(x) + nN(ne)⊆σ(n)x+mN(n)⊆σ(n)◦φ(U).
Mo eo e , he Q-cha s U, σ(n)◦φ, neand V, σ(n)◦ψ, nea e compa ible
(no e ha he compa ibili y is a equi emen only when U∩V6=∅,so in iew o
Co olla y 1.30, he cha s mus be o equal dimension). In ac ,
σ(n)◦φ◦σ(n)◦ψ−1=σ(n)◦φ◦ψ−1◦σ−1(n),
and since φ◦ψ−1is R-analy ic, he esul ollows by he nex lemma:
Lemma 1.43 (c . [24, Exe cise 13.4]).Le F∈Λ0(R)[[X1, . . . , Xn]], hen
σ◦F◦σ−1(n):n(en)→Q(e)
is s ic ly Q-analy ic.
P oo . As cus oma y Λ(R)s ands o he ac ion ield F ac(R)i Ris PID and
o Ro he wise, and he same o Λ(Q).
Obse e ha when Ris PID, since Ris an in eg al ing ex ension o Q, hen
dimK ull(Q) = dimK ull(R) = 1,so he p o-pdomain Qis also a disc e e alua ion
ing. The e o e, i πand ρa e uni o mise s o espec i ely Rand Q, hen ρ=πN
o some N∈N.
36
Le { 1, . . . , e}be he basis o Ras ee Q-module ha co esponds o σand
le us deno e σ−1by τ. Mo eo e , suppose ha
F(X1, . . . , Xn) = X
α∈N(n)
0
aαXα1
1. . . Xαn
n∈Λ0(R)[[X1, . . . , Xn]].
We will show ha he e exis some powe se ies F∗
l∈Λ0(Q)[[X1, . . . , Xen]],
l∈ {1, . . . , e},such ha
F◦τ(n)(y1, . . . , yen) =
e
X
l=1
lF∗
l(y1, . . . , yen)∀yj∈n.
Tha is, whene e xj=Pe
i=1 iyij o some yij ∈n, j ∈ {1, . . . , n}, hen
F(x1, . . . , xn) =
e
X
l=1
lF∗(y11, . . . , yen).
Fi s o all, he e exis some elemen s he e exis some elemen s aα(k, l)∈Λ(Q)
such ha
aα k=
e
X
l=1
aα(k, l) l.
In o de o p o e he p eceding when Ris a PID, we should ha e aken in o
accoun ha since F∈Λ0(R)[[X]] and since ρ=πN, he e exis s a big enough
in ege L∈Nsuch ha aαρ|α|L∈R o all α. In pa icula ,
aα(k, l)ρ|α|L∈Q∀k∈ {1, . . . , e}.(1.11)
Fu he mo e, by an applica ion o he mul inomial heo em, o any uple α=
(α1, . . . , αn)∈N(n)
0 he e exis some elemen s γk(β)∈Qsuch ha
n
Y
j=1 e
X
i=1
iYij!αj
=X
|β|=|α|
e
X
k=1
γk(β) k
e
Y
i=1
n
Y
j=1
Yβij
ij ,
whe e he Yij’s a e inde e mina es. Finally, he desi ed powe se ies a e de ined
as
F∗
l(Y11, . . . , Yen) = X
α∈N(d)
0X
|β|=|α|
e
X
k=1
aα(k, l)γk(β)
e
Y
i=1
n
Y
j=1
Yβij
ij .
When Ris no a PID hese powe se ies a e clea ly in Q[[X]].In con as , when
Ris a PID, F∗
l∈Λ0(Q)[[X]] by i ue o (1.11).
37
Fu he mo e, he p eceding Q-mani old s uc u e is independen o he isomo -
phism σchosen, i.e. o he Q-basis o Rchosen. Ac ually, i suffices o p o e ha
o he R-analy ic mani old mNany wo Q-module isomo phisms σand ˜σgi e
ise o equi alen Q-cha s. Fo ha no e ha
˜σ◦σ−1:σmN→˜σmN
is no hing bu he linea map de ined by he change o basis ma ix A∈GLe(Q),
and so, i is a Q-bianaly ic map. Thus, we will e e o his p ocedu e o gene a ing
new mani olds, simply as es ic ion o scala s, wi h no need o speci ying he
isomo phism. To inish, le us obse e he ollowing pa icula case:
Co olla y 1.44. Le Rbe a p o-pdomain o K ull dimension one. Then any R-
analy ic g oup is by es ic ion o scala s ei he a p-adic analy ic g oup i cha (R) =
0,o an Fp[[ ]]-analy ic g oup i cha (R) = pis posi i e.
Unde ce ain condi ions he con e se o his esul is also ue.
Theo em 1.45. Le Gbe a non-disc e e R-analy ic g oup.
(i) (c . [24, Theo em 13.23]) I Gadmi s a p-adic analy ic g oup s uc u e, hen
Ris a ini ely gene a ed Zp-module.
(ii) (c . [45, Theo em 1.1]) Suppose ha Gis ini ely gene a ed (as opological
g oup). I Gadmi s an Fp[[ ]]-analy ic g oup s uc u e, hen Ris a ini ely
gene a ed Fp[[ ]]-module.
In he la e a icle, he au ho s al eady obse ed ha he esul does no hold
o non ini ely gene a ed g oups, on accoun o he addi i e opological g oups
Fp[[ 1]] and Fp[[ 1, 2]] being isomo phic o one ano he . Ne e heless, whe he
o ini ely gene a ed g oups he analy ic s uc u e de e mines he base ing is a
sensible ques ion, in his di ec ion i is emp ing o specula e ha :
Conjec u e 1.46. Le Gbe a non-disc e e ini ely gene a ed opological g oup
ha admi s bo h an R-analy ic and a Q-analy ic g oup s uc u e. Then, Rand
Qsha e he K ull dimension and he cha ac e is ic.
1.5.2 Submani olds
The concep o submani old has o be de eloped when ea ing mani olds o e
gene al p o-pdomains. The heo y o analy ic mani olds o e local ields supplies
a ious equi alen de ini ions (c . [68, pg. 89]). He e we ep oduce some o hose,
unde sco ing ha any de ini ion should e lec he ac ha he se has R-analy ic
mani old s uc u e by i sel .
38

De ini ion 1.47. Le F:N→Mbe an injec i e weak imme sion. Then F(N)
is an imme sed subse o M.
Fo example, le mbe he maximal ideal o Zp[[ ]] wi h he na u al mani old
s uc u e. I we endow he se mwi h he global cha ψ: m→m, x 7→ x, he
inclusion ι: m→mis a weak imme sion, as Jx(Id ◦ι◦ψ−1) = ( ) o all x∈ m.
Howe e , in his example, he opology o m,which is chosen pu posely o make
ψa homeomo phism, does no coincide wi h he subspace opology. Since i is
na u al o ask o compa ibili y be ween opological s uc u es, we de ine:
De ini ion 1.48. Le Mbe an R-analy ic mani old. Then S⊆Mis an R-analy ic
submani old when o each s∈S he e exis ks∈N0,an open neighbou hood Us
o sin Sand an R-cha (Vs, φs, ds)o sin Msuch ha
•Us⊆Vsand
•φs(Us) = φs(Vs)∩R(ks)×{0}(ds−ks).
The in ege ksis he dimension o Sa s, deno ed by dimsS.
Wi h he subspace opology Sis an R-analy ic mani old, as each poin s∈S
can be endowed wi h he R-cha (Us,(φs,1, . . . , φs,ks), ks).
An immedia e applica ion o Lemma 1.38 yields:
P oposi ion 1.49. Le F:N→Man injec i e R-analy ic map. Then F(N)is
an R-analy ic submani old o Mi and only i Fis an imme sion.
Rema k 1.50. No e ha R-analy ic submani olds a e locally closed. Indeed, le
Mbe an R-analy ic mani old and S⊆Ma submani old. Fo each s∈S he e
exis s an wo subse s Us⊆oSand Vs⊆oMcon aining ssuch ha φs(Us)is
de ined by some linea equa ions in φ(Vs).In pa icula , Usis closed in Vs, ha
is, Vs Usis open in Vs.Hence, i V=∪s∈SVs, hen V S=Ss∈SVs Usis open
in V, so Sis closed in he open se V.
Ano he s aigh o wa d obse a ion yields:
Lemma 1.51. Le Mbe an R-analy ic mani old and Sa submani old. Suppose
ha dimsM= dimsS o all s∈S. Then Sis open in M.
P oo . In keeping wi h he no a ion o De ini ion 1.48: since ks=dsand φis a
homeomo phism, hen Us=Vsand so S=Ss∈SVsis open in M.
39
De ini ion 1.48 s eng hens he de ini ion o R-analy ic submani old used in [27]:
De ini ion 1.52. Le Mbe an R-analy ic mani old. Then S⊆Mis a weak
R-analy ic submani old i o each s∈S he e exis an open neighbou hood Us
o sin S, an R-cha (Vs, φs, ds)o sin Mand a K- ec o space Es≤K(ds)
(K= F ac R) such ha
•Us⊆Vsand
•φs(Us) = φs(Vs)∩Es.
O e ields De ini ions 1.48 and 1.52 a e equi alen . Ce ainly, by a con enien
change o basis we can assume ha Es=K(k)× {0},whe e k= dimK(Es).
None heless, as we shall illus a e wi h an example, in gene al hey a e no equal.
Le R=Z2[[ ]] wi h maximal ideal m= (2, )R, K =Q2(( )), he R-analy ic
mani old M=m(2) and he K- ec o space:
E=(x, y)∈K(2) |2x− y = 0.
Then, M∩E={( a, 2a)|a∈R}is clea ly a weak R-analy ic submani old, and
i is endowed wi h he global R-cha
ψ:M∩E→R, ( a, 2a)7→ a.
Howe e , Js(Id ◦ι◦ψ−1) = 
2 o all s∈M∩E, and he e o e ιis no an
imme sion bu a weak imme sion. In ac , he change o basis o K(2) om he
canonical basis o β={( , 2),(0,1)}is he linea map LA:K(2) →K(2) de ined
by he change o basis ma ix
A=1/ 0
−2/ 1,
and i maps E o K(1) × {0}.Howe e , LAis no an R-bianaly ic map. This
illus a es o by he difficul y o de e mine a alid coo dina e sys em o a weak
submani old. O e ields, he na u al way o doing so is as be o e, ha is, by
ixing a basis β={ 1, . . . , k} o Es,and conside ing he map
ψ:φs(Vs)∩Es→K(k),
k
X
i=1
αi i7→ (α1, . . . , αk)∈K(k).
40
Howe e , o e gene al p o-pdomains depending on he choice o β, i migh occu
ha im ψis no con ained in R(k),and so ψmigh no be an R-cha .
In classical Lie heo y, au ho s al eady dis inguish be ween imme sed ( he
equi alen o De ini ion 1.47) and embedded ( he equi alen o De ini ion 1.48)
submani olds (see [51, Chap e 5]), he la e being a s onge condi ion. Ne -
e heless, when he mani old is compac bo h concep s coincide, and so do hey
when wo king wi h analy ic mani olds o e p incipal ideal p o-pdomains (c . [68,
Pa II, Sec ion III.11.2]). O e gene al p o-pdomains, howe e , i is appa en
now ha he no ion o submani old we wo k wi h mus be ca ego ically speci ied.
In [45, Sec ion 4], Fq[[ ]]-analy ic submani olds a e cha ac e ized as ibe s o
analy ic maps. Mo e p ecisely, hey p o e he ollowing:
P oposi ion 1.53 (see [45, Co olla y 4.2]).Le Mbe an Fq[[ ]]-analy ic mani old
and a subse S⊆M. Suppose ha
(i) Sis homogeneous, i.e. i is con ained in a single o bi o he ac ion o he
g oup o R-bianaly ic au omo phisms o he mani old M, and
(ii) Sis an analy ic subse , i.e. o each s∈S he e exis an open neighbou hood
Uand some Fq[[ ]]-analy ic maps { i:U→Fq[[ ]]}i∈Isuch ha
S∩U={x∈U| i(x) = 0 ∀i∈I}.
Then Sis an Fq[[ ]]-analy ic submani old.†
Obse e ha when Mis an R-analy ic g oup, he ac ion o he le mul iplica-
ion maps is ansi i e, and hus in his si ua ion, e e y subse is homogeneous.
In o de o eplica e a e sion o his o gene al p o-pdomains, gi en an R-
analy ic mani old Mand S⊆M, we say ha Sis an R-analy ic subse o Mwhen
o e e y s∈S he e exis an open neighbou hood Uo sand some R-analy ic
maps { i:U→R|i= 1, . . . , s}such ha
S∩U={y∈U| i(y) = 0 ∀i= 1, . . . , s}.
In o he wo ds, an analy ic subse is locally he nullse o some analy ic unc ions.
This de ini ion ob iously ex ends ha o submani old. Mo eo e , no e ha since
R[[X1, . . . , Xn]] is Noe he ian (see [49, Theo em IV.9.4]), he de ini ion can be in
p inciple elaxed allowing s o be in ini e.
†This esul is alid ega dless o he de ini ion o submani old, as he base ing is a PID.
41
De ini ion 1.54. Le Sbe an R-analy ic subse o an R-analy ic mani old M.
A poin s∈Sis a egula poin when he e exis s an open neighbou hood Uo s
and some R-analy ic maps { i:U→R|i= 1, . . . , s}such ha
(i) S∩U={y∈U| i(y) = 0 ∀i= 1, . . . , s}and
(ii) i F:= ( 1, . . . , s), hen es. k (JsF) = s.
The in ege sis e med he co ank o Sa s, and by Lemma 1.38, we know ha
s≤dimsS.
Lemma 1.55. Le Mbe an R-analy ic mani old and S⊆M. Then, Sis a
submani old i and only i all he poin s o Sa e egula poin s.
P oo . The only i is immedia e om he de ini ion and Lemma 1.38. Fo he i ,
le s∈Sand d= dimsS, suppose ha sis a egula poin o co ank , hen he e
exis s an open neighbou hood Uo ssuch ha
S∩U={y∈U| i(y) = 0 ∀i= 1, . . . , }.
I F= ( 1, . . . , ), hen es. k JsF= , so we can ex end F o a coo dina e
sys em { i}d
i=1 o sin M, and wi h espec o ha coo dina e sys em o a possibly
smalle open se U′⊆Uwe ha e
S∩U′={y∈U′| 1(y) = ··· = (y) = 0},
and hus Mis a submani old o dimension d− a s.
Addi ionally, in R-analy ic g oups, we can blend he no ions o analy ic and
g oup subs uc u e. Acco dingly,
De ini ion 1.56. Le Gbe an R-analy ic g oup. An analy ic subg oup is a sub-
g oup H≤G ha is besides an R-analy ic submani old.
Lemma 1.57. An R-analy ic subg oup is closed.
P oo . I is a gene al ac ha in a opological g oup a locally closed subg oup
is ac ually closed, so he esul is s aigh o wa d om Rema k 1.50. To make i
explici , Hbeing locally closed means ha His open in i s opological closu e
H. Suppose by con adic ion ha H6=H, so ha he e exis s a non- i ial le
cose gH ⊆H. Since Hand gH a e open dense subse s o H, hen gH ∩H6=∅,
con adic ing he disjoin ness o he cose s.
42
Le Ibe he ideal o R[[X]] gene a ed by he o mal powe se ies
nπi,α |α∈N(d)
0, i ∈ {1, . . . , d}o.(2.1)
Since R[[X]] is Noe he ian (see [49, Theo em 9.4]), Iis gene a ed by a ini e subse
o (2.1), say F,and so conside m= max {|α| | πi,α ∈ F} ∈ N.Deno e by M he
ideal (X1, . . . , Xd)R[[X]], hen W=M/Mm+1 is a ee R-module o ini e ank,
and Sac s on W.
Clea ly Z(S)ac s i ially on W. Con e sely, suppose ha gac s i ially on
W. Then, π(g)=0, o all π∈ F.Hence, πi,α(g)=0 o all i∈ {1, . . . , d}and
α∈N(d)
0,so g·πi(X) = Xi, ha is, g∈Z(S).Thus, S/Z(S)ac s ai h ully on
W.
On o he hand, Laza d’s p oo – i will be b ie ly explained in Sec ion 2.1–, elies
on Ado’s Theo em in conjunc ion wi h he Bake -Campbell-Hausdo o mula in
o de o link he g oup ope a ion wi h he co esponding Lie algeb a ope a ion.
Based on ha p ocedu e, Camina and Du Sau oy [14] p o ed ha pe ec Zp[[ ]]-
s anda d g oups, namely Zp[[ ]]-s anda d g oups So le el Nsuch ha S′=S2N
(see (1.6)), a e linea . Fu he mo e, Jaikin-Zapi ain [43] exploi ed simila ideas o
p o e ha o e a p o-pdomain Ro cha ac e is ic ze o, e e y ini ely gene a ed
compac R-analy ic g oup is linea . The main esul o his chap e is o ex end
his, and p o e ha whene e Rhas cha ac e is ic ze o, any compac R-analy ic
g oup is linea .
2.1 Linea i y o compac p-adic analy ic g oups
In his sec ion, we will succinc ly desc ibe he cons uc ion o he a o emen ioned
ai h ul linea ep esen a ion o compac p-adic analy ic g oups, wi h a iew o-
wa ds con olling i s deg ee. We s a by ecalling Ado’s Theo em, mo e p ecisely
a gene alised e sion o i , due by Chu kin [19] and Weigel [74] (see Appendix A
o a de ailed p oo as well as o pe inen de ini ions on he opic).
Theo em 2.4. Le Lbe a Zp-Lie algeb a which is a ee Zp-module o ank .
The e exis s a Zp-Lie algeb a monomo phism φ:L,→Mn(Zp),whe e ndepends
only on .
Fix a p ime numbe p, and o simplici y o no a ion se p=pi pis odd and
p= 4 i p= 2.Any Zp-s anda d g oup (S, φ)is a so-called uni o mly powe ul
g oup, ha is, a ini ely gene a ed o sion- ee g oup such ha S′≤Sp(c . [24,
49

Thms 4.5 and 8.31]). The e is a ca ego ical isomo phism be ween he ca ego y
UG oup o d-dimensional uni o mly powe ul p o-pg oups and he ca ego y pLie o
d-dimensional powe ul Zp-Lie la ices, namely Zp-Lie algeb as L ha a e ee Zp-
modules o ank dsuch ha [L,L]≤pL.Mo e conc e ely, he e exis s a ca ego ical
isomo phism L:UG oup →pLie,wi h in e se E,and o each uni o mly powe ul
g oup So dimension d, i assigns a Zp-Lie algeb a L(S) ha is a ee Zp-module
o ank d, whe e he unde lying se is Si sel , and he module ope a ions a e gi en
in e ms o he g oup ope a ions as ollows: le ∈Zpand x, y ∈S hen
·x=x
x+y= lim
n→∞ xpnypnp−n
[x, y] = lim
n→∞ xpn, ypnp−2n,
we e e o [24, Sec ion 4.3] o he p ecise de ini ions o he igh -hand side
o mulas. Fo ins ance, pMn(Zp)is a powe ul Lie la ice wi h Lie b acke
[A, B] = AB −BA, and E(pMn(Zp)) ⊆GLn(Zp)( his can be iewed as he
usual p-adic ma ix exponen ia ion).∗
Con e sely, he Bake -Hausdo -Campbell o mula can be ega ded as a o mal
powe se ies H(x, y)in wo non-commu ing a iables sa is ying he iden i y
eH(x,y)=exey,
and gi en a powe ul Zp-Lie la ice S, he se Sis a uni o mly powe ul g oup
wi h he g oup ope a ion xy =H(x, y).
Consequen ly, i φ:L(S),→Mn(Zp)is he injec i e Zp-Lie algeb a homomo -
phism p o ided by Theo em 2.4, hen
φ|pL(S):pL(S)→pMn(Zp)
is an injec i e Lie algeb a homomo phism be ween powe ul Zp-Lie la ices, and
hence, since Eis a unc o , we ob ain a g oup monomo phism
E(φ): E(pL(S)) ,→ E(pMn(Zp)) ≤GLn(Zp).
Finally, he addi i e cose s o he Zp-Lie algeb a a e he same as he mul i-
plica i e cose s o he uni o mly powe ul g oup (c . [24, P oposi ion 4.31(iii)]),
so
|S:E(pL(S))|=|L(S) : L◦E(pL(S))|=|L(S) : pL(S)|=Z(d)
p:pZ(d)
p=pd,
∗Ac ually he image o pMn(Zp)by Eis he i s cong uence subg oup GL1
n(Zp).
50
and hus by aking he induced ep esen a ion we ob ain a g oup monomo phism
m:S ,→GLpdn(Zp).
We ga he all his in he ollowing esul :
Theo em 2.5. Le Sbe a Zp-s anda d g oup o dimension d. The e exis s a
ai h ul linea ep esen a ion m:S ,→GLn(Zp),whose deg ee ndepends only on
pand d.
Finally, since e e y compac p-adic analy ic g oup has a Zp-s anda d subg oup
o ini e index, he induced linea ep esen a ion leads o:
Co olla y 2.6 (Laza d).E e y compac p-adic analy ic g oup is linea o e Zp.
2.2 Change o p o-pdomains
The idea behind a ious esul s in his hesis is o educe he p oblem o analy ic
g oups o e p o-pdomains o K ull dimension one and use he al eady known
s uc u al esul s he e. Fo ha pu pose a homomo phism ϕ:R→Qbe ween
p o-pdomains can be used o cons uc Q-analy ic g oups ha a e na u al images
o R-analy ic g oups. Indeed, gi en a powe se ies F(X) = Pα∈N(d)
0aαXα∈R[[X]]
by applying ϕ o he coefficien s we ob ain he powe se ies
Fφ=X
α∈N(d)
0
ϕ(aα)Xα∈Q[[X]].
This is no mo e han a speci ic ins ance o he wide uni e sal p ope y o
opological powe se ies ings (see Sec ion 1.1).
To ans o m coefficien s, we es ic o na u al ing homomo phisms be ween
local ings, namely local ing homomo phisms. Those a e ing homomo phisms
ϕ: (R, m)→(Q, n)be ween local ings such ha ϕ(m)⊆n.Obse e ha , e e y
local ing homomo phism is con inuous, conside ing ha ϕ(mn)⊆nn o any
n∈N.
The ollowing lemma shows ha he o egoing change o ings commu es wi h
he composi ion o powe se ies.
Lemma 2.7. Le ϕ: (R, m)→(Q, n)be a local ing homomo phism. Le F∈
R[[X1, . . . , Xn]](m)and G∈R[[X1, . . . , Xm]](l)be o mal powe se ies, and assume
ha F(0)∈m(m).Then (G◦F)φ(X1, . . . , Xn) = Gφ◦Fφ(X1, . . . , Xn).
51
P oo . Using he uni e sal p ope y o powe se ies ings, he e exis s a unique
con inuous ing homomo phism
Φφ:R[[X1, . . . , Xn]] →Q[[X1, . . . , Xn]]
such ha Φφ(H(X)) = Hφ(X) o all H∈R[[X]],whe e X= (X1, . . . , Xn).
Le F(X)=(F1(X), . . . , Fm(X)). Since F(0)∈m(m)and ϕis local, hen
Φφ(F1(X)) , . . . , Φφ(Fm(X)) a e in (n, X1, . . . , Xn)Q[[X]], he maximal ideal o
Q[[X]], he e o e using he uni e sal p ope y o powe se ies ings we can de ine
he con inuous map
Φ1:R[[Y1, . . . , Ym]] →Q[[X1, . . . , Xn]]
such ha Φ1( ) = ϕ( ) o all ∈R, and Φ1(Yi) = (Fi)φ(X) o all i∈ {1, . . . , m}.
Simila ly, since F1(X), . . . , Fm(X)a e in he maximal ideal o R[[X]], de ine he
map
Φ2:R[[Y1, . . . , Ym]] →R[[X1, . . . , Xn]]
such ha Φ2|R= IdR,and Φ2(Yi) = Fi(X) o all i∈ {1, . . . , m}.
No ice ha Φ1( )=Φφ◦Φ2( ) = ϕ( ) o all ∈R, and ha Φ1(Yi) =
Φφ◦Φ2(Yi) = (Fi)φ(X) o all i∈ {1, . . . , m}.The e o e, by he uniqueness o he
uni e sal p ope y we ha e ha Φ1= Φφ◦Φ2,and so, in pa icula ,
(G◦F)φ(X) = Φφ◦Φ2(G(Y)) = Φ1(G(Y)) = Gφ◦Fφ(X).
Hence, he change o ings p ese es o mal powe se ies iden i ies, so in pa -
icula :
Co olla y 2.8. Le Rand Qbe p o-pdomains, le F∈R[[X1, . . . , X2d]](d)be
a o mal g oup law wi h o mal in e se Iand le ϕ:R→Qbe a local ing
homomo phism. Then Fφis a o mal g oup law wi h o mal in e se Iφ.
P oo . Le X,Yand Zbe d- uples o a iables. Since F(0) = 0and using Lemma
2.7 we ha e:
(i) since F(F(X,Y),Z) = F(X,F(Y,Z)), hen
Fφ(Fφ(X,Y),Z) = Fφ(X,Fφ(Y,Z)),
(ii) and since F(X,0) = F(0,X) = X, hen Fφ(X,0) = Fφ(0,X) = X.
52
The e o e Fφ∈Q[[X1, . . . , X2d]](d)is a o mal g oup law. Finally, since I(0) = 0
and F(I(X),X) = F(X,I(X)) = 0,by Lemma 2.7:
Fφ(Iφ(X),X) = Fφ(X,Iφ(X)) = 0,
and so Iφis he o mal in e se co esponding o Fφ.
Le now ϕ: (R, m)→(Q, n)be a local ing homomo phism be ween p o-p
domains and le Sbe an R-s anda d g oup, which can be iden i ied o simplici y
wi h mN(d),whose o mal g oup law is F.Using he abo e-cons uc ed o mal
g oup law Fφ, L := nN(d)can be endowed wi h a g oup ope a ion making i
in o a Q-s anda d g oup. Indeed, he g oup ope a ion is simply
x∗y=Fφ(x, y),(2.2)
o all x, y ∈L. In he ollowing lemma we keep all he p eceding no a ion.
Lemma 2.9. The map
ϕ(d):mN(d),F→nN(d),Fφ,( 1, . . . , d)7→ (ϕ( 1), . . . , ϕ( d))
is a g oup homomo phism.
P oo . I we w i e F= (F1, . . . , Fd)and
Fi(X,Y) = X
α,β∈N(d)
0
aα,βXαYβ∈R[[X,Y]],
hen
ϕ(Fi(x, y)) = ϕ

X
α,β∈N(d)
0
aα,βxα1
1. . . xαd
dyβ1
1. . . yβd
d


=X
α,β∈N(d)
0
ϕ(aα,β)ϕ(x1)α1. . . ϕ(xd)αdϕ(y1)β1. . . ϕ(yd)βd
= (Fi)φϕ(d)(x), ϕ(d)(y)∀x, y ∈mN(d),
using he con inui y o ϕin he second equali y; and consequen ly ϕ(d)is a g oup
homomo phism.
53
Suppose now ha Ris no a PID. Le Gbe a compac R-analy ic g oup and le
(S, F)be an open no mal R-s anda d g oup such ha he conjuga ion maps a e
s ic ly analy ic (we can assu e i s exis ence by i ue o Lemma 1.23). We shall
cons uc a compac Q-analy ic g oup, whose open no mal Q-s anda d subg oup
will be he Q-s anda d g oup L=nN(d),Fφbuil upon Sas in (2.2). Mo e
p ecisely, le Tbe a le ans e sal o Sin G, and assume ha 1∈T. Fo
no a ional con enience, we will use he ollowing: whene e g∈G hen ˜gis he
ep esen a i e o gS in T. By (1.3.1), and using he no a ion he ein, i x∈ S
and y∈ S, hei p oduc is gi en by
φe
(xy) = Ae
(F(C (φ (x)), φ (y))) .
De ine H:= T×Land he homeomo phisms ψ : ( , L)→L, ψ ( , l)7→ l. I
x∈( , L)and y∈( , L), imi a ing he p e ious o mula de ine he ope a ion:
x∗y=e
, Ae
φFφ(C )φ(ψ (x)), ψ (y).(2.3)
Rema k. We can iden i y (1, L)wi h L o plainness, and hen (2.3) ex ends
(2.2).
Lemma 2.10. Wi h he no a ion abo e, (H, ∗)is a compac Q-analy ic g oup wi h
open no mal Q-s anda d subg oup L.
P oo . Fi s ly, His a compac Q-analy ic mani old wi h a las {( L, ψ )} ∈T–
abusing he no a ion we will use L o deno e ( , L)–. Mo eo e , His a g oup.
Indeed,
(i) le , , p ∈T, om Lemma 1.24 and he associa i i y o Gwe know ha
as o mal powe se ies:
A
p
·e p FCe p(X), A e p
p(F(Cp(Y),Z))=
A
p
e
·pFCpAe
(F(C (X),Y)),Z.
Thus, le x∈ L, y ∈ L and z∈pL, hen by Lemma 2.7 and (2.3):
ψ
p(x∗(y∗z))
=A
p
·e pφFφ(Ce p)φ(ψ (x)),Ae p
pφFφ(Cp)φ(ψ (y)), ψp(z)
=A
p
e
·pφFφ(Cp)φAe
φFφ(C )φ(ψ (x)), ψ (y), ψp(z)
=ψ
p((x∗y)∗z).
54

(ii) The neu al elemen is (1,0)∈L.
(iii) The in e se o x∈ L is gi en by
y=
−1,Ag
−1
−1φ◦(C −1)φ◦Iφ(ψ (x)).
Indeed, clea ly x∗y∈Land by Lemma 1.24, we know ha
A1
·g
−1FCg
−1(X), Ag
−1
−1(C −1(I(X)))=0.
Hence, by Lemma 2.7 and (2.3),
0=A1
·g
−1φFφCg
−1φ(ψ (x)),Ag
−1
−1φ(C −1)φ(Iφ(ψ (x)))
=A1
·g
−1φFφCg
−1φ(ψ (x)), ψg
−1(y)
=ψ1(x∗y).
In a simila ashion, y∗x= (1,0).
Finally, Lis no mal in H, as 1 = 1 o all ∈T.
We will inish wi h a i ial (bu help ul in upcoming chap e s) obse a ion:
Rema k 2.11. By de ini ion, i we ha e ha
φe
(x·y) = M(φ (x), φ (y)) ∀(x, y)∈ S × S.
o a sui able uple o powe se ies M∈R[[X1, . . . , X2d]](d).Then,
ψe
(¯x∗¯y) = Mφ(ψ (¯x), ψ (¯y)) ∀(¯x, ¯y)∈ L × L.
Mo eo e , he same holds o he in e sion map.
2.2.1 E alua ion epimo phisms
In p ac ise, o he a o esaid change o p o-pdomains we will chie ly use e alua-
ion epimo phisms. Mo e p ecisely, le (R, m)be a p o-pdomain and a∈m(m),
he e alua ion epimo phism a ais he con inuous local ing homomo phism
sa:R[[ 1, . . . , m]] →R, F 7→ F(a).Those epimo phisms can be ex ended o
any in eg al ex ension o R[[ 1, . . . , m]],by i ue o he ollowing classical esul :
55
Theo em 2.12 (Going Up Theo em (c . [75, Theo em V.2.3])).Le A⊆Bbe
an in eg al ing ex ension. Fo e e y p ime ideal p⊆A he e exis s a p ime ideal
q⊆Bsuch ha q∩A=p.
Co olla y 2.13. Le A⊆Bbe an in eg al ing ex ension, le Pbe an in eg al
domain and le ϕ:A→Pbe a ing epimo phism. The e exis s an in eg al domain
Qsuch ha ϕex ends o a ing epimo phism ˜ϕ:B→Q.
P oo . Le p= ke ϕ, by he Going Up Theo em, he e exis s a p ime ideal q⊆B
such ha q∩A=p.Thus, he ollowing diag am is commu a i e:
B B/q
A A/p
˜φ
φ
ψ
whe e ψ(x+p) = x+qis injec i e. Iden i ying A/pwi h P, hen ˜ϕex ends ϕ.
Rema k 2.14. The p oo abo e gi es mo e in o ma ion abou Q. Ac ually, since
Bis an in eg al ex ension o A,Qis also an in eg al ex ension o P. Indeed,
any A-in eg al dependence in B, emains so modulo q: i is now an A/p-in eg al
dependence in B/q.
Fu he mo e, i Bis a p o-pdomain, so is Qas a quo ien o Bby a p ime
ideal.
Finally, i A⊆Bis besides a ini ely gene a ed ing ex ension, hen B/qis a
ini ely gene a ed A/p-module. Indeed, educing he gene a o s o Bas A-module
modulo q,we ob ain a gene a ing se o B/qas A/p-module.
These ex ended e alua ion epimo phisms appea in any p o-pdomain R. Recall
ha by i ue o Cohen’s S uc u e Theo em, Ris a ini ely gene a ed, and so
in eg al, ex ension o P[[ 1, . . . , m]] whe e m= dimK ull(R)−1and (P, n)is a
p o-pdomain o K ull dimension one – ac ually we can speci y e en mo e by
ecollec ing ha Pis ei he Zpo Fp[[ ]] depending on he cha ac e is ic o R–,
and he e o e o each a∈n(m)we ob ain a con inuous epimo phism ˜sa:R→Qa.
Las ly, we shall p esen a p ope y ha esembles Lemma 1.8. In keeping wi h
p io no a ion:
Co olla y 2.15. Le U⊆on(m)and Da dense subse o U, hen ∩a∈Dke ˜sa={0}.
P oo . We w i e A=P[[ 1, . . . , m]],pa= ke saand qa= ke ˜sa o all a∈n(m).
Fi s o all, e alua ing a powe se ies F∈P[[ 1, . . . , m]] is con inuous, so i
56
F(a)=0 o all a∈D, hen F(a)=0 o all a∈U. Mo eo e , by cons uc ion,
pa=qa∩A, so Lemma 1.8 yields ha
(∩a∈Dqa)∩A=∩a∈Dpa=∩a∈Upa={0}.
Suppose by con adic ion ha he e exis s ∈ ∩a∈Dqa {0}.Since he ing
ex ension A⊆Ris in eg al, he e exis s a monic polynomial (X)∈A[X],such
ha
( ) = n+
n−1
X
i=0
ai i= 0.
We can addi ionally assume o be o minimal deg ee. Bu since ∩a∈Dqais an
ideal o R, we ha e ha a0∈ ∩a∈Dqa∩A={0},which con adic s he minimali y
o n, since we could conside (X) = Xn−1+Pn−1
i=1 aiXi−1.
2.3 Disc imina ion
The pu pose o his sec ion is o s udy he ollowing concep :
De ini ion 2.16. Le Aand Bbe wo ins ances o he same algeb aic s uc u e,
Ais said o be ully esidually Bo Ais disc imina ed by B– equi alen ly, B
disc imina es A– i o any ini e subse S⊆A, he e exis s a homomo phism
h:A→Bin he co esponding ca ego y such ha he es ic ion h|Sis injec i e.
Mo e gene ally, a amily B={Bi}i∈Idisc imina es Ao Ais ully esidually-B,
i o each ini e subse S⊆A he e exis s a mo phism h:A→Bi, o some i∈I,
such ha h|Sis injec i e.
Example 2.17. Le us exempli y his wi h ings: le Aand B={Bi}i∈Ibe ings
and suppose ha Ais an in eg al domain. Then Abeing ully esidually-Bis
equi alen o he exis ence o a collec ion o homomo phisms F ⊆ ∪i∈IHom(A, Bi)
such ha
∈F
ke ={0}.
The necessi y is clea , since o he wise he e would be a non-ze o elemen x∈A
such ha (x) = 0 o any ing homomo phism :A→Bi, and so no homomo -
phism would be injec i e when es ic ed o {0, x}.
Fo he sufficiency, le S={ i}i∈I⊆Abe a ini e se , and de ine =Qi=j( i−
j)∈R. No ice ha 6= 0,as he i’s a e dis inc and Ais an in eg al domain.
57
The e o e, he e exis s ∈Hom(A, Bi)such ha
06= ( ) = Y
i=j
( ( i)− ( j)) ,
and hus ( i)6= ( j) o all i6=j∈I.
Fo ins ance, Lemma 1.8 yields ha he p o-pdomain Rdisc imina es R[[X]]
o any ini e numbe o a iables in X.
Lemma 2.18. Le Rbe a p o-pdomain. Fo each ini e S⊆R he e exis s a
p o-pdomain Qo K ull dimension one and cha ac e is ic cha R, and a local
ing epimo phism ϕ:R→Q ha is injec i e when es ic ed o S. In pa icula ,
a p o-pdomain Ris disc imina ed by he se o p o-pdomains o K ull dimension
one and cha ac e is ic cha R.
P oo . Le m= dimK ull(R)−1and le (P, n)be he p o-pdomain Zpi cha (R) =
0o Fp[[ ]] i cha (R) = pis posi i e. Acco ding o Subsec ion 2.2.1, o each
a∈n(m), he e exis s a p o-pdomain Qaand a local ing epimo phism ˜sa:R→Qa
ha ex ends he e alua ion homomo phism sa:P[[ 1, . . . , m]] →P. Mo eo e , by
Rema k 2.14, Qais an in eg al ex ension o P, and hus
dimK ull Qa= dimK ull P= 1
and
cha Qa= cha P= cha R.
Finally, by Co olla y 2.15, ∩a∈n(m)ke ˜sa={0},and Example 2.17 yields he
esul .
As men ioned in he p eceding p oo , i Rhas cha ac e is ic ze o, i is a ini ely
gene a ed ex ension o Zp[[ 1, . . . , m]].Le us deno e by µ(R) he minimum num-
be o elemen s ha is necessa y o gene a e Ras a Zp[[ 1, . . . , m]]-module. We
ecall om Rema k 2.14 ha Qais a ee Zp-module o ank a mos µ(R).
P oposi ion 2.19. Le (R, m)be a p o-pdomain o cha ac e is ic ze o, and le
Gbe an R-s anda d g oup. The e exis s an in ege n∈N,depending on R, he
dimension o Gand he le el o G, such ha Gis disc imina ed by GLn(Zp).
P oo . We iden i y Gwi h mN(d),whe e he g oup ope a ion is gi en by he
o mal g oup law
F(X,Y) = X
α,β∈N(d)
0
aα,βXαYβ∈R[[X,Y]].
58
3
Hausdo dimension in compac
R-analy ic g oups
F ac al dimensions a ose as a gene alisa ion o he no ion o opological di-
mension and he e a e se e al al e na i e de ini ions o ha pu pose. Howe e ,
he bulk o hem depends on some so o measu emen . Amongs all hese ac-
al dimension, he mos p ominen ones a e he Hausdo dimension and he
Minkowski-Bouligand dimension (also known as box dimension).
These dimensions can be de ined in any me ic space, and in he speci ic g oup
heo e ical con ex , he s udy o he Hausdo dimension in he se ing o p o ini e
g oups has a ac ed conside able a en ion. Ac ually, i Gis a coun ably based
p o ini e in ini e g oup, he e exis s a il a ion se ies o G, ha is, a amily
{Gn}n∈No descending open subg oups which is a neighbou hood sys em o he
iden i y, i.e. Tn∈NGn={1}.Such il a ion de ines a me ic on Gby le ing
d(x, y) = in |G:Gn|−1|xy−1∈Gn.
This no ion o dis ance makes Gin o a me ic space. Thus, one can de ine
he Hausdo and he Minkowski-Bouligand dimensions o a subse X⊆G(see
Sec ion 3.1 o he p ecise de ini ions), which will be deno ed, espec i ely as
hdim(X)and lbdim(X).The e is a unique measu e on a p o ini e g oup, namely
65

he Haa measu e, whe eas he e migh be se e al non-equi alen me ics; and
usually ac al dimensions depend on he me ic –o equi alen ly on he il a ion
se ies employed o de ine i –. Fu he mo e, o a ixed il a ion se ies {Gn}n∈Nwe
can conside he collec ion o alues hdim{Gn}(H)whe e H anges o e he closed
subg oups o G, ha is
hspec{Gn}(G) := hdim{Gn}(H)|H≤cG,
which is called he Hausdo spec um o Gwi h espec o he il a ion se ies
{Gn}n∈N.Al hough we could de ine he Minkowski-Bouligand spec um simila ly,
o na u al il a ion se ies we ha e ha hdim(H) = lbdim(H) o e e y closed
subg oup H≤cG(see upcoming Theo em 3.7). Consequen ly, in keeping wi h
classical e minology, we will me ely use he name ”Hausdo ”.
I u ns ou ha hese spec a migh ha e li le o no esemblance as one
changes he il a ion. Fo ins ance, conside he addi i e p o-pg oup Zp⊕Zp.
Fo ini ely gene a ed p o-pg oups o his kind, he e exis s a na u al il a ion
se ies, namely he p-powe il a ion se ies, gi en by Gn=Gpn.Wi h espec o
his se ies hspec{Gn}(G) = {0,1/2,1},so i is ini e; whils , by [48, Theo em 1.3]
he e exis s a il a ion se ies {Hn}n∈Nsuch ha hspec{Hn}(G)con ains he eal
in e al h1
p+1,p−1
p+1i,so, whene e p > 2, i is uncoun able.
The a icle [6] o Ba nea and Shale is one o he ea lies wo ks conce ning
Hausdo dimension in p o ini e g oups, and amid o he esul s, he e is shown
ha hspec{Gpn}(G)is ini e o any p-adic analy ic p o-pg oup G. None heless
he con e se emains open:
Ques ion 3.1 (c . [6, P oblem 1]).Le Gbe a ini ely gene a ed p o-pg oup
such ha hspec{Gpn}(G)is ini e. Is G p-adic analy ic?
I is wo hwhile men ioning ha he ques ion has posi i e answe when Gis
besides a soluble g oup (see [48, Theo em 1.7]).
Howe e , he p-powe il a ion se ies ypically can no be used in he se ing
o p o ini e R-analy ic g oups, as Gpnis no no mally an open subg oup o a com-
pac R-analy ic g oup G. Ne e heless, hose g oups possess a canonical il a ion
se ies, which depends on he g oup’s analy ic s uc u e. By a way o example, in
Sec ion 3.2 we s udy he connec ion be ween he analy ic dimension o a closed
submani old Mand he a o esaid ac al dimensions compu ed wi h espec o
his na u al il a ion se ies, and we ob ain he ollowing iden i y:
hdim(M) = lbdim(M) = max{dimxM|x∈M}
dim H.(3.1)
66
The Hausdo dimension ela i e o his il a ion se ies, which is in oduced
insigh ully in Subsec ion 3.1.3, is called R-s anda d Hausdo dimension. The
co esponding Hausdo spec um, he R-s anda d Hausdo spec um, is deno ed
as hspecs .Fo p-adic analy ic p o-pg oups, he ini eness o he Hausdo spec-
um wi h espec o he p-powe il a ion can be s a ed di e en ly:
Theo em 3.2 (c . [6, Co olla y 1.2] and [27, Co olla y 3.4]).Le Gbe a compac
p-adic analy ic g oup. Then hspecs (G)is ini e and a ional.
Ne e heless, he si ua ion is adically dissimila when he base ing is dis inc
om a ini ely gene a ed ex ension o Zp( ecall ha an R-analy ic g oup is p-adic
analy ic i and only i Ris a ini ely gene a ed ing ex ension o Zp). In his
chap e , we shall mos ly es ic o he case R=Fp[[ ]],and he main indings o
ou in es iga ion can be summa ised as ollows:
Theo em 3.3. Le Gbe a compac Fp[[ ]]-analy ic g oup.
(i) The s anda d Hausdo spec um o Gcon ains he eal in e al [0,1/dim G].
(ii) I Gis soluble, hen hspecs (G) = [0,1].
These sugges ha he s anda d spec um o an R-analy ic g oup migh be
sufficien o isola e p-adic analy ic g oups, as he p io esul s a e consonan wi h
he nex conjec u e:
Conjec u e 3.4. Le Gbe a compac R-analy ic g oup such ha hspecs (G)is
ini e. Then Gis p-adic analy ic.
3.1 Hausdo and box dimension
In his sec ion, we b ie ly desc ibe he abo e-men ioned ac al dimensions. Fu -
he mo e, we collec hei basic p ope ies and some p elimina y esul s, ocusing
chie ly on he se ing o p o ini e g oups.
3.1.1 Basic de ini ions and p ope ies
Le us sho ly p esen he ac al dimensions alluded o h oughou he p e ious
in oduc o y sec ion. Le (M, d)be a me ic space, le X⊆Mand le δand zbe
posi i e numbe s. We de ine
Hz
δ(X) := in
∞
X
n=1
diam(Un)z,
67
whe e {Un}n∈Nis a δ-co e ing, namely a co e ing o Xconsis ing o se s o diam-
e e a mos δ, and he in imum is aken o e all hose co e ings. Obse e ha
he limi
Hz(X) := lim
δ→0Hz
δ(X)
exis s, since Hz
δ(X)is non-dec easing as δ ends o ze o. Besides, Hzis an ou e
measu e (see [28, P oposi ion 11.17]), called he z-Hausdo measu e in M. The
ollowing esul holds:
Lemma 3.5 (c . [26, Sec ion 3.2]).Suppose ha Hs(X)<∞and ≥s. Then
H (X) = 0.
Acco dingly, we can de ine he Hausdo dimension o Xwi h espec o he
me ic das
hdimd(X) := in {s| Hs(X) = 0}= sup {s| Hs(X) = ∞}
–o e p o ini e g oups, he me ic depends on a il a ion se ies {Gn}n∈N,and
consequen ly, we will use he no a ion hdim{Gn}–.
I is s aigh o wa d o e i y ha he Hausdo dimension is
•mono one, ha is, hdimd(X)≤hdimd(Y)whene e X⊆Yand
•coun ably s able, ha is, hdimd(∪n∈NXn) = supn∈Nhdimd(Xn)
(compa e wi h [26, pp. 48-49]). Fu he mo e, we highligh he ollowing p ope y:
P oposi ion 3.6 (c . [26, P oposi ion 3.3]).Le : (M1, d1)→(M2, d2)be a
bi-Lipschi z map be ween me ic spaces, i.e. he e exis wo posi i e cons an s
C, c ∈R≥0such ha
c·d1(x, y)≤d2( (x), (y)) ≤C·d1(x, y)∀x, y ∈M1.
Then hdimd2( (X)) = hdimd1(X) o all X⊆M1.In pa icula , isome ies
p ese e Hausdo dimension.
The o he ac al dimension we will ea wi h is he Minkowski-Bouligand
dimension o he (lowe ) box dimension. Le Nδ(X)be he minimal numbe o
se s o diame e a mos δ ha a e equi ed o co e X, and de ine espec i ely
he lowe box dimension and he uppe box dimension (albei we will mainly ocus
on he o me ) as ollows:
lbdimd(X) := lim in
δ→0+
log Nδ(X)
−log δand ubdimd(X) := lim sup
δ→0+
log Nδ(X)
−log δ.
68
Obse e ha hese exp essions a e independen o he base o which we ake he
loga i hm. When hese wo alues coincide, and he e o e he unde lying sequence
o eal numbe s is con e gen , ha common alue is e e ed o as bdimd(X), he
box dimension o X.
In he con ex o coun ably based p o ini e g oups, we can ob ain a pu ely g oup
heo e ical o mula o he abo e exp essions. Indeed, in a p o ini e g oup G, he
sole alues ha ake he me ic de ined by using he il a ion se ies {Gn}n∈Na e
|G:Gn|−1,and he ball o cen e xand adius δ=|G:Gn|−1is simply he coclass
xGn.Thus, o e e y X⊆Gwe ha e Nδ(X) = |XGn:Gn|( his exp ession s ands
o he numbe o cose s o he o m xGn o some x∈X), and consequen ly he
p eceding de ini ions can be ew i en as
lbdim{Gn}(X) = lim in
n→∞
log |XGn:Gn|
log |G:Gn|
and
ubdim{Gn}(X) = lim sup
n→∞
log |XGn:Gn|
log |G:Gn|
–we will subs i u e he subsc ip wi h {Gn},as he me ic is comple ely de e -
mined by he il a ion se ies–. Mo eo e , i easily ollows om hese de ini-
ions ha bo h box dimensions a e mono one and bi-Lipschi z in a ian (see [26,
P oposi ion 2.5]). In addi ion, he uppe box dimension is ini ely s able, ha is,
ubdim(X∪Y) = max{ubdim(X),ubdim(Y)}.Howe e , he lowe box dimension
migh no ha e his p ope y.
In his pionee ing wo k [1], Abe c omb ie p o ed ha o closed subg oups –and
some il a ion se ies– he Hausdo and he lowe box dimension coincide. In
o he wo ds,
Theo em 3.7 (c . [6, Theo em 2.4]).Le Gbe a coun ably based p o ini e g oup
wi h no mal il a ion se ies {Gn}n∈N, ha is, Gn⊴G o all n∈N.Fo e e y
closed subg oup H≤cGwe ha e
hdim{Gn}(H) = lbdim{Gn}(H) = lim in
n→∞
log |HGn:Gn|
log |G:Gn|.(3.2)
In ac , in he e e ed wo k he au ho p o ed ha lbdim(H)≤hdim(H),as
he o he inequali y is ue in any me ic space. In acco dance wi h ma hema ical
li e a u e, we will use he name Hausdo dimension inasmuch as we will mainly
be conce ned abou he dimension o closed subg oups.
69
3.1.2 Fo mulae: subg oups and quo ien s
Th oughou his chap e , ela ing he Hausdo dimension o a coun ably based
p o ini e g oup o ha o i s subg oups and quo ien s will be o i al impo ance.
The e o e, i is some imes con enien o use he no a ion hdimG
{Gn} o emphasize
ha he dimension, wi h espec o he il a ion se ies {Gn}n∈N, is calcula ed
wi hin he g oup G.
Lemma 3.8 (c . [48, Lemma 5.3]).Le Gbe a coun ably based p o ini e g oup,
{Gn}n∈Na no mal il a ion se ies o Gand H≤cGa closed subg oup whose
Hausdo dimension is gi en by a p ope limi . Then
hdimG
{Gn}(K) = hdimG
{Gn}(H) hdimH
{H∩Gn}(K)
o all K≤cH.
Rema k 3.9. The Hausdo dimension o Habo e being a p ope limi means
ha
hdim{Gn}(H) = lim
n→∞
log |HGn:Gn|
log |G:Gn|.
P oo . A simple compu a ion shows ha
hdimG
{Gn}(K) = lim in
n→∞
log |K:K∩Gn|
log |G:Gn|
= lim
n→∞
log |H:H∩Gn|
log |G:Gn|lim in
n→∞
log |K:K∩H∩Gn|
log |H:H∩Gn|
= hdimG
{Gn}(H) hdimH
{H∩Gn}(K).
Mo eo e , o quo ien s o coun ably based p o ini e g oups we ha e he ollow-
ing esul :
Lemma 3.10 (c . [47, Lemma 2.2]).Le Gbe a coun ably based p o ini e g oup,
{Gn}n∈Na no mal il a ion se ies o Gand N⊴cGa closed no mal subg oup.
Assume ha he Hausdo dimension o Nis gi en by a p ope limi . Then o
e e y subg oup H≤cGcon aining None has
hdimG
{Gn}(H) = 1−hdimG
{Gn}(N)hdimG/N
{GnN/N}(H/N) + hdimG
{Gn}(N).
70

P oo . We obse e ha
log |HGn:NGn|
log |G:Gn|=log |G:NGn|
log |G:Gn|·log |HGn:NGn|
log |G:NGn|
=log |G:Gn|−log |NGn:Gn|
log |G:Gn|·log |HGn:NGn|
log |G:NGn|
=1−log |NGn:Gn|
log |G:Gn|log |HGn:NGn|
log |G:NGn|.
The e o e, since hdimG
{Gn}(N) = ηis gi en by a p ope limi
hdimG
{Gn}(H) = lim in
n→∞
log |HGn:Gn|
log |G:Gn|
= lim in
n→∞ log |HGn:NGn|
log |G:Gn|+log |NGn:Gn|
log |G:Gn|
= lim in
n→∞ 1−log |NGn:Gn|
log |G:Gn|log |HGn:NGn|
log |G:NGn|+η
= (1 −η) lim in
n→∞ log |HGn/N :NGn/N|
log |G/N :NGn/N|+η
= (1 −η) hdimG/N
{NGn/N}(H/N) + η,
as equi ed.
Co olla y 3.11. Le Gbe a coun ably based p o ini e g oup wi h no mal il a ion
se ies {Gn}n∈Nand le N⊴Gbe a ini e no mal subg oup. Then
hspec{Gn}(G) = hspec{GnN/N}(G/N).
P oo . Since hdimG
{Gn}(N) = 0 is gi en by a p ope limi , he inclusion
hspec{GnN/N}(G/N)⊆hspec{Gn}(G)
is a di ec consequence o he Co espondence Theo em and Lemma 3.10.
Fo he con e se, conside η∈hspec{Gn}(G); hen he e exis s H≤cGsuch
ha hdimG
{Gn}(H) = η. Since Nis ini e and he igh mul iplica ion is an isome y
by Lemma 3.10 one has
hdimG
{Gn}(H) = hdimG
{Gn} [
n∈N
Hn!
= hdimG
{Gn}(HN) = hdimG
{GnN/N}(HN/N),
as equi ed.
71
Finally, he combina ion o he abo e esul s yields he ollowing co olla y.
Co olla y 3.12. Le Gbe a coun ably based p o ini e g oup, {Gn}n∈Na no mal
il a ion se ies and le N⊴K≤Gbe closed subg oups such ha hdimG
{Gn}(N) = η
and hdimG
{Gn}(K) = κa e gi en by p ope limi s. I hspec{(K∩Gn)N
N}(K/N) = [0,1]
hen [η, κ]⊆hspec{Gn}(G).
P oo . Fi s ly, by Lemma 3.8 i ollows ha hdimK
{K∩Gn}(N) = η/κ, and using he
Co espondence Theo em and Lemma 3.10 we ob ain
[η/κ, 1] = (1 −η/κ)α+η/κα∈hspecn(K∩Gn)N
No(K/N)⊆hspec{K∩Gn}(K).
By ano he applica ion o Lemma 3.8, one concludes [η, κ]⊆hspec{Gn}(G).
We conclude his subsec ion by s a ing he ollowing esul , due o Klopsch, Thi-
llaisunda am and Zugadi-Reizabal in [48], ha will be o u ili y o ind p o ini e
g oups wi h ull Hausdo spec um:
Theo em 3.13 (c . [48, Theo em 5.4]).Le Gbe a coun ably based p o-pg oup
and le {Gn}n∈Nbe a no mal il a ion se ies. Suppose ha e e y ini ely gene a ed
closed subg oup H≤cGsa is ies hdim{Gn}(H) = 0.Then hspec{Gn}(G) = [0,1].
3.1.3 R-s anda d Hausdo dimension
In he con ex o compac R-analy ic g oups a na u al il a ion is a ailable. In-
deed, le Gbe a compac R-analy ic g oup o dimension dand le (S, φ)be an open
R-s anda d subg oup o le el N. As al eady p esen ed in (1.6), he R-s anda d
il a ion se ies induced by Sis he il a ion se ies {Sn}n∈Nde ined as
Sn:= φ−1mN+n(d),∀n∈N0.
These a e ob iously open subg oups and, by i ue o he K ull In e sec ion
Theo em (loc. ci .) an R-s anda d il a ion se ies is indeed a il a ion se ies.
Fu he mo e, om (1.5) one has ha Sn⊴S o e e y n∈N,and hus o mula
(3.2) holds o R-s anda d g oups wi h he abo e il a ion.
Because o he dependence o hdim on he chosen il a ion we should no assume
a p io i ha he Hausdo dimension ( esp. lowe box dimension) o a subg oup
o a compac R-analy ic g oup is he same when compu ed wi h espec o wo
di e en R-s anda d il a ions. To p o e his ac ual independence, we s a by
ecalling his consequence o (1.7):
72
Rema k 3.14. The Hilbe unc ion o (R, m)is de ined as H:N0→N, n 7→
dimR/m(mn/mn+1),and o la ge enough ni coincides wi h a polynomial p(n)
o deg ee dimK ull(R)−1,called he Hilbe polynomial o R(c . [25, Chap e 6,
Theo em C]). Hence, acco ding o he Eule -Maclau in o mula he sum Pn−1
i=1 p(i)
is asymp o ically equi alen o a polynomial (n)o deg ee dimK ull(R),i.e. hei
a io ends o 1as n ends o in ini y.
Le qbe he size o he esidue ield R/mand le (S, φ)be an R-s anda d g oup
o dimension dand le el N. In iew o (1.7),
logq|S:Sn|=d
N+n−1
X
i=N
H(i)
is asymp o ically equi alen o d (n).
The ollowing esul shows ha he lowe box dimension is independen o he
s anda d il a ion chosen.
Lemma 3.15 (c . [27, Theo em 3.1]).Le Gbe a compac R-analy ic g oup and
le (S, φ)and (T, ψ)be wo open R-s anda d subg oups o G. Fo e e y X⊆Gwe
ha e ha
lbdim{Sn}(X) = lbdim{Tn}(X).
P oo . Le us deno e by N(S)and N(T) espec i ely he le els o Sand T. Fi s ly,
we shall p o e he exis ence o wo in ege s a, b ∈Nsuch ha o e e y in ege n
sa is ying n−b∈N
Sn+a≤Tn≤Sn−b.(3.3)
Indeed, since he R-analy ic map ψ◦φ−1is con e gen in φ(S∩T)⊆mN(S)(d)
and since ψ◦φ−1(0) = 0, acco ding o (1.1) he e exis s L≥N(S)such ha
ψ◦φ−1mL+n(d)⊆(mn)(d)
o any n∈N.Hence, by se ing a=L−N(S)+N(T),we ob ain ha Sn+a≤Tn.
A guing simila ly wi h φ◦ψ−1we ob ain (3.3). Consequen ly,
lbdim{Tn}(X) = lim in
n→∞
log |XTn:Tn|
log |G:Tn|
≤lim in
n→∞
log |XSn+a:Sn+a|
log |G:Sn+a|·log |G:Sn+a|
log |G:Tn|
= lim in
n→∞
log |XSn+a:Sn+a|
log |G:Sn+a|·log |G:Sn+a|
log |G:Sn+a|−log |Tn:Sn+a|
= lbdim{Sn}(X),
73
using in he ul ima e equali y ha
lim
n→∞
log |G:Sn+a|
log |G:Sn+a|−log |Tn:Sn+a|= 1.
Indeed, acco ding o he p e ious ema k
log |Tn:Sn+a| ≤ log |Sn−b:Sn+a|=
N+n+a−1
X
i=N+n−b
H(i).
Hence, o la ge enough n he igh hand side e m is he sum o a+bpolyno-
mials o deg ee dimK ull(R)−1,while log |G:Sn+a|= log |G:S|+ log |S:Sn+a|
is asymp o ically equi alen o a polynomial o deg ee dimK ull(R).We inish he
p oo by swapping Sand T.
This allows us o de ine he s anda d o R-s anda d lowe box dimension, de-
no ed lbdims ,and he R-s anda d uppe box dimension, deno ed ubdims ,dis e-
ga ding he chosen s anda d il a ion. Besides, i is wo h no ing as an aside ha
when Ris a PID he analogue o Lemma 3.15 can be p o ed o he Hausdo
dimension.
Lemma 3.16. Le Rbe p o-pdomain ha is a PID, le Gbe a compac R-analy ic
g oup and le (S, φ)and (T, ψ)be wo open R-s anda d subg oups o G. Fo e e y
X⊆G, we ha e
hdim{Sn}(X) = hdim{Tn}(X).
P oo . Le qbe he size o he esidue ield R/mand d= dim G. Since Ris a PID,
|Sn:Sn+1|=|Tn:Tn+1|=qd o e e y n∈N.F om (3.3) he e exis wo in ege s
a, b ∈Nsuch ha Sn+a≤Tn≤Sn−b o e e y in ege nsuch ha n−b∈N.
Thus,
|G:Tn|−1≥ |G:Sn+a|−1=q−d(a+b)|G:Sn−b|−1.
Hence, i we deno e by δSand δT he dis ances in Ginduced espec i ely by he
il a ion se ies {Sn}n∈Nand {Tn}n∈N, hen
δT(x, y)≥q−d(a+b)δS(x, y).
By swapping Sand Twe ob ain he exis ence o a cons an C > 0such ha
C·δS(x, y)≥δT(x, y).Hence, he iden i y map be ween he me ic spaces (G, δS)
and (G, δT)is bi-Lipschi z, and he esul ollows by P oposi ion 3.6.
74
whe e k= dimxM. Rep oducing he a gumen s o (3.6) e ba im, i δSis he
dis ance in Ginduced by S, we ob ain ha
hdimδS(Ux) = dimxM
dim G.
Le (T, ψ)be ano he open R-s anda d subg oup o Gand le δTbe he dis ance
de ined in Gby using T. On he one hand, om [26, P oposi ion 3.4] and (3.6),
hdimδT(Ux)≤ubdims (Ux) = dimxM
dim G.
Le us de ine ˜
Ux=x−1Ux∩T, so in pa icula ψ(˜
Ux)⊆ψ(S∩T).Then
φ(˜
Ux) = φ(x−1Ux∩T) = mN(k)×{0}(d−k)∩φ(S∩T),
and since φ(S∩T)is open in m(d), he e exis s K∈Nsuch ha
mK(k)×{0}(d−k)⊆φ(˜
Ux).
Le us deno e by d he dis ance de ined on m(d)using he s anda d il a ion se ies
n(mn)(d)on∈N.Since φand ψa e isome ies, hen φ◦ψ−1is also an isome y om
(ψ(S∩T), d) o (φ(S∩T), d).Hence, in iew o Lemma 3.17, P oposi ion 3.6 and
he mono onici y,
hdimG
δT(Ux)≥hdimT
δT˜
Ux= hdimm(d)
δψ(˜
Ux)= hdimm(d)
δφ◦ψ−1◦ψ(˜
Ux)
= hdimm(d)
δ(φ(˜
Ux)) ≥hdimm(d)
{(mn)(d)}mK(k)×{0}(d−k)=k/d,
using Lemma 3.23 in he las equali y. The e o e, hdim(Ux) = dimxM/dim G,un-
ega ding he s anda d il a ion chosen. We inish as in (3.7) and (3.8), bu
eplacing he box dimension wi h he Hausdo dimension.
Conside ing ha R-analy ic subg oups a e closed pu e R-analy ic submani olds,
we eco e he p incipal esul in [27]:
Co olla y 3.27 (c . [27, Main Theo em]).Le Gbe a compac R-analy ic g oup
and le Hbe an R-analy ic subg oup. Then
bdims (H) = hdims (H) = dim H
dim G.
In pa icula , bo h dimensions a e a p ope limi .
81

When R=Zpe e y closed subg oup is p-adic analy ic (see [24, Theo em 9.6]),
so we ob ain an al e na i e p oo o Theo em 3.2, and in passing we ge a clea -cu
exp ession o s anda d spec a in his se ing. In ac , i Gis a d-dimensional
compac p-adic analy ic g oup, hen
hspecs (G)⊆0,1
d, . . . , d−1
d,1.
3.3 Abelian compac R-analy ic g oups
Hence o h, we will ca y on desc ibing he s anda d spec a o p o ini e R-analy ic
g oups ha a e no p-adic analy ic, i.e. Ris no a ini e ex ension o Zp.In he
i s place, we will deal wi h he abelian case.
Rema k. F om now on we will wo k wi h closed subg oups, and Theo em 3.7
will be implici ly used.
P oposi ion 3.28. Le Rbe a p o-pdomain o cha ac e is ic po K ull dimension
a leas 2,and le (S, φ)be an abelian R-s anda d g oup. Then hspecs (S) = [0,1].
P oo . By Theo em 3.13 i suffices o p o e ha e e y ini ely gene a ed closed
subg oup H≤cSsa is ies hdims (H) = 0.Le dbe he dimension o Sand le
H≤Sbe a opologically -gene a ed closed subg oup.
I Rhas cha ac e is ic p, since he g oup ope a ion in Sis gi en by a o mal
g oup law, by (1.3) whene e x∈Snwe ha e
φ(xp)≡pφ(x) = 0mod m2n(d),
and hus xp≡1 (mod S2n).The e o e Sn/S2nis an elemen a y abelian p-g oup.
Since Sis abelian, H/(H∩Sn)is an abelian p-g oup o exponen pewhe e
e≤ dlog2(n)e.Mo eo e , His opologically -gene a ed, so H/(H∩Sn)is -
gene a ed, and hus
|H:H∩Sn| ≤ pe ≤p⌈log2(n)⌉ .
Acco ding o Rema k 3.14, i |R/m|=q=pc, hen |S:Sn|is asymp o ically
equi alen o qd (n)whe e (n)is a polynomial o deg ee dimK ull(R).Consequen ly,
hdims (H) = lim in
n→∞
logp|H:H∩Sn|
logp|S:Sn|≤lim in
n→∞
dlog2(n)e
cd (n)= 0,
as desi ed.
82
Simila ly, i Rhas K ull dimension o a leas 2,by (1.3), whene e x∈Snwe
ha e
φ(xp)≡pφ(x)≡0mod mn+1(d),
so Sn/Sn+1 is an elemen a y abelian p-g oup. Consequen ly, H/(H∩Sn)is an
-gene a ed abelian g oup o exponen pe,whe e e≤n−1.The e o e
|H:H∩Sn| ≤ p e ≤p (n−1),
so, acco ding o Rema k 3.14,
hdims (H) = lim in
n→∞
logp|H:H∩Sn|
logp|S:Sn|≤lim in
n→∞
(n−1)
cd (n)= 0,
as (n)is a polynomial o deg ee dimK ull(R)≥2.
Clea ly, in iew o o Co olla y 3.20, his esul can be gene alised o abelian
compac R-analy ic g oups.
Co olla y 3.29. Le Rbe a p o-pdomain o cha ac e is ic po K ull dimension
a leas 2.I Gis an abelian compac R-analy ic g oup, hen hspecs (G) = [0,1].
Fu he mo e, i is known ha any R-s anda d g oup o dimension one is abelian
(see [35, Theo em 1.6.7]), and we hus ha e he ollowing:
Co olla y 3.30. Le Rbe a p o-pdomain o cha ac e is ic po K ull dimension
a leas 2and le Gbe a compac R-analy ic g oup o dimension one. Then
hspecs (G) = [0,1].
3.4 Compac Fp[[ ]]-analy ic g oups
Sec ion 3.3 in i es o su mise ha when Gis a soluble compac R-analy ic g oup
ha is no p-adic analy ic, i s R-s anda d spec um is he whole eal in e al [0,1].
The main s a egy o p o e ha would lie in adding successi e in e als o he
spec um, using he consecu i e abelian quo ien s o a subno mal se ies. In ac ,
we ha e he ollowing esul :
Lemma 3.31. Le Gbe a compac R-analy ic g oup and le N⊴K≤Gbe
R-analy ic subg oups such ha hspecs (K/N) = [0,1].Then
dim N
dim G,dim K
dim G= [hdims (N),hdims (K)] ⊆hspecs (G).
83
P oo . The equali y is a di ec consequence o Co olla y 3.27, namely hdims (H) =
dim H/dim G o e e y analy ic subg oup H≤Gand such dimension is gi en by a
p ope limi , and he inclusion is s aigh o wa d om Co olla y 3.12, Lemma
3.21 and Lemma 3.22.
Thus, we shall es ablish a use ul c i e ion o inding R-analy ic subg oups o a
compac R-analy ic g oup. The main obs acle compa ed wi h classical Lie heo y
a ises he e: i is well known ha any closed subg oup o a eal (p-adic) Lie g oup
is a eal (p-adic) Lie subg oup; ne e heless o R-analy ic g oups, closeness is a
necessa y condi ion (see Lemma 1.57), bu no sufficien . Fo example, he addi-
i e g oup Fp[[ ]] is an Fp[[ ]]-analy ic g oup and Fp[[ 2]] is a closed subg oup wi h
i s own Fp[[ ]]-analy ic g oup s uc u e. Howe e , hose mani old s uc u es a e
no compa ible, so Fp[[ 2]] is no an Fp[[ ]]-analy ic subg oup o Fp[[ ]].
Now we will u n o he case when R=Fp[[ ]].The ask o inding Fp[[ ]]-
analy ic subg oups can be ca ied ou by using P oposi ion 1.53, which shows ha
analy ic subse s ha e a mani old s uc u e o e Fp[[ ]].Acco ding o he de ini ion
he ein, a se X⊆Mis an analy ic subse i o each x∈X he e exis an open
neighbou hood Uo xand some Fp[[ ]]-analy ic unc ions 1, . . . , de ined on U
( o some = x) such ha
X∩U={y∈U| i(y) = 0 ∀i= 1, . . . , }.
We hen ha e:
Theo em 3.32 (c . [45, Co olla y 4.2]).Le Gbe an Fp[[ ]]-analy ic g oup and
le Hbe bo h a subg oup o Gand an analy ic subse o G. Then His an Fp[[ ]]-
analy ic subg oup o G.
Le us see some examples o applica ions o he p eceding heo em:
Co olla y 3.33. Le Sbe an Fp[[ ]]-s anda d g oup and ain S. Then Z(S)and
CS(a)a e Fp[[ ]]-analy ic subg oups.
P oo . By he p e ious heo em i is enough o show ha Z(S)and CS(a)a e
analy ic subse s. The o me is p o ed in [45, Co olla y 4.3], while he la e
ollows he same spi i . Indeed, since Sis Fp[[ ]]-s anda d o le el say Nand
dimension say d, hen i can be iden i ied wi h  N(d).Since he g oup ope a ion
is gi en by a o mal g oup law, by (1.5) he e exis some gi,α ∈Fp[[ ]][[X1, . . . , Xd]]
such ha
πiy−1ay=ai+X
|α|≥1
gi,α(a)yα1
1. . . yαd
d=ai+hi(y)
84
o all yin S, whe e he map πi: N(d)→ Nis he p ojec ion o he i h
coo dina e. Mo eo e , he maps hi(y) = P|α|≥1gi,α(a)yα1
1. . . yαd
da e clea ly Fp[[ ]]-
analy ic. The e o e
CS(a) = y∈S|πiy−1ay=ai∀i= 1, . . . , d
={y∈S|hi(y) = 0 ∀i= 1, . . . , d},
and CS(a)is an analy ic subse .
The second applica ion in ol es he gene al linea g oup GLn(R).In addi ion
o he usual opology in GLn(R),namely he one induced by he ing opology o
R, we also ha e he so-called Za iski opology, in which he closed subse s a e he
affine se s, i.e. subse s o he o m
{A∈GLn(R)| (A) = 0 ∀ ∈ F},
whe e F ⊆ R[X]is a subse o polynomials in n2 a iables. No e as well ha any
subg oup H≤GLn(R)can be likewise endowed wi h bo h he usual subspace
opology o he weake (polynomial maps a e con inuous wi h espec o he m-
adic opology) Za iski opology.
Le us p esen some gene al ac s conce ning he Za iski opology:
P oposi ion 3.34 (c . [73, Lemma 5.9 and Theo em 5.11]).Le H≤GLn(R)
and le Hbe i s Za iski closu e.
(i) Then H ≤ GLn(R).
(ii) I His no mal in GLn(R),so is H.
(iii) Suppose ha His nilpo en o class c. Then, Hhas a cen al se ies o leng h
cconsis ing o Za iski closed subg oups. In pa icula , His nilpo en o class
c.
(i Suppose ha His soluble o leng h c. Then Hhas a subno mal se ies o
leng h cconsis ing o Za iski closed subg oups whose quo ien g oups a e
abelian. In pa icula , His soluble o leng h c.
(i ) Le Kbe a subg oup o Hsuch ha |H:K|is ini e and le Kbe he Za iski
closu e o Kin GLn(R).Then Khas ini e index in H.
Co olla y 3.35. Le G⊆GLn(Fp[[ ]]) be a linea Fp[[ ]]-analy ic g oup and le
Hbe a Za iski closed subg oup o GLn(Fp[[ ]]).Then H∩Gis an Fp[[ ]]-analy ic
subg oup o G.
85
P oo . Since His closed in he Za iski opology, i is an affine se , ha is, he e
exis s a subse Fo Fp[[ ]][X],whe e Xis a uple o n2 a iables, such ha
H={A∈GLn(Fp[[ ]]) | (A) = 0 ∀ ∈ F}.
Bu since Fp[[ ]][X]is Noe he ian we can assume F o be ini e, and hus
H∩G={A∈G| (A) = 0 ∀ ∈ F}
is an analy ic subse , so i is an Fp[[ ]]-analy ic subg oup by Theo em 3.32.
We a e now in a posi ion o p o e pa o Theo em 3.3 by using he p e ious
esul s:
Theo em 3.36. Le Gbe a soluble compac Fp[[ ]]-analy ic g oup. Then,
hspecs (G) = [0,1].
P oo . By Co olla y 3.20, we can assume wi hou loss o gene ali y ha Gis
Fp[[ ]]-s anda d. We i s p o e he heo em o he case when Gis linea o e
Fp[[ ]], ha is, G⊆GLn(Fp[[ ]]).Le Gbe he Za iski closu e o Gin GLn(Fp[[ ]]).
Acco ding o P oposi ion 3.34, Gis a soluble g oup, and he e exis s a subno mal
se ies
G=H1⊵H2⊵···⊵Hℓ−1⊵Hℓ={1}
consis ing o Za iski closed subg oups whose quo ien g oups a e all abelian. Then
G=H1∩G⊵H2∩G⊵···⊵Hℓ−1∩G⊵Hℓ∩G={1}
is a soluble se ies o Ggi en by Fp[[ ]]-analy ic subg oups by Co olla y 3.35.
Deno e Hi=Hi∩G. Since each Hiis an Fp[[ ]]-analy ic subg oup o G hen
Hi−1/Hiis a compac abelian Fp[[ ]]-analy ic g oup o all i∈ {2, . . . , `}, so by
Co olla y 3.29 i ollows ha hspecs (Hi/Hi−1) = [0,1].Hence by Lemma 3.31 one
has ha [hdims (Hi),hdims (Hi−1)] ⊆hspecs (G) o all i∈ {2, . . . , `},and hus
hspecs (G) = [0,1].
Le us inally u n o he gene al case. By Co olla y 3.33, Z(G)is an abelian
Fp[[ ]]-analy ic subg oup o Gand hus by Co olla y 3.29 and Lemma 3.31
[0,hdims Z(G)] ⊆hspecs (G).
86

Mo eo e , by P oposi ions 1.59 and 2.3 one has ha G/Z(G)is a compac soluble
Fp[[ ]]-analy ic g oup ha is linea o e Fp[[ ]]. Hence, acco ding o Lemma a 3.21
and 3.22,
hspec{SnZ(G)/Z(G)}(G/Z(G)) = hspecs (G/Z(G)) = [0,1],
and so by Co olla y 3.12
[hdims Z(G),1] ⊆hspecs (G),
hus ob aining he whole in e al in he spec um.
Mo e gene ally, a sui able way o ind an in e al in he Fp[[ ]]-s anda d Haus-
do spec um o a compac Fp[[ ]]-analy ic g oup Gis looking o a soluble Fp[[ ]]-
analy ic subg oup. This sea ch will ely hea ily on he opological analogue o
he Ti s al e na i e. Bu we i s obse e he ollowing:
Lemma 3.37. Le Gbe an Fp[[ ]]-s anda d g oup. Suppose ha ei he
(i) Z(G)is in ini e o
(ii) Gcon ains an elemen xo in ini e o de .
Then [0,1/dim G]⊆hspec(G).
P oo . Unde he i s hypo hesis, by Co olla y 3.33, Z(G)is an abelian in ini e
Fp[[ ]]-analy ic subg oup. Simila ly, unde he second hypo hesis Z(CG(x)) is an
abelian Fp[[ ]]-analy ic subg oup which is in ini e, because hxi ≤ Z(CG(x)).In
bo h cases, he e exis s an in ini e abelian Fp[[ ]]-analy ic subg oup H≤G. Since
Gis compac , Hhas s ic ly posi i e analy ic dimension, and acco ding o Co ol-
la y 3.29 he Fp[[ ]]-s anda d spec um o His he whole in e al [0,1].Finally,
by Lemma 3.31, [0,dim H/dim G]is con ained in he Fp[[ ]]-s anda d spec um.
Theo em 3.38. Le Gbe a compac Fp[[ ]]-analy ic g oup. Then,
[0,1/dim G]⊆hspecs (G).
P oo . We can assume, in iew o Co olla y 3.20, ha Gis R-s anda d. Fi s ly,
obse e ha when Z(G)is in ini e he esul ollows by Lemma 3.37(i), so we shall
deal wi h he case when Z(G)is ini e. Bu hen G/Z(G)is an Fp[[ ]]-analy ic
g oup o dimension dim Gand acco ding o Co olla y 3.11 i ollows ha
hspecs (G) = hspecs (G/Z(G)).
87
Fu he mo e, by P oposi ion 2.3, G/Z(G)is an Fp[[ ]]-analy ic g oup ha is linea
o e Fp[[ ]].Hence by he opological Ti s al e na i e (c . [12, Theo em 1.3]) i
ollows ha G/Z(G)con ains ei he an open soluble subg oup, say H, o con ains
a dense ee subg oup. In he o me case, His a soluble Fp[[ ]]-analy ic g oup o
dimension dim G/Z(G) = dim Gand hus
hspecs (G/Z(G)) = [0,1] .
In he la e case G/Z(G)con ains an elemen o in ini e o de and he s a emen
ollows by Lemma 3.37(ii).
3.5 Classical Che alley g oups
The spec um o a compac Fp[[ ]]-analy ic g oup need no be he whole in e al
[0,1].Fo ins ance, conside he special linea g oup SLn(Fp[[ ]]).I is well-known
ha SLn(Fp[[ ]]) is a compac Fp[[ ]]-analy ic g oup o dimension n2−1,con aining
as an open subg oup he Fp[[ ]]-s anda d g oup
SL1
n(Fp[[ ]]) := ke {SLn(Fp[[ ]] →SLn(Fp[[ ]]/ Fp[[ ]])}.
In [6, Co olla y 1.5], he s anda d spec um o SL2(Fp[[ ]]) is comple ely es ab-
lished when p > 2, o wi
hspecs (SL2(Fp[[ ]])) = [0,2/3]∪{1}.
Mo eo e , in [6, Theo em 1.4] i is p o ed ha when p > 2,
hspecs (SLn(Fp[[ ]])) ∩1−1
n+ 1,1=∅,
and 1is an isola ed poin o he spec um he eo . We will p o ide u he exam-
ples o compac Fp[[ ]]-analy ic g oups whose spec um is no he whole in e al,
by p o ing an analogous esul o he o he classical Che alley g oups. Fo ha
pu pose, we will ollow he same echniques al eady used in [6] and wo k in he
co esponding g aded Lie algeb a. We s a wi h a b ie summa y o hose ma ix
g oups. Fo basic de ini ions ega ding oo sys ems and a comp ehensi e analysis
on he opic we e e o [15]. Le Rbe a gene al p o-pdomain.
• The Che alley g oup o e Rassocia ed o a oo sys em o ype An(n≥1)
is SLn+1(R).
88
• A oo sys em o ype Bn(n≥2) de ines he odd special o hogonal g oup
SO2n+1(R) := A∈M2n+1(R)A K2n+1A=K2n+1,
whe e Kn=




0. . . 0 1
0. . . 1 0
.
.
.
...
.
.
..
.
.
1. . . 0 0




∈Mn(R),which is an R-analy ic g oup o
dimension n(2n+ 1).
• A oo sys em o ype Cn(n≥3) de ines he symplec ic g oup
Sp2n(R) := A∈M2n(R)|A J2nA=J2n,
whe e J2n=0Kn
−Kn0,which is an R-analy ic g oup o dimension
n(2n+ 1).
• A oo sys em o ype Dn(n≥4) de ines he e en special o hogonal g oup
SO2n(R) := A∈M2n(R)|A K2nA=K2n,
which is an R-analy ic g oup o dimension n(2n−1).
All hose g oups a e compac , being closed subse s o he compac space Mn(R)
∼
=R(n2).Tha is, classical Che alley g oups o e Ra e ac ually compac R-analy ic
g oups. Fu he mo e, he ollowing esul desc ibes hei associa ed Lie algeb as.
Theo em 3.39 (c . [24, Exe cise 13.11(iii)]).Le Xnbe a oo sys em o ype An
(n≥1), Bn(n≥2), Cn(n≥3) o Dn(n≥4). Le G(R)be he Che alley g oup
associa ed o Xno e a p o-pdomain R.
(i) I Xn=An, he e exis s an open R-s anda d g oup Ssuch ha
L(S)∼
=sln+1(R) = {A∈Mn+1(R)| (A) = 0}.
(ii) I Xn=Bn, he e exis s an open R-s anda d subg oup Ssuch ha
L(S)∼
=so2n+1(R) = A∈M2n+1(R)|A =−A.
89
(iii) I Xn=Cn, he e exis s an open R-s anda d subg oup Ssuch ha
L(S)∼
=sp2n(R) = A∈M2n(R)|J2nA+A J2n= 0,
whe e J2nis de ined as be o e.
(i ) I Xn=Dn, he e exis s an open R-s anda d subg oup Ssuch ha
L(S)∼
=so2n(R) = A∈M2n(R)|A =−A.
He e L(S)is an abb e ia ion o he Lie algeb a associa ed o S(compa e wi h
Sec ion 1.4).
We s a by p esen ing he cons uc ion in [54, De ini ion 2.9]. Gi en an R-
s anda d g oup (S, φ)and he co esponding R-s anda d il a ion {Sn}n∈Nwe
de ine he g aded Lie R/m-algeb a g L(S) = Ln≥0Sn/Sn+1,which is so wi h he
Lie b acke ob ained ex ending by bilinea i y he ule
[xSn+1, ySm+1]g L(S):= [x, y]Sn+m+1
( he igh -hand side b acke s s and o he g oup commu a o in S).
On he one hand, [,]g L(S)is a Lie b acke . Indeed, om (1.5),
φ([x, y]) = B(φ(x), φ(y)) −B(φ(y), φ(x)) mod φ(Sn+m+1),
and he e o e [·,·]g L(S)sa is ies Jacobi’s iden i y by i ue o Lemma 1.25. On he
o he hand, whene e x∈Snand y∈Sm, hen [x, y]∈Sn+m,so he abo e-de ined
algeb a is g aded o e he na u al numbe s. Finally, le qbe he ca dinali y o
R/m.Since each Sn/Sn+1 is an R/m- ec o space, g L(S)is a g aded Fq-Lie al-
geb a.
Any closed subg oup H≤cSde ines a g aded subalgeb a o g L(S), which by
abuse o no a ion we will deno e by g L(H)and is gi en by
g L(H) := M
n≥0
(H∩Sn)Sn+1
Sn+1
.
Al hough e e y closed subg oup de ines a g aded subalgeb a, he e migh be
g aded subalgeb as ha do no a ise in his way.
90
Examples 4.5. Le Gbe a g oup.
(i) Bo h Gi sel and he i ial subg oup {1}a e e bal subg oups co espond-
ing, espec i ely, o he wo d w(x) = xand he emp y wo d.
(ii) One o he mos common wo ds is he commu a o γ2(x, y)=[x, y] =
x−1y−1xy. I s e bal subg oup is he de i ed subg oup γ2(G) = G′and i s
ma ginal subg oup is he cen e γ∗
2(G) = Z(G).
(iii) Lowe cen al wo ds a e de ined ecu si ely as
γn(x1, . . . , xn) := [γn−1(x1, . . . , xn−1), xn]∀n≥3,
and de i ed wo ds a e de ined ecu si ely as δ1(x1, x2) := γ(x1, x2)and
δn(x1, . . . , x2n) := [δn−1(x1, . . . , x2n−1), δn−1(x2n−1+1, . . . , x2n)] ∀n≥2.
(i ) The Bu nside wo d wm(x) = xmde ines he e bal subg oup Gm, he sub-
g oup gene a ed by he m h powe s o elemen s o G, and o ins ance, he
ma ginal subg oup w∗
2(G)consis s on all cen al elemen s o o de di iding
2, ha is, w∗
2(G) = {g∈Z(G)|g2= 1}.
P. Hall [33] posed se e al ques ions ega ding he ela ion be ween he se o
w- alues and he co esponding e bal and ma ginal subg oups. The ollowing
de ini ions will se e o summa ise some o hose:
De ini ion 4.6. Le wbe a wo d and le Cbe a class o g oups.
(i) A wo d wis concise in Ci o e e y G∈ C we ha e ha w(G)is ini e
whene e w{G}is ini e.
(ii) A wo d wis obus in Ci o e e y G∈ C we ha e ha w(G)is ini e
whene e |G:w∗(G)|is ini e.
Consonan ly, wis concise ( esp. obus ) i i is concise ( esp. obus ) in he class
o all g oups. Likewise, i e e y wo d is concise in a g oup G, we will say ha G
is e bally concise.
In gene al, since |w{G}| ≤ |G:w∗(G)|k(being k he numbe o a iables o
he wo d w), obus ness is s onge han conciseness: i wis concise in C, hen w
is obus in C.Howe e , o e esidually ini e g oups, and all he g oups we a e
conce ned abou a e so, bo h concep s a e equi alen :
97

Lemma 4.7 (c . [67, Lemma 1.4.1]).Le Gbe a g oup and le wbe a wo d.
(i) I |G:w∗(G)|is ini e, hen w{G}is ini e.
(ii) I Gis esidually ini e and w{G}is ini e, hen |G:w∗(G)|is ini e.
One o he o iginal p edic ions o P. Hall was ha all g oups we e e bally con-
cise. Howe e , his conjec u e was e u ed almos h ee decades la e by I ano
[39], by inding a g oup Gand a wo d wsuch ha w{G}has wo elemen s, bu
w(G)is in ini e cyclic. None heless, his coun e example, o he compa able coun-
e example cons uc ed by Ol’shanskiĭ (see [62, Theo em 39.7]), is no esidually
ini e. This leads us o he ollowing conjec u e, p oposed by Jaikin-Zapi ain [44]
and Segal [67]:
Conjec u e 4.8 (Conciseness conjec u e o esidually ini e g oups).E e y wo d
is concise in he class o esidually ini e g oups.
The e a e ew known examples o classes o e bally concise g oups. Apa
om he ob ious examples o abelian (see Lemma 4.2) and pe iodic (see upcom-
ing Lemma 4.15) g oups; in he decade o 1960s, Me zjalko [57] and Tu ne -Smi h
[72] p o ed, espec i ely, ha linea g oups and g oups all o whose quo ien s a e
esidually ini e (e.g. i ually nilpo en g oups) a e e bally concise.
When he se o w- alues is in ini e, he compa able no ion o ha o conciseness
is e bal ellip ici y. In o de o de ine i , we will use he ollowing no a ion: o a
subse X⊆G, we deno e by X∗ℓ he se o all p oduc s o a mos `elemen s o
Xand hei in e ses.
De ini ion 4.9. Le wbe a wo d and le Gbe a g oup. We say ha wis ellip ic
in G, i he e exis s `∈Nsuch ha w(G) = w{G}∗ℓ.
The smalles o such in ege s `is he e bal wid h o win G. In consonance, a
g oup Gis e bally ellip ic when e e y wo d is ellip ic in G. Mo eo e , ellip ici y
is s onge han conciseness: whene e wis ellip ic in all he g oups o a class C,
hen wis also concise in C.
I ollows om Lemma 4.2 ha abelian g oups a e e bally ellip ic and in hem
all wo ds ha e e bal wid h equal o 1. Aside om hem, i is known ha lin-
ea algeb aic g oups∗(c . [56]), ini ely gene a ed i ually abelian-by-nilpo en
∗by a linea algeb aic g oup we mean a Za iski closed subg oup o GLn(K) o some n∈N
and an algeb aically closed ield K.
98
g oups (c . [29] and [71]) o , di ec ly ela ed o he opic o his hesis, compac
p-adic analy ic g oups (c . [44]) a e e bally ellip ic.
None heless, he e a e na u al examples o non- e bally ellip ic g oups. Fo
ins ance, Roman’ko [66] p esen ed a ini ely gene a ed soluble p o-pg oup whe e
he second de i ed wo d δ2(x1, . . . , x4) = [[x1, x2],[x3, x4]] has in ini e wid h.
Rega ding p o ini e g oups, e bal wid h is ela ed o whe he he co esponding
e bal subg oup is closed.
P oposi ion 4.10. Le Gbe a compac Hausdo opological g oup and le wbe
a wo d. Then wis ellip ic in Gi and only i w(G)is closed.
P oo . Fo he only i implica ion, no e ha o e e y in ege n he se w{G}∗n
is closed, as i is he con inuous image o a compac se . Thus, i `is he e bal
wid h o w, hen w(G) = w{G}∗ℓis closed.
Fo he i , no e ha
w(G) = [
n∈N
w{G}∗n
whe e each w{G}∗nis closed. The e o e, since w(G)is a compac Hausdo
opological space, by he Bai e Ca ego y Theo em (c . [59, Theo em 48.2]), he e
exis s an in ege msuch ha w{G}∗mhas non-emp y in e io in w(G), i.e. i
con ains a non-emp y open subse U⊆ow(G).Hence,
w(G) = [
g∈w(G)
gU,
and by he compac ness o w(G)
w(G) =
[
i=1
giU,
o some elemen s g1, . . . , g ∈w(G).Take k∈Nsuch ha gi∈w{G}∗k o all
i∈ {1, . . . , }, hen
w(G) =
[
i=1
giU⊆w{G}∗(k+m),
as desi ed.
99
Gene ally, knowing ha a ( e bal) subg oup is closed can be help ul when
wo king wi h p o ini e g oups. Tha is why we should inish his in oduc ion by
poin ing ou he ollowing c i ical esul due o Jaikin-Zapi ain.
Theo em 4.11 (c . [44, Theo em 1.1]).Le w∈Fkbe a wo d in k a iables.
Then whas ini e wid h in all ini ely gene a ed p o-pg oups i and only i w /∈
δ2(Fk) (F′
k)p.
This chap e is de o ed o p o ing ha compac R-analy ic g oups a e e bally
concise. I is wo h men ioning ha when cha R= 0, his esul is a di ec
consequence o Theo em 2.27 –e e y compac R-analy ic g oup is linea – oge he
wi h Me zjalko ’s Theo em –linea g oups a e e bally concise–. Bu in spi e o
ha , i is in e es ing o p o ide an independen p oo , which is alid o any p o-p
domain ega dless o he cha ac e is ic.
Fu he mo e, his gene al esul p o ides ye ano he e idence o a posi i e
answe o he ques ion o whe he compac analy ic g oups a e linea in posi i e
cha ac e is ic.
4.1 Conciseness in R-s anda d g oups
In he class o R-s anda d g oups, conciseness is s aigh o wa d:
P oposi ion 4.12. Le Sbe an R-s anda d g oup and le wbe a wo d such ha
w{S}is ini e. Then w(S) = {1}.
P oo . Fi s ly, Scan be iden i ied wi h mN(d),whe e Nis he le el and d he
dimension o S, such ha he mul iplica ion and he in e sion a e de ined by wo
uples o powe se ies and he iden i y is 0.Consequen ly, he wo d map wis a
single powe se ies W∈R[[X1, . . . , Xdk]](d),whe e kis he numbe o a iables
o he wo d. Since w{S}is ini e and he wo d map is con inuous, Wis locally
cons an , so by Lemma 1.8 Wis cons an , i.e W(X1, . . . , Xdk) = W(0, . . . , 0) = 0,
and hus w{S}={0}.
We should ake no ice o a couple o consequences o he p e ious esul . On
he one hand, any R-analy ic g oup Gsa is ies a weake e sion o he conciseness
conjec u e:
Co olla y 4.13. Le Gbe an R-analy ic g oup and le wbe a wo d such ha
w{G}is ini e. The e exis s an open R-s anda d subg oup Swhe e wis a law,
ha is, w(S) = {1}.
100
P oo . Acco ding Lemma 1.21, he e exis s an open R-s anda d subg oup So G.
Since |w{S}| ≤ |w{G}|, om P oposi ion 4.12 i ollows ha w(S) = {1}.
On he o he hand, i Gis a compac R-analy ic g oup such ha w{G}is
ini e, we can ob ain he se o w- alues jus by looking a a ans e sal o a
con enien R-s anda d subg oup. Indeed, le Gbe a compac R-analy ic g oup,
S⊴oGan open no mal R-s anda d subg oup whose conjuga ion maps a e s ic ly
analy ic, which exis s by Lemma 1.23 (p o ided ha Ris no a PID), and le
Tbe a le ans e sal o Sin G. We should b ing he a las induced by S o
mind, namely he a las {( S, φ )} ∈Twhe e φ (x) = φ( −1x).As a consequence o
Lemma 1.24, he R-analy ic wo d map w:G(k)→G, which is no hing bu an
adequa e composi ion o mul iplica ion and in e sion maps, is gi en by a single
uple o powe se ies on he open subse 1S×···× kS( i∈T), i.e. he e exis s
a uple o powe se ies W 1,..., k∈R[[X1, . . . , Xdk]](d)such ha
φp(w(x1, . . . , xk)) = W 1,..., k(φ 1(x1), . . . , φ k(xk)) ∀xj∈ jS, (4.2)
whe e pis he elemen o Tsuch ha w( 1, . . . , k)p−1∈S.
I we u he assume ha w{G}is ini e, he con inuous map wis locally con-
s an , so by Lemma 1.8, W 1,..., kis cons an , i.e.
W 1,..., k(X1, . . . , Xdk) = c∈R(d).
Tha is, φp(w(x1, . . . , xk)) = c o all xj∈ jS. In o he wo ds,
P oposi ion 4.14. Le Gbe a compac R-analy ic g oup, le wbe a wo d and
le Sbe an open no mal R-s anda d subg oup whose conjuga ion maps a e s ic ly
analy ic. I w{G}is ini e, hen Sis ma ginal o w.
4.2 Conciseness in compac Fp[[ ]]-analy ic g oups
The demons a ion echnique will consis in educing he p oblem o a g oup ha
is analy ic o e a p o-pdomain o K ull dimension one, by using he change o
ings desc ibed in Sec ion 2.2. Hence, i s ly we shall deal wi h he 1-dimensional
case. Compac p-adic analy ic g oups a e e bally concise, as hey a e linea by
i ue o Co olla y 2.6. Hence, in iew o Co olla y 1.44 we can es ic o he
case R=Fp[[ ]].
We shall p oceed ia se e al echnical esul s. Some o hem will be p o ed,
whils o he s will be simply s a ed. We s a wi h he ollowing ” olklo e” esul :
101
Lemma 4.15. Le Gbe a g oup and le wbe a wo d such ha w{G}is ini e.
Then w(G)′is ini e, and w(G)is ini e i and only e e y w- alue in Ghas ini e
o de .
P oo . Le g∈G. By Lemma 4.3, w{G}g⊆w{G},i.e. o all x∈w{G} he
conjugacy class xGis con ained in w{G}.Hence
|G:CG(x)|=wG≤ |x{G}|,
so CG(x)has ini e index in G. The e o e, CG(w(G)) = ∩x∈w{G}CG(x)has ini e
index in G, and so |w(G) : Z(w(G))|is also ini e. Thus, by Schu ’s Theo em (c .
[65, Theo em 10.1.4]), w(G)′is ini e.
Finally, i w(G)is ini e, i mus ha e ini e exponen . Con e sely, suppose ha
he elemen s o w{G}ha e ini e o de , hen w(G)/w(G)′is an abelian g oup
ini ely gene a ed by elemen s o ini e o de , in pa icula , i is ini e; and he
ini eness o w(G)′yields he esul .
Mo eo e , we will need he ollowing Schu - ype esul :
Lemma 4.16 (c . [45, P oposi ion 5.1]).Le Gbe a g oup wi h a nilpo en
no mal subg oup N. Suppose ha NZ(G)
Z(G)has ini e exponen . Then [N, G]has
ini e exponen .
P oo . The Hall-Pe escu o mula (see [38, III.9.4]) s a es ha o all m∈N,
xmym= (xy)mc2(x, y)(m
2). . . cm(x, y)(m
m),(4.3)
whe e c (x, y)∈γ (hx, yi).
Le mbe he exponen o NZ(G)
Z(G),acco ding o (4.3), o all n∈Nand g∈G:
[n, g]m≡n−m(n[n, g])m=n−m(ng)m= [nm, g] = 1 mod γ2(K),(4.4)
whe e K:= hn, [n, g]i ≤ N. Mo eo e , by (4.3) and (4.4), o e e y l∈Nwe ha e
([n1, g1]. . . [nl, gl])m≡[n1, g1]m. . . [nl, gl]m≡1 mod γ2(N).
Le η(m)be† he o de o he la ges 2-gene a ed nilpo en g oup o exponen
m. I suffices o p o e ha γ2(N)η(m)={1}.Fo ha , le x, y ∈Nand H=
hx, yi ≤ N. Since H/Z(H)is a 2-gene a ed nilpo en g oup o exponen di iding
†This in ege exis s because, as Bae [4] p o ed, nilpo en g oups sa is y he Bu nside p oblem
(see [20, Theo em 2.23]).
102

m, i is ini e and k=|H:Z(H)|di ides η(m).Mo eo e , θ:H→Z(H),h7→ hk
is he ans e o Hin o Z(H)(compa e wi h he p oo o [65, Theo em 10.1.3]).
In pa icula , θis a homomo phism, so (xy)k=xkyk.Since xkand ykcommu e
and kdi ides η(m),we ha e ha
(xy)η(m)=xη(m)yη(m)∀x, y ∈N.
In pa icula , θ′:N→Z(N), n 7→ nη(m),is a homomo phism, and since im θ′
is abelian, γ2(N)mus be con ained in ke θ′, ha is, γ2(N)η(m)={1}.
The upcoming auxilia y esul s use ideas om he heo y o linea algeb aic
g oups. The eade is di ec ed o [37] o a ho ough accoun o he heo y behind
hese esul s.
Le Kbe an algeb aically closed ield. Fo ou pu poses a linea algeb aic g oup
will be a closed subg oup Go GLn(K)endowed wi h he Za iski opology, and
he iden i y componen o Gis he connec ed componen o he iden i y.
P oposi ion 4.17 (c . [37, P oposi ion 7.3]).Le Gbe a linea algeb aic g oup.
(i) The iden i y componen o Gis a no mal subg oup o ini e index.
(ii) Le H ≤ G be a closed connec ed subg oup o ini e index, hen H=G◦.
The iden i y componen o a linea algeb aic g oup Gis he unique no mal closed
connec ed subg oup o ini e index in G.
Fu he mo e, a ma ix is unipo en i i s unique eigen alue is 1,and a sub-
g oup o GLn(K)is a unipo en subg oup i all i s elemen s a e unipo en . The
key s uc u al esul abou unipo en g oups is ha any unipo en subg oup is a
conjuga e o a subg oup o Un(K), he g oup o uppe iangula ma ices wi h
1’s along he diagonal (see [37, Co olla y 17.5]). In pa icula , e e y unipo en
g oup Gis nilpo en , and when he base ield is o posi i e cha ac e is ic, Ghas
ini e exponen , compa e wi h [38, Chap e III, Lemma 16.2 and Theo em 16.5]
(al hough in his e e ence he base ield is ini e, he a gumen s a e s ill alid o
ields o posi i e cha ac e is ic).
Addi ionally, gi en an a bi a y linea algeb aic g oup G, he unipo en adical
o G, deno ed Ru(G),is he subg oup consis ing o all he unipo en elemen s o
G,and i is also cha ac e ised as he la ges connec ed unipo en subg oup o G.
Since Ru(G)is connec ed and nilpo en , i is con ained in he soluble adical o G,
namely he iden i y componen o he la ges soluble subg oup o G.A non- i ial
103
linea algeb aic g oup Gis said o be educ i e, when i is connec ed and Ru(G)
is i ial.
Al hough all hose cons uc ions play an impo an ôle in he heo y o alge-
b aic g oups, we ha e jus summa ised he de ini ions and he ela ions be ween
hem, conside ing ha we will simply use he ollowing echnical esul :
P oposi ion 4.18 (c . [37, Lemma 17.9]).Le Gbe a connec ed linea algeb aic
g oup and le Nbe i s soluble adical. Then [N,G]is unipo en .
P oo . Le Rube he unipo en adical o G.Then G/Ruis educ i e, so acco ding
o [37, Lemma 17.9], N/Ru⊆Z(G/Ru),and he e o e [N,G]⊆ Ru.
Now, we can p o e he desi ed esul :
Theo em 4.19. Compac Fp[[ ]]-analy ic g oups a e e bally concise.
P oo . Le Gbe a compac Fp[[ ]]-analy ic g oup and le wbe a wo d such ha
w{G}is ini e. Fi s ly, by Lemma 4.15, w(G)′is ini e, and hus up o a quo ien
we can assume ha w(G)is a ini ely gene a ed abelian subg oup.
Acco ding o Co olla y 4.13, he e exis s an open Fp[[ ]]-s anda d g oup Swhe e
wis a law. Fo abb e ia ion, Zs ands o Z(S)and K o he algeb aic closu e
o he local ield Fp(( )).By P oposi ion 2.3, S/Z is a linea g oup o e Fp[[ ]],so
i is also linea o e he ields Fp(( )) and K. Acco ding o he opological Ti s
al e na i e (loc. ci .) S/Z con ains ei he an open soluble subg oup o a dense
ee subg oup. Bu S/Z sa is ies an iden i y, so i mus be i ually soluble.
Le Sbe he Za iski closu e o S/Z in GLn(K),which is also i ually soluble
by P oposi ion 3.34. Le Nbe he la ges no mal soluble subg oup o Sand
N◦i s iden i y componen , i.e. he soluble adical o he algeb aic g oup S.In
iew o P oposi ion 4.17(i), N◦has ini e index in S, so acco ding o P oposi ion
4.17(ii), N◦=S◦.Le N/Z be he in e sec ion o S/Z wi h N◦, hen, passing o
he no mal co e i necessa y, we can assume ha Nis a no mal subg oup o ini e
index in G.
Besides, acco ding o P oposi ion 4.18, [N◦,N◦]is unipo en . In pa icula ,
[N◦,N◦]is nilpo en and, since Khas cha ac e is ic p, i has ini e exponen .
The e o e, [N, N]Z/Z is nilpo en o ini e exponen . Thus [N, N]Zis nilpo en
and acco ding o Lemma 4.16, H:= [N, N, S]has ini e exponen .
On he one hand,
H= [N, N, S]≥[N, N, N],
and so G/H is i ually nilpo en o class a mos 2.Since |w{G/H}| ≤ |w{G}|
and G/H is i ually nilpo en , we conclude ha w(G/H)is ini e by Tu ne -
Smi h’s Theo em (c . [72, Co olla y 2]).
104
On he o he hand, w(G)∩His a ini ely gene a ed abelian g oup o ini e
exponen , so i is ini e. Finally, he isomo phism
wG
H=w(G)H
H∼
=w(G)
w(G)∩H
yields he esul .
Co olla y 4.20. Le Rbe a p o-pdomain o K ull dimension one. E e y compac
R-analy ic g oup is e bally concise.
4.3 Conciseness in compac R-analy ic g oups
Now, we a e p imed o p o e he p incipal esul .
Theo em 4.21. E e y compac R-analy ic g oup is e bally concise.
P oo . Suppose, in iew o Co olla y 4.20, ha Rhas K ull dimension a leas 2.
Le Gbe a compac R-analy ic g oup and le wbe a wo d in k a iables such ha
w{G}is ini e. By i ue o Lemma 4.15, i suffices o p o e ha e e y w- alue
is o ini e o de .
By Lemma 1.23, he e exis s an open no mal R-s anda d subg oup (S, φ)such
ha o e e y g∈G he conjuga ion map cg:S→S, x 7→ xgis s ic ly ana-
ly ic. I n=|G:S|, hen wn{G} ⊆ Sand, by Lemma 4.15, wn(G)is ini e i and
only i w(G)is ini e. The e o e, wi hou loss o gene ali y assume ha w{G} ⊆ S.
Le (P, m)be he p incipal ideal p o-pdomain Zpi cha R= 0 o Fp[[ ]] i
cha R=p. Acco ding o Cohen’s S uc u e Theo em (see Theo em 1.2), Ris a
ini ely gene a ed in eg al ex ension o P[[ 1, . . . , m]],whe e m= dimK ull(R)−1.
Fo each a∈m(m),le sabe he e alua ion epimo phism sa:P[[ 1, . . . , m]] →
P, F( 1, . . . , m)7→ F(a).By Co olla y 2.13, saex ends o a con inuous ing
epimo phism ˜sa:R→Q, whe e, in iew o Rema k 2.14, Q= (Q, n)is a p o-p
domain and a ini ely gene a ed in eg al ex ension o P, in pa icula , Qhas K ull
dimension 1.
Fix a∈m(m), h oughou his p oo we use Wa o deno e W˜sa o any uple
o powe se ies W(see Sec ion 2.2). In pa icula , i Fis he o mal g oup law o
S, Fais he o mal g oup law F˜sa(see Co olla y 2.8). Le Tbe a le ans e sal
o Sin G, and assume ha 1∈T. We will use he a las induced by S, namely
{( S, φ )} ∈Twhe e φ (x) := φ( −1x)(compa e wi h Sec ion 1.3).
105
Using Lemma 2.10, we de ine he Q-s anda d g oup L:= nN(d),whose g oup
ope a ion is gi en by Fa,and a compac Q-analy ic g oup H:= T×L, ha can be
ega ded as an o e g oup o Land whose g oup ope a ion, say ∗a,is de ined as in
(2.3). Recall ha he Q-analy ic s uc u e o His gi en by he a las {( L, ψ )} ∈T
whe e ψ ( , l) = l.
Fo he es o he p oo ix ( 1, . . . , k)∈T(k)and assume, by (4.2), ha o
any `∈N, he wo d map wℓis gi en in 1S×···× kSby he uple o powe se ies
Wℓ, ha is, ecalling ha w{G} ⊆ Swe ha e ha
φwℓ(x1, . . . , xk)=Wℓ(φ 1(x1), . . . , φ k(xk)) ∀xj∈ jS
(e en hough in o de o ligh en he no a ion i is no w i en explici ly, he powe
se ies Wℓalso depends on 1, . . . , k).
Le wℓ:H(k)→Hbe he wo d map wℓwi h espec o he ope a ion ∗ao H.
By Lemma 2.7 and Rema k 2.11,
ψ1wℓ(x1, . . . , xk)=Wℓ
a(ψ 1(x1), . . . , ψ k(xk)) ∀xj∈ jL.
Fu he mo e, acco ding o P oposi ion 4.14, since w{G}is ini e, Sis ma ginal
o w. Tha is, wis cons an in each open subse 1S×···× kS. Hence, he wo d
map w:H(k)→His cons an in each 1L×···× kL, and hus |w{H}| ≤ |H:
L|k=nkis ini e. Acco ding o Co olla y 4.20, His e bally concise, so he e
exis s `a∈Nsuch ha wℓa(H) = {(1,0)}.In pa icula , by Lemma 1.8
Wℓa
a(X1, . . . , Xdk) = 0.(4.5)
De ine he ollowing pa i ion o he space m(m):
mℓ=a∈m(m)Wℓ
a=0, ` ∈N.
Since m(m)=Sℓ∈Nmℓand m(m)is comple e, by he Bai e Ca ego y Theo em
he e exis s `′such ha mℓ′con ains a non-emp y open subse V⊆om(m).Thus,
Wℓ′is a cons an uple o powe se ies, i.e.
Wℓ′(X1, . . . , Xdk) = (c1, . . . , cd)∈R(d),
ha whene e a∈mℓ′,sa is ies
(˜sa(c1), . . . , ˜sa(cd)) = Wℓ′
a(X1, . . . , Xdk) = 0.
Consequen ly, ˜sa(ci)=0 o all a∈mℓ′∩V. Besides, mℓ′∩Vis dense in V,
so by Co olla y 2.15, we ha e ha ci= 0 o all i∈ {1, . . . , d}.Finally, e-
pea ing he p ocess o all he uples in T(k),we ob ain an in ege `such ha
φwℓ(x1, . . . , xk)=0 o all xi∈G, and hus wℓ(G) = {1}.
106
The e a e se e al p oo s o Ado’s Theo em (see, o ins ance, [41, Chap e
VI, Sec ion 2]), and om mos o hem we can conclude ha he deg ee o he
ep esen a ion depends only on he ec o space dimension o he Lie algeb a.
Mo e p ecisely, le
deg L:= min{deg φ|φis a ai h ul ep esen a ion o L}.
I Ris a ield o cha ac e is ic ze o and Lis an R-Lie algeb a o dimension , hen
deg L≤ ( ) o some non-dec easing unc ion :N0→N.Fo ins ance, om
[13] and [60], we know ha
deg L≤ +α2
√ ,(A.1)
o some α≈2.763 (we e e o [58, Sec ion 1.1.2] o a de ailed p oo ). The e a e
a ious o he wo ks s udying deg Lo e ields, and we ough o men ion [8], [31]
and [63].
Fo gene al PIDs, meanwhile, we can no di ec ly make he same deduc ion
om he ini ially-men ioned wo e e ences. In ac , in [74, P oposi ion 3.4], he
ini eness o he deg ee- o-be ollows om he ac ha since Ris Noe he ian any
ascending chain o ideals mus be s a iona y, al hough we can no de e mine he
numbe o ideals in he chain.
In iew o his, we shall p esen a quan i a i e way o cons uc ing a ai h ul
ep esen a ion o an R-Lie la ice. This p ocedu e will be g ounded on he ideas
ha appea in [8] and [63].
Theo em A.4. Le Lbe an R-Lie la ice o ank . Then
deg L≤ + + 1
4 .
Las ly, i is wo h men ioning ha he posi i e cha ac e is ic coun e pa o
Ado’s Theo em is also ue, as p o ed by Iwasawa [40]. In ac , he gene al e sion
o Theo em A.2, wi hou any es ic ion on he cha ac e is ic o he base ield, is
e e ed o as he Ado-Iwasawa Theo em. Ne e heless, in posi i e cha ac e is ic
he esul can be s a ed wi h much mo e gene ali y:
Theo em A.5 (c . [19, Theo em 3]).Le Ra ing o posi i e cha ac e is ic and
Lan R-Lie la ice. Then he e exis a ee R-module Wo ini e ank and an
injec i e R-Lie algeb a homomo phism φ:L,→EndR(W).
113

In o de o p o e his heo em i suffices o ep oduce he o iginal p oo wo d by
wo d, and hus we ob ain o deg L he same bound we al eady knew o e ields,
namely
deg L≤n k3L,
whe e n= cha R(compa e wi h [5, Sec ion 6.2.4]).
Rema ks. Thoughou he p oo s we will use some basic p ope ies o ee R-
modules o e PIDs. He e is wha we should ake in o accoun :
(i) submodules o a ee R-module Ma e ee, and hey ha e ank a mos
k M.
Le Mbe an R-module and N≤Ma submodule. The isola o o Nin Mis he
submodule
IsoM(N) = {x∈M| ∃ ∈R {0}such ha x ∈N},
and Nis isola ed in Mi IsoM(N) = N.
(ii) M/ IsoM(N)is a o sion- ee R-module.
(iii) I Mis a ee R-module, N≤Mis an isola ed submodule and M/N is
ini ely gene a ed, hen M/N is a ee R-module and
k(M) = k(N) + k(M/N).
The in ege k(M/N)is e e ed o as he co ank o Nin M.
(i ) I Mis a ee R-module, N≤Mis an isola ed submodule and M/N is
ini ely gene a ed, hen Nhas a complemen a y in M, i.e. he e exis s a
ee R-module Lsuch ha M=N⊕L.
A.2 Adjoin and egula ep esen a ions
We shall in oduce a couple o ep esen a ions ha a ise na u ally in e e y R-Lie
la ice L. Fi s ly, x∈Lde ines he linea endomo phism adx:L→L, y 7→ [x, y].
This assigna ion gi es he adjoin ep esen a ion o deg ee k L,namely
Ad: L→EndR(L), x 7→ adx,
114
which is an R-Lie algeb a homomo phism in iew o Jacobi’s iden i y. None heless,
he adjoin ep esen a ion is no ai h ul in gene al, as i s ke nel is he cen e o
he algeb a,
Z(L) := {x∈L|[x, y] = 0 ∀y∈L}.
The e o e, i Lis a semisimple R-Lie algeb a –i has no non- i ial abelian ideal–,
he adjoin ep esen a ion is ai h ul, and, in his si ua ion, deg L≤ k L.
In o de o p esen he second ep esen a ion, we mus de ine he uni e sal
en eloping algeb a:
De ini ion A.6 (c . [41, Chap e V, Sec ion 1]).Le Lbe an R-Lie algeb a. The
R- enso algeb a o Lis
TR(L) = R⊕L1⊕L2⊕···⊕Li⊕. . . ,
whe e Li:= L⊗(i)
. . . ⊗L.Each Liis an R-module wi h he na u al R-module
s uc u e o enso ial modules and he mul iplica ion in TR(L)is de ined by he
ule
(x1⊗···⊗xi)⊗(y1⊗···⊗yj) = x1⊗···⊗xi⊗y1⊗···⊗yj.
Le Rbe he ideal o TR(L)gene a ed by he elemen s
[x, y]−(x⊗y−y⊗x), x, y ∈L.
The uni e sal en eloping algeb a o Lis he associa i e R-algeb a wi h iden i y
UR(L) := TR(L)
R.
Iden i ying Lwi h L1,we ob ain a homomo phism ι:L→ UR(L),which, since
Ris a PID, is injec i e whene e Lis ini ely gene a ed (see [74, Theo em 3.2]).
Hence, o simplici y we can assume ha L⊆ UR(L).Besides, in o de o simpli y
he no a ion, he elemen xi1⊗···⊗xikwill be w i en as he monomial xi1. . . xik.
The uni e sal algeb a is desc ibed by he Poinca é-Bi kho -Wi Theo em:
Theo em A.7 (c . [74, Theo em 3.2]).Le Lbe an R-Lie la ice wi h basis
{x1, . . . , x }.Then UR(L)is a ee R-module, and he monomials
{xα1
1. . . xα
|αi∈N0}(A.2)
o m a basis.
115
Indeed, gi en wo monomials hei p oduc can be exp essed as a linea com-
bina ion o monomials o he o m (A.2) by successi ely applying he iden i y
xjxi=xixj−[xi, xj] o eo de he e ms un il all he in ol ed monomials ha e
he equi ed o de .
Any R-Lie la ice Lac s by le mul iplica ion on UR(L).Indeed, o each x∈L
we ge he R-linea endomo phism `x:UR(L)→ UR(L), u 7→ xu. On he one hand,
o e e y x, y ∈L
[x, y] = xy −yx
in UR(L),and he e o e L:L→EndR(UR(L)), x 7→ `xis an R-Lie algeb a ho-
momo phism. On he o he hand, since he uni e sal en eloping algeb a has an
iden i y, whene e x6=ywe ha e ha
`x(1) = x6=y=`y(1),
and so Lis a ai h ul ep esen a ion, called (le ) egula ep esen a ion.
Howe e , since UR(L)is o in ini e ank, he egula ep esen a ion is no i-
ni e. Ne e heless, he e is a p ope y ha cha ac e ises he uni e sal en eloping
algeb a, and as a consequence o i any ini e ep esen a ion ac o s hough UR(L).
Theo em A.8 (Uni e sal p ope y, c . [11, Chap e I, § 2.1, P oposi ion 1]).Le
Lbe an R-Lie la ice, Aan associa i e R-algeb a wi h iden i y oge he wi h he Lie
b acke [a, b] = ab −ba (a, b ∈A) and an R-Lie algeb a homomo phism ψ:L→
(A, [,]).Then, he e exis s a unique R-algeb a homomo phism ψ∗:UR(L)→A
such ha ψ=ψ∗◦ι. Tha is, he ollowing diag am commu es:
UR(L)
LA.
ψ∗
ι
ψ
Ac ually, he name o UR(L)comes om his uni e sal p ope y. Fu he , i can
be p o ed ha any R-algeb a ha sa is ies he uni e sal p ope y is isomo phic
o UR(L)(compa e wi h [41, Chap e V, Theo em 1.1]).
A.3 Ado’s Theo em
In o de o p o e Ado’s Theo em we will usually mo e om he R-Lie la ice L
o he K- ec o space LK:= L⊗RK. No e ha LKis a K-Lie algeb a, whose Lie
b acke is no hing bu he K-linea applica ion induced by he Lie b acke o L.
We should bea in mind he ollowing ac s:
116
•LKis K- ec o space o dimension k L,
• i L=hx1, . . . , x iR hen LK=hx1, . . . , x iK,and
• whene e I⊴LKis an ideal, hen I∩L⊴Lis an isola ed ideal.
We will p o e he heo em in h ee s eps:
A.3.1 Nilpo en Lie la ices
Fi s o all, le us suppose ha Lis a nilpo en R-Lie la ice o ank . Recall
ha he lowe cen al se ies o Lis de ined as:
γ1(L) := L, γi(L) := [γi−1(L),L]∀i≥2,
and ha Lis nilpo en i he e exis s an in ege c∈Nsuch ha γc+1(L) = {0}.
The smalles o such in ege s, when i exis s, is he nilpo ency class o he R-Lie
la ice. Conside ing ha he enso p oduc is linea , we can easily p o e he
ollowing:
Lemma A.9. Le Lbe an R-Lie la ice and le I,H⊴Lbe ideals. Then,
[I⊗RK, H⊗RK] = [I,H]⊗RK.
Thus, i Lis a nilpo en R-Lie la ice o nilpo ency class c,LKis a nilpo en
K-Lie algeb a o nilpo ency class c.
The e o e, LKis nilpo en o nilpo ency class say c, i.e
LK=γ1(LK)>··· > γi(LK)>··· > γc+1(LK) = {0}
is a s ic ly descending chain o K- ec o spaces and hus c≤dimKLK= . De ine
now he isola ed ideals Li:= γi(LK)∩L⊴L,and choose a basis {x1, . . . , x }o
Lsuch ha he i s x1, . . . , x 1elemen s a e a basis o Lc, he i s x1, . . . , x 2
( 2> 1) elemen s o m a basis o Lc−1and so o h. Acco ding o Theo em A.7,
he monomials
xα:= xα1
1. . . xα
, α = (α1, . . . , α )∈N( )
0
o m a basis o he uni e sal en eloping algeb a UR(L),and acco dingly we can
de ine a weigh unc ion ω:UR(L)→N0∪{∞} in he ollowing ashion:
117
ω(xi) = max{m|xi∈Lm}, ω(xα) = P
i=1 αiω(xi),
ω(Pαcαxα) = min {ω(xα)|cα6= 0}and ω(0) = ∞.
Fo each m∈N0,we de ine
Um(L) := {u∈ UR(L)|ω(u)> m}
–when he la ice is clea om he con ex , we will simply w i e Um–.
Le us show ha Um(L)⊴UR(L)is an isola ed ideal:
(i) No e ha 0∈Um o all m∈Nand ha
ω( x) = ω(x)and ω(x+y)≥min {ω(x), ω(y)},
o all ∈R {0}and x, y ∈ UR(L).Addi ionally, since ω([xi, xj]) ≥
ω(xi) + ω(xj) o e e y i, j ∈ {1, . . . , },we ha e ha
ω(xy)≥ω(x) + ω(y).(A.3)
Consequen ly, Umis an ideal.
(ii) Le ∈R {0},
x ∈Um=⇒ω(x) = ω( x)> m =⇒x∈Um,
i.e. Umis isola ed in UR(L).
Mo eo e , UR(L)/Umis a ini ely gene a ed R-module, as i is gene a ed by
Bm={xα+Um|ω(xα)≤m}.
The e o e, Umis an isola ed ideal o ini e co ank, and in iew o (A.3), o
e e y x∈Lwe ob ain ha `x(Um)⊆Um.In consequence, o any m he egula
ep esen a ion induces he ini e ep esen a ion
Lm:L→EndR(UR(L)/Um), x 7→ `x,
whose ke nel is L∩Um–wi h an abuse o no a ion, whene e ∈EndR(UR(L))
sa is ies (X)⊆X o some ideal X⊴UR(L),we will s ill call o he elemen in
118

EndR(UR(L)/X) ha sends x+X o (x) + X–.
Besides, L∩Uc(L) = {0}.Indeed, whene e x=P
i=1 αixi∈L, hen
ω(x) = ω
X
i=1
αixi!≤max
i=1,..., ω(xi) = c.
The e o e, Lcis a ini e ai h ul ep esen a ion o L,and in o de o bound i s
deg ee i suffices o de e mine an uppe bound o |Bc|.All he monomials in Bc
ha e weigh a mos c, so hey ha e polynomial deg ee a mos c. Mo eo e , he
numbe o monomials o polynomial deg ee a mos cin a iables is exac ly
he numbe o monomials o polynomial deg ee cin + 1 a iables (by adding
an auxilia y a iable, homogenise he monomials such ha hey ha e polynomial
deg ee c).
Lemma A.10. The numbe o monomials in a iables and o polynomial deg ee
cis  +c−1
c.
This obse a ion gi es us a simple bound (compa e wi h [31, Co olla y 5.1]):
deg L≤ k UR(L)
Uc(L)≤ +c
c.
We conclude his sec ion by gi ing a no e y sha p bound o deg Lin e ms o
. In ac , c∈ {1, . . . , }, ha is, c=α whe e α∈ {1/ , . . . , 1}.Remembe ha
acco ding o he S i ling app oxima ion o mula,
√2π ( /e) ≤ !≤√2π ( /e) e1
12 ∀ ∈N,
and he e o e,
 +α
α ≤p2π( +α )( +α ) +α
√2πα (α )α √2π e1
12(1+α)
≤e1/12(1 + α)
√2π
√1 + α
√α(1 + α)1+α
αα
.
In addi ion, he le -hand side and he igh -hand side e ms a e asym o ically
equi alen as ends o in ini y. Fu he , since α∈[1/ ,1] , hen
√1 + α
√α≤√ + 1 and (1 + α)1+α
αα≤4.
119
Tha is,
e1/12(1 + α)
√2π
1
√
√1 + α
√α(1 + α)1+α
αα
≤ + 1
4 .
Consequen ly,  +c
c≤ + 1
4 ∀c∈ {1, . . . , }.(A.4)
A.3.2 Spli able R-Lie la ices
The second s ep consis s on ob aining a sui able R-Lie algeb a ep esen a ion o
he so-called spli able R-Lie la ices.
Since he sum o nilpo en ideals is again nilpo en (compa e wi h [41, Chap e
I, P oposi ion 7.6]), e e y R-Lie la ice Lhas a nilpo en ideal ha con ains any
o he nilpo en ideal. This is called he nilpo en adical o L,and i will be ep-
esen ed as Rn(L). Addi ionally, Rn(L)is an isola ed ideal, as i is no hing bu
Rn(LK)∩L.
We will say ha an R-Lie algeb a Lis spli able i he e exis s an R-Lie subal-
geb a S≤Lsuch ha L=Rn(L)⊕S, ha is, Lis he semidi ec p oduc o S
wi h he ideal Rn(L).Fo he ca ego icaly minded eade we poin ou ha his
condi ion is equi alen o he ac ha he sho exac sequence
0→Rn(L)→L→L/Rn(L)→0
spli s in he ca ego y o R-Lie algeb as.
To deal wi h he spli able case, we can blend he p eceding egula ep esen-
a ion and he ep esen a ions induced om de i a ions:
De ini ion A.11. Le Abe an R-algeb a. A de i a ion o Ais an R-module
endomo phism D∈EndR(A) ha sa is ies Leibniz iden i y, i.e.
D(ab) = aD(b) + D(a)b∀a, b ∈A.
The se o all de i a ions o Ais deno ed by De R(A).
Fo example, by i ue o Jacobi’s iden i y, o all x∈Lwe ha e ha adxis
a de i a ion o he R-Lie algeb a L.S a ing om a de i a ion Do Lwe can
induce a de i a ion o UR(L),which will be deno ed by D∗.Fo ha , we should
120
ex end Dby imposing Leibniz iden i y, i.e. by aking he linea ex ension o he
ule
D∗(x1. . . x ) =
X
i=1
x1. . . xi−1D(xi)xi+1 . . . x ,
oge he wi h D∗(1) = 0 as i mus happen o e e y de i a ion o an algeb a
wi h iden i y. Ac ually, his ex ension is a consequence o he uni e sal p ope y
(compa e wi h [41, Chap e V, Theo em 1.1(7)]).
Lemma A.12. Le Lbe a nilpo en R-Lie la ice and D∈De R(L).Then
D∗(Um(L)) ⊆Um(L) o e e y m∈N.
P oo . Le {x1, . . . , x }be he basis o Lwi h espec o which he weigh unc ion
ωis de ined (compa e wi h Subsec ion A.3.1). Since Dis a de i a ion, D(Li)⊆
Li o all i∈ {1, . . . , c},so ω(D(xi)) ≥ω(xi) o all i∈ {1, . . . , }.Hence, i
xi1. . . xi ∈Um(L),by (A.3)
ω(D∗(xi1. . . xi )) ≥min
j=1,..., ω(xi1. . . D(xij). . . xi )≥ω(xi1. . . xi )> m.
P oposi ion A.13 (Zassenhaus ex ension, c . [11, Chap e I, § 7.3, Theo em 1]
and [41, Chap e VI, Theo em 2.1]).Le Lbe a spli able R-Lie la ice and le c
be he nilpo ency class o Rn(L). Then, he e exis s a ini e ep esen a ion
Φ: L→EndRUR(Rn(L))
Uc(Rn(L)) 
o L ha is injec i e in Rn(L).In pa icula , deg Φ depends only on k Rn(L).
P oo . Deno e o simplici y Rn(L)as N.Then, L=N⊕S o some R-Lie
subalgeb a S≤L. F om Lemma A.12, ad∗
x(Uc(N)) ⊆Uc(N) o all x∈L,so we
can de ine he map
Φ: L=N⊕S→EndR(UR(N)/Uc(N)), n +s7→ `n+ ad∗
s.
In o de o show ha i is an R-Lie algeb a homomo phism, i suffices o con i m
ha
Φ ([s, n]) = [Φ(s),Φ(n)] = [ad∗
s, `n]
o all n∈Nand s∈S.Fo ha , no e ha o any n∈Nand any D∈
De R(UR(N)) :
[D, `n](u) = D◦`n(u)−`n◦D(u) = D(n)u=`D(n)(u),∀u∈N.
121
Mo eo e , since Nis an ideal, hen [s, n]∈N,so
Φ ([s, n]) = `[s,n]=`ad∗
s(n)= [ad∗
s, `n] = [Φ(s),Φ(n)] .
Consequen ly, Φis an R-Lie algeb a ep esen a ion. In addi ion, i s ke nel has
i ial in e sec ion wi h he nilpo en adical, as Φ|Nis no hing bu he ai h ul
ep esen a ion Lco N.
Finally, om (A.4) we conclude ha
deg Φ ≤s k Rn(L)+1
k Rn(L)·4 k Rn(L).(A.5)
A.3.3 Embedding heo em
Like nilpo ency in R-Lie algeb as, we can de ine solubili y: he de i ed se ies o
an R-Lie algeb a Lis de ined ecu si ely as
L(1) := L,L(i):= L(i−1),L(i−1)∀i≥2,
Lis said o be soluble when L(ℓ)={0} o some `∈N,and he smalles o such
in ege s is he de i ed leng h o L.In addi ion, he sum o soluble ideals is again
soluble (compa e wi h [41, Chap e I, P oposi ion 7.4]), and he e o e, i Lis
ini ely gene a ed, he e exis s a soluble adical o L,namely a soluble ideal Rs(L)
ha con ains any o he soluble ideal. Ob iously, any nilpo en R-Lie algeb a is
soluble and hus Rn(L)≤Rs(L).
Acco ding o Le i’s Theo em (see [41, Chap e III, Sec ion 9]), i Ris a ield o
cha ac e is ic ze o, e e y R-Lie algeb a Lspli s as Rs(L)⊕S o some semisimple
Lie subalgeb a S≤L, called Le i ac o o L.This decomposi ion plays a unda-
men al ôle in he majo i y o p oo s o Ado’s Theo em o ields, as hey i s ly
ob ain a ini e ai h ul ep esen a ion o Rs(L),and hen i is ex ended o Lusing
Zassenhaus ex ension (compa e wi h P oposi on A.13).
Howe e , Le i’s Theo em does no hold o gene al R-Lie algeb as: he Z-Lie
algeb a sl2(2Z)⊕ 2(2Z),i.e. he di ec sum o 2×2ma ices o ace 0and 2×2
uppe iangula ma ices o e he ing 2Z,is no decomposable in he desi ed
way (see [19, Example in pg. 838]).
In his hi d and las s ep we shall p o e he main heo em. Fo ha , we will
embed he ini ial R-Lie la ice Lin a spli able R-Lie la ice and make use o he
p e ious subsec ion. Ac ually he p oblem educes o ields:
122
Bibliog aphy
[1] Abe comb ie, A. G.: Subg oups and sub ings o p o ini e ings, Ma h. P oc.
Camb . Phil. Soc. 116, 209-222 (1994).
[2] Ado, I. D.: The ep esen a ion o Lie algeb as by ma ices, Ame . Ma h. Soc.
T ansla ions 21(2) (1949); ansla ion om Usehi Ma em. Nauk (NS) 6(22),
159-173 (1947).
[3] A iyah, M., F. and Macdonald, I., G.: In oduc ion o Commu a i e Algeb a,
Addison-Wesley, Reading (1969).
[4] Bae , R.: Nilpo en g oups and hei gene aliza ion, T ans. Ame . Ma h. Soc.
47, 393-434 (1940).
[5] Bah u in, Y.: Iden i ical ela ions in Lie algeb as, De G uy e Exposi ions in
Ma hema ics 68, De G uy e , Be lin-Bos on (2021).
[6] Ba nea, Y. and Shale , A.: Hausdo dimension, p o-pg oups and Kac-
Moody algeb as, T ans. Ame . Ma h. Soc. 349, 5073-5091 (1997).
[7] Ba nea, Y., Shale , A. and Zelmano , E. I.: G aded subalgeb as o affine
Kac-Moody algeb as, Is ael J. Ma h. 104, 321-334 (1998).
[8] Bi kho , G.: Rep esen a ibili y o Lie algeb as and Lie g oups by ma ices,
Ann. o Ma h. 38, 526-532 (1937).
[9] Bishmu h, R.: Ra ional o mal g oup laws, B. Sc. Thesis, McMas e Uni e -
si y, Hamil on, On a io (1976).
[10] Bou baki, N.: Commu a i e Algeb a, Chap e s 1-7, Sp inge -Ve lag, Be lin-
Heidelbe g (1989).
[11] Bou baki, N.: Lie G oups and Lie Algeb as, Chap e s 1-3, Sp inge -Ve lag,
Be lin-Heidelbe g (1989).
129

[12] B euilla d, E. and Gelande , T.: A opological Ti s al e na i e, Ann. o Ma h.
166, 427-474 (2007).
[13] Bu de, D.: On a e inemen o Ado’s Theo em, A ch. Ma h. 70 118-127,
(1998).
[14] Camina, R. and Du Sau oy, M.: Linea i y o Zp[[ ]]-Pe ec G oups, Geom.
Dedica a 107, 1-16 (2004).
[15] Ca e , R., W.: Simple g oups o Lie ype, John Wiley & Sons, London-New
Yo k (1989).
[16] Casals-Ruiz, M. and Zozaya, A.: Linea i y o compac R-analy ic g oups,
submi ed.
[17] Chang, C. C. and Keisle , H. J.: Model Theo y, No h Holland, New Yo k
(1990).
[18] Che alley, C.: Publica ions de l’Ins i u Ma héma ique de l’Uni e si é de
Nancago, Ac uali és Scien i iques e Indus ielles 1226, He mann, Pa is
(1968).
[19] Chu kin, V. A.: The ma ix ep esen a ion o Lie algeb as o e ings, Ma h.
No es 10(6), 835-839 (1971); ansla ion om: Ma . Zame ki 10, 671-678
(1971).
[20] Clemen , A. E., Majewicz, S. and Zyman, M.: The heo y o nilpo en g oups,
Bi khäuse , Cham (2017).
[21] Cohen, I. S.: On he s uc u e and ideal heo y o comple e local ings, T ans.
Ame . Ma h. Soc. 59, 54-106 (1946).
[22] De omi, E., Klopsch, B. and Shumya sky, P.: S ong conciseness in p o ini e
g oups, J. London Ma h. Soc. 102, 977-993 (2020).
[23] De omi, E., Mo igi, M. and Shumya sky, P.: On conciseness o wo ds in
p o ini e g oups, J. Pu e Appl. Algeb a 220, 3010-3015 (2016).
[24] Dixon, J., Du Sau oy, M., Mann, A. and Segal, D.: Analy ic P o-pG oups,
2nd ed., Camb idge Uni e si y P ess, Camb idge (1999).
130
[25] Eisenbud, D.: Commu a i e algeb a wi h a iew owa d algeb aic geome-
y, G adua e Tex s in Ma hema ics 150, Sp inge -Ve lag, Be lin-Heidelbe g-
New Yo k (1995).
[26] Falcone , K.: F ac al geome y: ma hema ical ounda ions and applica ions,
3 d edn. Wiley, Chiches e (2014).
[27] Fe nández-Alcobe , G. A., Giannelli, E. and González-Sánchez, J.: Hausdo
dimension in R-analy ic p o ini e g oups, J. o G oup Theo y 20, 579-587
(2017).
[28] Folland, G. B.: Real Analysis: mode n echniques and hei applica ions,
John Wiley & Sons, New Yo k (1999).
[29] Geo ge, K. M.: Ve bal p ope ies o ce ain g oups, D.Phil. Thesis, Uni e si y
o Camb idge, Camb idge (1976).
[30] González-Sánchez, J. and Zozaya, A.: S anda d Hausdo spec um o com-
pac Fp[[ ]]-analy ic g oups, Mona sh Ma h 195(3), 401-419 (2021).
[31] de G aa , W. A.: Cons uc ing ai h ul ma ix ep esen a ion o Lie algeb as,
P oceedings o he 1997 In e na ional Symposium on Symbolic and Algeb aic
Compu a ion, ISSAC’97, pp. 54-59, ACM P ess, New Yo k (1997).
[32] G eco, S. and Salmon, P.: Topics in m-adic opologies, E gebnisse de Ma he-
ma ik und ih e G enzgebie e 58, Sp inge -Ve lag, Be lin-Heidelbe g (1971).
[33] Hall, P.: Ve bal and ma ginal subg oups, J. Reine Angew. Ma h. 182, 156-
157 (1949).
[34] Ha ish-Chand a: On ep esen a ion o Lie algeb a, Ann. o Ma h. 50, 900-915
(1949).
[35] Hazewinkel, M.: Fo mal G oups and Applica ions, Academic P ess, New
Yo k-San F ancisco-London (1978).
[36] de las He as, I. and Zozaya, A.: Hausdo en dimen sioa alde p o ini ue an,
Ekaia 40, 291-314 (2021).
[37] Humph eys, J. E.: Linea Algeb aic G oups, Sp inge -Ve lag, New Yo k-
Be lin-Heidelbe g (1975).
131
[38] Huppe , B.: Endliche G uppen I, Sp inge -Ve lag, Be lin-Heidelbe g-New
Yo k (1967).
[39] I ano , S. V.: P. Hall’s conjec u e on he ini eness o e bal subg oups, So .
Ma h. 33, 71-76 (1989); ansla ion om Iz . Vyssh. Uchebn. Za ed. Ma . 6,
60–70 (1989).
[40] Iwasawa, K.: On he ep esen a ion o Lie algeb as, Jap. J. Ma h. 19, 405-426
(1948).
[41] Jacobson, N.: Lie Algeb as, Do e Publica ions, Inc., New Yo k (1979).
[42] Jaikin-Zapi ain, A.: On linea jus in ini e p o-pg oups, J. Algeb a 255,
392-404 (2002).
[43] Jaikin-Zapi ain, A.: On linea i y o ini ely gene a ed R-analy ic g oups,
Ma h. Z. 253, 333-345 (2006).
[44] Jaikin-Zapi ain, A.: On he e bal wid h o ini ely gene a ed p o-pg oups,
Re . Ma . Ibe oam. 24, 617-630 (2008).
[45] Jaikin-Zapi ain, A. and Klopsch, B.: Analy ic g oups o e gene al p o-pdo-
mains, J. London Ma h. Soc. 76, 365-383 (2007).
[46] Jo dan, C.: T ai é des sub i u ions e de équa ions algéb iques, Les G ands
Classiques Gau hie -Villa s, Pa is (1870).
[47] Klopsch, B. and Thillaisunda am, A.: A p o-pg oup wi h in ini e no mal
Hausdo spec a. Paci ic J. Ma h. 303, 569-603 (2019).
[48] Klopsch, B., Thillaisunda am, A. and Zugadi-Reizabal, A.: Hausdo dimen-
sion in p-adic analy ic g oups. Is ael J. Ma h. 231, 1-23 (2019).
[49] Lang, S.: Algeb a, G adua e Tex s in Ma hema ics 211, Sp inge -Ve lag,
New Yo k (2005).
[50] Laza d, M.: G oupes analy iques p-adiques, Publ. Ma h. I.H.E.S. 26, 5-219
(1965).
[51] Lee, J. M.: In oduc ion o Smoo h Mani olds, 2nd ed., Sp inge , New Yo k
(2013).
132
[52] Lubo zky, A.: G oup p esen a ion, p-adic analy ic g oups and la ices in
SL2(C),Ann. o Ma h. 118, 115-130 (1983).
[53] Lubo zky, A. and Mann, A.: On g oups o polynomial subg oup g ow h,
In en . Ma h. 104, 521-533 (1991).
[54] Lubo zky, A. and Shale , A.: On some Λ-analy ic p o-pg oups, Is ael J.
Ma h. 85, 307-337 (1994).
[55] Mal’ce , A.: On isomo phic ma ix ep esen a ions o in ini e g oups, Rec.
Ma h. N.S. 8(50), 405-422 (1940).
[56] Me zljako , Yu. I.: Algeb aic linea g oups as ull g oups o au omo phisms
and he closeness o hei e bal subg oups, Algeb a i Logika 6, 83-94 (1967).
[57] Me zljako , Yu. I.: Ve bal and ma ginal subg oups o linea g oups, Dokl.
Akad. Nauk SSSR 177, 1008-1011 (1967).
[58] Moens, W. A.: Fai h ul ep esen a ions o minimal deg ee o Lie algeb as
wi h an abelian adical, PhD Thesis, Uni e si ä Wein, Vienna (2009).
[59] Munk es, J.R.: Topology, 2nd ed., Pea son Educa ion, Ha low (2014).
[60] Ne e in Yu.: A cons uc ion o ini e-dimensional ai h ul ep esen a ion o
Lie algeb as. P oceedings o he 22nd Win e School on “Geome y and
Physics” (S ni, 2002). Rend. Ci c. Ma . Pale mo (2) Suppl. No. 71, 159–
161 (2003).
[61] Neumann, H.: Va ie ies o G oups, Sp inge -Ve lag, New Yo k (1967).
[62] Ol’shanskiĭ, A., Yu.: Geome y o de ining ela ions in g oups, ol. 70, Ma he-
ma ics and Applica ions (So ie Se ies), Kluwe Academic Publishe s G oup,
Do d ech (1991).
[63] Reed, B. E.: Rep esen a ions o sol able Lie algeb as, Michigan Ma h. J. 16,
227-233 (1969).
[64] Ri es, L. and Zalesskii, P.: P o ini e G oups, 2nd ed., A Se ies o Mode n
Su eys in Ma hema ics 40, Sp inge -Ve lag, Be lin-Heidelbe g (2000).
[65] Robinson, D. J. S: A cou se in he heo y o g oups, 2nd ed., Sp inge -Ve lag,
New Yo k (1982).
133
[66] Roman’ko , V. A.: Wid h o e bal subg oups in sol able g oups, Algeb a and
Logic 21, 41-49 (1982); ansla ion om Algeb a i Logika 21, 60–72 (1982).
[67] Segal, D.: Wo ds. No es on Ve bal Wid h in G oups, Camb idge Uni e si y
P ess, Camb idge (2009).
[68] Se e, J. P.: Lie Algeb as and Lie G oups, Lec u e No es in Ma hema ics ol.
1500, Sp inge -Ve lag, Be lin-Heidelbe g (2006).
[69] Simons, N. J.: The wid h o e bal subg oups in p o ini e g oups, PhD Thesis,
Uni e si y o Ox o d, Ox o d (2009).
[70] S ade, H.: Simple Lie Algeb as O e Fields o Posi i e Cha ac e is ic. S uc-
u e Theo y, ol. I, De G uy e , Be lin-Bos on (2004).
[71] S oud, P. W.: Topics in he heo y o e bal subg oups, PhD Thesis, Uni-
e si y o Camb idge, Camb idge (1966).
[72] Tu ne -Smi h, R. F.: Fini eness condi ions o e bal subg oups, J. London
Ma h. Soc. 41, 166-176 (1966).
[73] Weh i z, B. A. F.: In ini e Linea G oups, Sp inge -Ve lag, Be lin-
Heidelbe g-New Yo k (1973).
[74] Weigel, T.: A Rema k on he Ado-Iwasawa Theo em, J. Algeb a 212, 613-625
(1999).
[75] Za iski, O. and Samuel, P.: Commu a i e Algeb a, Vol. I and II, Sp inge -
Ve lag, Be lin-Heidelbe g-New Yo k (1958).
[76] Zozaya, A.: Conciseness in compac R-analy ic g oups, submi ed.
[77] Zozaya, A.: A ema k on Ado’s Theo em o p incipal ideal domains, in
p epa a ion.
134

Index
δ-co e ing, 68
m-adic opology, 11
p-adic analy ic, 19
Ado
Theo em, 112
affine se , 85
algeb a
cen al simple ∼, 92
semisimple ∼, 115
spli able ∼, 120
enso ∼, 115
analy ic
∼dimension, 16, 24
∼g oup, 18
∼mani old, 16
∼map, 15, 17
∼subg oup, 42
∼submani old, 39
∼subse , 41
s ic ly ∼map, 17
a las, 16
compa ible ∼, 16
maximal ∼, 16
Bai e
Ca ego y Theo y, 99
Bake -Hausdo
o mula, 50
bi-Lipschi z, 68
bianaly ic map, 30
box dimension, 68, 69
s anda d lowe ∼, 74
uppe ∼, 68
lowe ∼, 68
s anda d ∼, 79
cen oid, 92
chain ule, 27
cha , 16
adap ed ∼, 43
compa ible ∼, 16
egula ∼, 16
Che alley
classical g oup, 88
Cohen
S uc u e Theo em, 12, 35
conciseness
s ong ∼, 107
e bal ∼, 97
coo dina e
∼change map, 16
∼sys em, 17
canonical ∼sys em, 18
change o ∼, 31
co ank, 42, 114
de i a ion, 120
135
de i ed
∼lengh , 122
di e en ial, 27
dimension
∼submani old, 39
analy ic ∼, 16, 24
box ∼, 68
Hausdo ∼, 68
local analy ic ∼, 17
Minkowski-Bouligand ∼, 68
disc imina ion, 57
e alua ion
∼epimo phism, 55
∼map, 14
il e , 60
il a ion se ies, 65
p-powe ∼, 66
no mal ∼, 69
s anda d ∼, 21, 72
ini e in e sec ion p ope y, 61
o mal g oup law, 19
addi i e ∼, 20
mul iplica i e ∼, 20
o mal mo phism, 27
ac ion ield, 12
ull esiduali y, 57
Going Up Theo em, 56
Hall-Pe escu
o mula, 102
Hall-Wi
iden i y, 26
Hausdo
∼densi y, 91
∼dimension, 68
∼measu e, 68
∼spec um, 66
s anda d ∼dimension, 67, 76
s anda d ∼spec um, 67, 76
Hilbe
∼ unc ion, 73
∼polynomial, 73
iden i y componen , 103
imme sed subse , 39
imme sion, 32
weak ∼, 30
in e se
o mal ∼, 19
in e sion
∼map, 18
isola ed module, 114
isola o , 114
Iwasawa
Theo em, 113
Jacobi
iden i y, 112
Jacobian, 27
K ull
In e sec ion Theo em, 11
Leibniz
iden i y, 120
Le i
∼ ac o , 122
Theo em, 122
Lie algeb a, 27, 111
∼cen e, 115
∼homomo phism, 112
∼ ep esen a ion, 112
associa ed ∼, 26
conc e e ∼, 112
g aded ∼, 90
nilpo en ∼, 117
pe ec ∼, 92
136
soluble ∼, 122
Lie b acke , 26, 111
Lie la ice, 112
powe ul ∼, 50
linea
∼algeb aic g oup, 98
∼ ep esen a ion, 47
∼algeb aic g oup, 98, 103
∼g oup, 47
gene al ∼g oup, 18, 85
special ∼g oup, 88
local
∼ ing, 11
∼ ing homomo phism, 51
mani old
analy ic ∼, 16
pu e ∼, 17
ma ginal
∼subg oup, 96
Minkowski-Bouligand
∼dimension, 68
mono onici y, 68
mul iplica ion
∼map, 18
le ∼map, 29
nil- ep esen a ion, 127
nilpo ency
∼class, 117
nilpo en
∼ adical, 120
nowhe e cons an map, 107
Poinca é-Bi kho -Wi
Theo em, 115
powe se ies
∼uni e sal p ope y, 14
∼ ing, 13
con e gen ∼, 14
p incipal ideal domain (PID), 12
p o-pdomain, 12
p o ini e space, 107
adical
nilpo en ∼, 120
soluble ∼, 103, 122
unipo en ∼, 103
educ i e g oup, 104
egula poin , 42
ep esen a ion, 112
∼deg ee, 112
adjoin ∼, 114
ai h ul ∼, 112
ini e ∼, 112
induced ∼, 48
linea ∼, 47
ma icial ∼, 112
egula ∼, 116
esidue ield, 11
esidue ank, 31
es ic ion o scala s, 38
Schu
Theo em, 102
se ies
de i ed ∼, 122
lowe cen al ∼, 117
soluble
∼ adical, 103, 122
special o hogonal g oup, 89
s abili y
coun able ∼, 68
ini e ∼, 69
s anda d
∼ il a ion se ies, 21
∼g oup, 19
s ong iangle inequali y, 12
137
subimme sion, 33
submani old
∼dimension, 39
analy ic ∼, 39
weak ∼, 40
subme sion, 32
weak ∼, 30
symplec ic g oup, 89
Ti s
opological al e na i e, 88
opologically nilpo en , 14
ansla ion, 18
ul a il e , 60
∼ heo em, 61
non-p incipal ∼, 60
ul ame ic space, 12
ul apowe , 61
ul ap oduc , 61
uni o mise , 13
uni o mly powe ul g oup, 49
unipo en
∼ma ix, 103
∼ adical, 103
∼subg oup, 103
unique ac o isa in domain (UFD),
45
uni e sal en eloping algeb a, 115
uni e sal p ope y, 116
e bal
∼conciseness, 97
∼ellip ici y, 98
∼ obus ness, 97
∼subg oup, 96
∼wid h, 98
wo d, 95
∼map, 95
∼ alue, 95
Bu nside ∼, 97
commu a o ∼, 97
de i ed ∼, 97
ellip ic ∼, 98
emp y ∼, 95
equi alen ∼, 95
lowe cen al ∼, 97
Zassenhaus
ex ension, 121
Zo n
Lemma, 92
138
an isime ia i xu azkoa baino ez da. Izan e e, ha u g∈Gelemen ua e a demagun
w(x1, . . . , xig,...,xk) = w(x1, . . . , xk)dela xj∈Gguz ie a ako, o duan
w(x1, . . . , gxi, . . . , xk) = w(x1, . . . , xigxi, . . . , xk)
=w(x1, . . . , (xig)xi, . . . , xk)
=wxx−1
i
1, . . . , xig,...,xx−1
i
kxi
=wxx−1
i
1, . . . , xi, . . . , xx−1
i
kxi=w(x1, . . . , xi, . . . , xk)
da.
Adibideak 4.5. Izan bedi G aldea.
(i) G alde o ala e a {1}azpi alde ibiala hi zezko azpi aldeak di a, hu enez
hu en, w(x) = xe a hi z hu sa i dagozkionak.
(ii) Hi z a un e a ezagunena kommu ado e hi za da, hau da, γ2(x, y) = [x, y] =
x−1y−1xy. Ha en hi zezko azpi aldea γ2(G) = G′azpi alde de iba ua da e a
dagokion azpi alde ma jinala γ∗
2(G) = Z(G)zen oa.
(iii) Hi z behe zen alak e eku siboki
γn(x1, . . . , xn) := [γn−1(x1, . . . , xn−1), xn]∀n≥3
gisa de ini zen di a, e a hi z de iba uak e eku siboki δ1(x1, x2) := γ2(x1, x2)
e a
δn(x1, . . . , x2n) := [δn−1(x1, . . . , x2n−1), δn−1(x2n−1+1, . . . , x2n)] ∀n≥2
moduan de ini zen di a.
(i ) Bu nside en hi zak wm(x) = xmdi a. Ho iek Gmhi zezko azpi aldeak de ini-
zen di uz e, hau da, G-ko elemen uen mga en be e u ek so u iko aldeak.
E a, adibidez, w∗
2(G)azpi alde ma jinala gehienez 2o denako elemen u zen-
alek osa zen du e, hau da, w∗
2(G) = {g∈Z(G)|g2= 1}da.
P. Hallek [33] hainba galde a egin zi uen w-balioen mul zoa en e a ha en
hi zezko azpi aldea en e a azpi alde ma jinala en a eko e lazioa en ingu uan.
Hu engo de inizioak i aun ho iek labu zeko balioko du:
De inizioa 4.6. Izan bi ez whi za e a C alde klasea.
241

(i) whi za labu a da C-n, G∈ C guz ie a ako w{G} ini ua iza eak w(G)e e
ini ua dela inplika zen badu.
(ii) whi za sendoa da C-n, G∈ C guz ie a ako |G:w∗(G)| ini ua iza eak w(G)
ini ua dela inplika zen badu.
Ho ela, wlabu a (sendoa) da alde guz ien klasean labu a (sendoa) denean.
An zeko moduan, G aldean hi z guz iak labu ak badi a, Ghi zez labu a dela
diogu.
O oko ean, |w{G}| ≤ |G:w∗(G)|kdenez (kzenbaki osoa whi za en aldagai
kopu ua da), sendo asuna labu asuna baino gogo agoa da: wlabu a bada C-n,
o duan wsendoa da C-n. Haa ik, alde e esidualki ini ue an, e a gu i a du a
zaizkigun aldeak ho elakoak di a, bi kon zep uak baliokideak di a:
Lema 4.7 (c . [67, Lema 1.4.1]).Izan bi ez G aldea e a whi za.
(i) |G:w∗(G)| ini ua bada, o duan w{G} ini ua da.
(ii) Ge esidualki ini ua e a w{G} ini ua badi a, o duan |G:w∗(G)| ini ua
da.
P. Hallek ba hi z guz iak labu ak zi ela i aga i zuen. Ai zi ik, ia hi u ha-
ma kada en os ean, I ano ek [39] aie u ho i e e uxa u zuen, G alde ba e a w
hi z ba opa u bai zi uen non w{G}mul zoak bi elemen u di uen, baina w(G)
alde zikliko in ini ua den. Hala e e, kon adibide ho i ez da e esidualki ini ua,
ez a Ol’shanskiĭk (ikusi [62, Teo ema 39.7]) e aiki zuen an zeko kon adibidea
e e. Ho ek Jaikin-Zapi ainek [44] e a Segalek [67] p oposa u iko aie u hone a a
ga ama za:
Aie ua 4.8 (Labu asuna en aie ua alde e esidualki ini ue an).Hi z guz iak
labu ak di a alde e esidualki ini uen klasean.
Talde hi zez labu en klase gu xi ba zuk baino ez di a ezagu zen. Talde abelda-
en (ikusi Lema 4.2) e a pe iodikoen (ikusi da o en Lema 4.15) age iko adibideez
gain; 1960ko hama kadan, Me zjalko ek [57] e a Tu ne -Smi hek [72] hu enez
hu en oga u zu en alde linealak e a za idu a guz iak e esidualki ini uak di-
uz en aldeak (e.g. alde bi ualki nilpo en eak) hi zez labu ak di ela.
Hi z balioen mul zoa in ini ua denean, labu asuna en pa eko kon zep ua hi z
elip iko asuna da. Ho i de ini zeko no azio hau e abiliko da: X⊆Gazpimul zo
ba e ako, izenda u X∗ℓmoduan X∪X−1∪{1}mul zoko `elemen u en bide ke ek
osa zen du en mul zoa.
242
De inizioa 4.9. Izan bi ez G aldea e a whi za. O duan, welip ikoa da G-n
exis i zen bada `∈Nnon w(G) = w{G}∗ℓden.
Au eko baldin za be e zen du en `zenbaki osoe an xikiena i w- en hi z za-
bale a de i zo. Ho ela, G aldea hi zez elip ikoa da hi z guz iak G-n elip ikoak
di enean. Elip iko asuna labu asuna baino gogo agoa da: welip ikoa bada C
alde klaseko alde guz ie an, o duan wlabu a da C-n.
Lema 4.2 en a abe a, alde abelda ak hi zez elip ikoak di a e a 1hi z zabale a
du e. Ho iez gain, alde aljeb aiko linealak∗(ikusi [56]), ini uki so u ako alde
abelda -bide -nilpo en eak (ikusi [29] e a [71]) edo, esi honen gaia ekin ze ikusi
zuzena duena, alde p-adiko anali iko inkoak (ikusi [44]) hi zez elip ikoak di a.
Haa ik, hi zez elip ikoak ez di en aldeen adibide na u alak daude (kon a-
dibideak ez di a hi zezko labu asuna enak bezain konplexuak bede en). Esa e
ba e ako, Roman’ko ek [66] ini uki so u ako p o-p alde ebazga i ba au kez u
zuen non δ2(x1, . . . , x4) = [[x1, x2],[x3, x4]] hi z de iba uak zabale a in ini ua duen.
Talde p o ini uei dagokionez, hi z zabale a e a hi zezko azpi aldea i xia iza ea
lo u a daude.
P oposizioa 4.10. Izan bi ez GHausdo alde opologiko inkoa e a whi za.
O duan, welip ikoa da G-n baldin e a soilik baldin w(G)i xia bada.
F oga. Soilik baldin no an zan, oha u edozein n- a ako w{G}∗ni xia dela, mul zo
inko ba en i udi ja ai ua bai a. Ho az, w- en hi z zabale a `bada, w(G) =
w{G}∗ℓi xia da.
Bes e ik baldina oga zeko, oha u
w(G) = [
n∈N
w{G}∗n
dela e a w{G}∗nguz iak i xiak di ela. Ho i dela e a, w(G)Hausdo e a inkoa
denez, Bai e en Ka ego ia Teo ema en (ikusi [59, Teo ema 48.2]) a abe a, ex-
is i zen da mzenbaki osoa non w{G}∗m-k ba nealde ez-hu sa duen, hau da,
U⊆ow(G)azpimul zo i eki ez-hu s ba du ba uan. Ho az,
w(G) = [
g∈w(G)
gU
∗ alde aljeb aiko lineal diogunean, GLn(K)- en azpi alde Za iski i xi ba esan nahi dugu, K
go pu z aljeb aikoki i xia dela ik.
243
da, e a w(G)- en inko asunaga ik
w(G) =
[
i=1
giU
da g1, . . . , g ∈w(G)elemen u ba zue a ako. Ha u k∈Nnon gi∈w{G}∗kden
i∈ {1, . . . , }guz ie a ako, o duan
w(G) =
[
i=1
giU⊆w{G}∗(k+m),
da, nahi genuen moduan.
O oko ean, (hi zezko) azpi alde ba i xia den ala ez jaki ea e abilga ia da
alde p o ini uekin ja du e akoan. Ho ega ik, Jaikin-Zapi ainen emai za naba -
men hau enun zia u beha dugu:
Teo ema 4.11 (c . [44, Teo ema 1.1]).Izan bedi w∈Fkhi za kaldagai an.
O duan, whi zak zabale a ini ua ini uki so u ako p o-p alde guz ie an baldin
e a soilik baldin w /∈δ2(Fk) (F′
k)pbada.
Kapi ulu honen xedea alde R-anali iko inkoak hi zez labu ak di ela oga zea
da. Aipa u beha ekoa da cha R= 0 denean, emai za ho i Teo ema 2.27 en – alde
R-anali iko inkoak linealak di a– e a Me zjalko en Teo ema en – alde linealak
hi zez labu ak di a– ondo io zuzena dela. Ai zi ik, in e esga ia da ho en oga
independen ea ema ea, zeina p o-pdomeinu guz ie a ako, ka ak e is ika edozein
dela ik e e, be e zen den.
A e gehiago, emai za o oko hau alde R-anali iko inko guz iak linealak iza-
ea en aldeko bes e ebiden zia ba da.
4.1 Labu asuna alde R-es anda e an
Talde R-es anda en klasean labu asuna be ehalakoa da:
P oposizioa 4.12. Izan bi ez S alde R-es anda a e a whi za. Demagun w{S}
ini ua dela, o duan w(S) = {1}da.
F oga. Lehenik e a behin, S aldea mN(d)- ekin iden i ika u dai eke, non N
aldea en maila e a ddimen sioa di en. Ho ela, bide ke a e a alde an zizkoa bi
244
be e u a se ie o malen uplak de ini zen di uz e e a e a iden i a ea 0da. Ho-
enbes ez, whi z un zioa W∈R[[X1, . . . , Xdk]](d)be e u a se ie upla baka a
da (khi zeko inde e mina u kopu ua da). Ho ela, w{S} ini ua e a hi z un zioa
ja ai ua di enez, Wlokalki kons an ea da; e a, be az, Lema 1.8 en a abe a,
Wkons an ea da. Ho s, W(X1, . . . , Xdk) = W(0, . . . , 0) = 0da, e a, be az,
w{S}={0}.
Emai za ho en pa e ba ondo io aipa u beha di ugu. Alde ba e ik, alde R-
anali iko guz iek labu asuna en aie ua en be sio ahulago hau be e zen du e:
Ko ola ioa 4.13. Izan bi ez G alde R-anali ikoa e a whi za. Demagun w{G}
ini ua dela. O duan, exis i zen da Sazpi alde R-es anda i ekia non wlegea den,
hau da, w(S) = {1}da.
F oga. Lema 1.21en a abe a, badago Sazpi alde R-es anda i eki ba G-n. Ho-
ela, |w{S}| ≤ |w{G}| denez, P oposizioa 4.12 dela e a, w(S) = {1}da.
Bes e alde ba e ik, G alde R-anali iko inkoa bada e a w{G} ini ua, w-
balioen mul zoa soilik azpi alde R-es anda jakin ba en ezke koklaseei begi a
kalkula dai eke. Ho s, izan bi ez G alde R-anali iko inkoa e a S⊴oGazpi-
alde R-es anda i ekia zeina en konjokazio un zioak he siki anali ikoak di en
(biga ena Lema 1.23 dela e a exis i zen da Rez denean IND ba ), e a izan bedi
Tezke ansbe sal ba S- en za G-n. O oi u S- ik e a o i ako a lasa, hau da,
{( S, φ )} ∈Tnon φ (x) = φ( −1x)den. Lema 1.24 en ondo ioz, w:G(k)→G
un zio R-anali ikoa 1S×···× kS( i∈T) mul zo i ekian be e u a se ie upla
baka ak emanda dago –wez da bide ke a e a alde an zizko un zioen konposake a
egokia bes e ik–. Alegia, exis i zen da W 1,..., k∈R[[X1, . . . , Xdk]](d)be e u a se-
ie o malen upla non
φp(w(x1, . . . , xk)) = W 1,..., k(φ 1(x1), . . . , φ k(xk)) ∀xj∈ jS(4.2)
den, hemen pelemen ua w( 1, . . . , k)p−1∈Sbe e zen duen T-ko elemen u baka-
a da.
Gaine a, w{G} ini ua bada, w un zio ja ai ua lokalki kons an ea da, e a Lema
1.8 en e aginez, W 1,..., kkons an ea da, hau da,
W 1,..., k(X1, . . . , Xdk) = c∈R(d).
Ho s, φp(w(x1, . . . , xk)) = cda xj∈ jSguz ie a ako. Bes e e a ba e a esanda:
P oposizioa 4.14. Izan bi ez whi za, G alde R-anali iko inkoa e a Sazpi-
alde no mal R-es anda a zeina en konjokazio un zioak he siki anali ikoak di en.
O duan, w{G} ini ua bada, Sma jinala da w- en za .
245
4.2 Labu asuna alde Fp[[ ]]-anali iko inkoe an
F ogapen eknika hasie ako p oblema ba K ull dimen sioko p o-pdomeinu ba en
gainean anali ikoa den alde ba e a mu iz ean da za, e a ho e a ako A ala 2.2n
desk iba u iko e az un aldake a e abiliko da. Ho i dela e a, lehenbiziko ba di-
men sioko kasua az e u beha da. O oba , alde p-adiko anali iko inkoak hi zez
labu ak di a, linealak bai i a Ko ola ioa 2.6 en a abe a. Ho az, Ko ola ioa 1.44
kon uan ha u a, R=Fp[[ ]] kasu a mu iz gai ezke.
Hainba emai za ekniko e abiliko di ugu. Ho ie ako ba zuk oga uko di a,
baina bes e ba zuk enun zia u baino ez di ugu eginen. Lehenik, gogo a dezagun
emai za ezagun hau:
Lema 4.15. Izan bi ez G aldea e a whi za. Demagun w{G} ini ua dela. O duan,
w(G)′ ini ua da, e a w(G) ini ua da baldin e a soilik baldin w-balio guz iek o dena
ini ua badu e G-n.
F oga. Izan bedi g∈G. Lema 4.3 en a abe a, w{G}g⊆w{G}da, hau da,
x∈w{G}guz ie a ako xGkonjokazio klasea w{G}-n dago. Ho az,
|G:CG(x)|=xG≤ |w{G}|
da, e a CG(x)-k indize ini ua du G-n. Ho enbes ez, CG(w(G)) = ∩x∈w{G}CG(x)
azpi aldeak indize ini ua du G-n, e a, ondo ioz, |w(G) : Z(w(G))|e e ini ua da.
Be az, Schu en Teo ema en a abe a (ikusi [65, Teo ema 10.1.4]), w(G)′ ini ua
da.
Azkenik, w(G) ini ua bada, exponen e ini ua eduki beha du. Alde an ziz,
demagun w{G}-ko elemen uek o dena ini ua du ela, o duan w(G)/w(G)′ alde
abelda a o dena ini uko elemen u kopu u ini u ba ek so zen du, be eziki, ini-
ua da; e a w(G)′ ini ua denez emai za e dies en dugu.
Schu mo ako emai za hau e e beha ko dugu:
Lema 4.16 (c . [45, P oposizioa 5.1]).Izan bi ez G aldea e a Nazpi alde no mal
nilpo en ea. Demagun NZ(G)
Z(G)za idu ak exponen e ini ua duela. O duan, [N, G]
azpi aldeak exponen e ini ua du.
F oga. Hall-Pe escu en o mula en a abe a (ikusi [38, III.9.4]), m∈Nguz i-
e a ako
xmym= (xy)mc2(x, y)(m
2). . . cm(x, y)(m
m)(4.3)
246

da, non c (x, y)∈γ (hx, yi)den.
Izan bedi mzenbakia NZ(G)
Z(G)za idu a aldea en exponen ea, o duan (4.3) dela
e a, n∈Ne a g∈Gguz ie a ako:
[n, g]m≡n−m(n[n, g])m=n−m(ng)m= [nm, g] = 1 mod γ2(K)(4.4)
da, non K:= hn, [n, g]i ≤ Nden. A e gehiago, (4.3) e a (4.4) di ela e a, l∈N
guz ie a ako
([n1, g1]. . . [nl, gl])m≡[n1, g1]m. . . [nl, gl]m≡1 mod γ2(N)
da.
Izan bedi η(m)zenbakia 2so zaileko e a mexponen eko alde nilpo en e han-
diena en o dena†. Nahikoa da γ2(N)η(m)={1}dela oga zea. Ho e a ako,
izan bi ez x, y ∈Ne a H=hx, yi ≤ N. Ho ela, H/Z(H) aldea 2so zaileko
alde nilpo en ea denez e a ha en exponen eak mza i zen duenez, ini ua da e a
k=|H:Z(H)|zenbakiak η(m)za i zen du. Halabe , θ:H→Z(H),h7→ hk
un zioa H- en ans e a da Z(H)- a (konpa a u [65, Teo ema 10.1.3] en oga-
ekin). Be eziki, θhomomo ismoa da e a (xy)k=xkykda. Ho ela xke a yk
elka ekin uka zen di enez e a kzenbaki osoak η(m)za i zen duenez,
(xy)η(m)=xη(m)yη(m)∀x, y ∈N
da. Be eziki, θ′:N→Z(N), n 7→ nη(m) alde homomo ismoa da. Be az, im θ′
abelda a denez, γ2(N)≤ke θ′da, hau da, γ2(N)η(m)={1}.
Da ozen emai zek alde aljeb aiko linealen eo iako ideiak da abil za e. I aku -
leak [37] a jo dezake emai za ho ien a zeko eo ian sakondu nahi badu.
Izan bedi Kgo pu z aljeb aikoki i xia. Tes u hone an zeha alde aljeb aiko
lineal ba GLn(K)- en Gazpi alde Za iski i xi ba izanen da, e a G- en iden i a e
osagaia iden i a ea en osagai konexua da.
P oposizioa 4.17 (c . [37, P oposizioa 7.3]).Izan bedi G alde aljeb aiko lineal
konexua.
(i) O duan, G◦indize ini uko azpi alde no mala da.
(ii) Izan bedi, H ≤ G indize ini uko azpi alde i xi konexua, o duan H=G◦da.
†Zenbaki hau ini ua da, Bae ek [4] oga u zuenez, alde nilpo en eek Bu nside en p oblema
be e zen du elako (ikusi [20, Teo ema 2.23]).
247
Ma ize ba unipo en ea da ha en au obalio baka a 1bada, e a GLn(K)- en
azpi alde ba azpi alde unipo en ea da elemen u guz iak unipo en eak badi a.
Talde ho ien ingu uko egi u azko emai za nagusia alde unipo en e o o Un(K)-
en –diagonalean 1ak di uz en ma ize goi iangelua en aldea– azpi alde ba en
konjoka ua dela da (ikusi [37, Ko ola ioa 17.5]). Be eziki, alde unipo en e guz-
iak nilpo en eak di a, e a oina iko Kgo pu za ka ak e is ika posi ibokoa bada
e a G ⊆ GLn(K) alde unipo en ea, G-k exponen e ini ua du, konpa a u [38,
Kapi ulua III, Lema 16.2 e a Teo ema 16.5] (nahiz e a e e e en zia go pu z ini-
ue a ako izan, a gumen uek exponen e posi iboko go pu ze a ako be din-be din
balio du e).
Bes alde, G alde aljeb aiko lineala emanda, ha en e adikal unipo en ea,Ru(G)
izenda uko duguna, G-ko elemen u unipo en e guz iek osa u ako azpi aldea da,
edo baliokideki G- en azpi alde unipo en e konexu handiena. Ho ela, Ru(G)
konexua e a nilpo en ea denez, G- en e adikal ebazga ian dago, hau da, G- en
azpi alde ebazga i handiena en iden i a e osagaia en ba uan. Halabe , G alde
aljeb aiko lineala e eduk iboa dela diogu konexua bada e a Ru(G) ibiala bada.
E aikun za ho iek guz iak alde aljeb aiko linealen eo ian ga an zia handi-
koak di a, haa ik soilik de inizio nagusiak labu u e a haien a eko e lazioak
enun zia uko di ugu. Izan e e, honako emai za eknikoa baino ez dugu beha :
P oposizioa 4.18 (c . [37, Lema 17.9]).Izan bi ez G alde aljeb aiko lineal
konexua e a Nha en e adikal ebazga ia. O duan, [N,G]unipo en ea da.
F oga. Izan bedi Ru(G)e adikal unipo en ea. O duan, G/Ru(G)e eduk iboa
da. Be az, [37, Lema 17.9] en a abe a, N/Ru⊆Z(G/Ru)da e a, ondo ioz, [N,G]⊆
Ru(G)da.
O ain a al hone an bila genbil zan emai za oga dezakegu:
Teo ema 4.19. Talde Fp[[ ]]-anali iko inkoak hi zez labu ak di a.
F oga. Izan bi ez G alde Fp[[ ]]-anali iko inkoa e a whi za. Demagun w{G}
ini ua dela. Lehenik e a behin, Lema 4.15 dela e a, w(G)′ ini ua da. Ondo ioz,
beha izanez ge o za idu a ba e a pasa a, o oko asunik galdu gabe w(G) ini uki
so u ako alde abelda a dela suposa dezakegu.
Ko ola ioa 4.13 en a abe a, exis i zen da S alde Fp[[ ]]-es anda a non wlegea
den. Labu zea en izenda di zagun Z=Z(S)e a K=Fp(( ))alg,Fp(( )) go pu z
lokala en i xi u a aljeb aikoa. P oposizioa 2.3ga ik, S/Z lineala da Fp[[ ]]- en
gainean, e a, ondo ioz, lineala da Fp(( )) e a Kgo pu zen gainean e e. Gaine a,
Ti sen al e na iba opologikoa (loc. ci .) dela e a, S/Z-k azpi alde ebazga i
248
i eki ba edo azpi alde aske den so ba du ba uan. Baina S/Z aldeak lege ba
be e zen duenez, bi ualki ebazga ia izan beha da.
Izan bedi S aldea S/Z- en Za izki i xi u a GLn(K)-n, o udan Sbi ualki ebaz-
ga ia den P oposizioa 3.34 en a abe a, e a izan bi ez N S- en e adikal ebazga -
ia, hau da, azpi alde ebazga i konexu handiena e a N◦be e osagai konexua, hau
da, S- en e adikal ebazga ia. P oposizioa 4.17(i) dela e a, Nindize ini uko
azpi aldea da S-n, be az, P oposizioa 4.17(ii) en e aginez, N◦=S◦da.
Izan bedi N/Z aldea S/Z- en ebakidu a N◦- ekin, o duan, beha izanez ge o
muina no male a pasa a, Nindize ini uko azpi alde no mala da G-n. P opo-
sizioa 4.18 en a abe a, [N◦,N◦]unipo en ea da. Be eziki, [N◦,N◦]nilpo en ea
da e a, Kgo pu zak ka ak e is ika posi iboa duenez, exponen e ini ua du. Ho -
az, [N, N]Z/Z exponen e ini uko alde nilpo en ea da. Ho enbes ez, [N, N]Z
nilpo en ea da e a Lema 4.16 en a abe a, H:= [N, N, S]-k exponen e ini ua du.
Alde ba e ik,
H= [N, N, S]≥[N, N, N]
da, e a, be az, G/H bi ualki gehienez 2klaseko alde nilpo en ea da. Ho ela,
|w{G/H}| ≤ |w{G}| denez e a G/H bi ualki nilpo en ea denez, w(G/H) ini ua
da Tu ne -Smi hen Teo ema en a abe a (ikusi [72, Ko ola ioa 2]).
Bes e alde ba e ik, w(G)∩H ini uki so u ako alde abelda a da e a exponen e
ini ua du, be az ini ua da. Azkenik,
wG
H=w(G)H
H∼
=w(G)
w(G)∩H
isomo ismoak ema en du emai za.
Ko ola ioa 4.20. Izan bedi Rba K ullen dimen sioko p o-pdomeinua. O duan,
alde R-anali iko inkoak hi zez labu ak di a.
4.3 Labu asuna alde R-anali iko inkoe an
P es gaude emai za nagusia oga zeko.
Teo ema 4.21. Talde R-anali iko inkoak hi zez labu ak di a.
F oga. Ko ola ioa 4.20 ain za ha u a, demagun Rp o-pdomeinuak gu xienez 2
K ull dimen sioa duela. Izan bi ez G alde R-anali iko inkoa e a whi z ba k
aldagai an non w{G} ini ua den. Lema 4.15 dela e a, nahikoa da w-balio guz iek
o dena ini ua du ela oga zea.
249
Lema 1.23 en a abe a, exis i zen da (S, φ)azpi alde R-es anda i eki no mala
non g∈Gguz ie a ako cg:S→S, x 7→ xgkonjokazio aplikazioak he siki anali-
ikoak di en. Ho ela, n=|G:S|bada, wn{G} ⊆ Sda e a, Lema 4.15en a abe a,
wn(G) ini ua da baldin e a soilik baldin w(G)is ini ua bada. Ho az, o oko a-
sunik galdu gabe suposa dezagun w{G} ⊆ Sdela.
Izan bedi (P, m)ideal nagusie ako p o-pdomeinua hau: P=Zp,cha R= 0
denean, e a P=Fp[[ ]],cha R=pposi iboa denean. Cohenen Egi u a Teo e-
ma en a abe a (ikusi Teo ema 1.2), Re az una P[[ 1, . . . , m]]- en ini uki so u a-
ko e az un hedadu a in eg ala da, m= dimK ull(R)−1izanik. Ho ela, a∈m(m)
bakoi ze ako, izan bedi sa:P[[ 1, . . . , m]] →P, F( 1, . . . , m)7→ F(a)ebaluazio
homomo ismoa. Ko ola ioa 2.13k saepimo ismoa ˜sa:R→Qe az un epimo -
ismo a heda zen du, non Oha a 2.14 en a abe a, Q= (Q, n)p o-pdomeinua
P- en ini uki so u ako e az un hedadu a in eg ala den, be eziki Q- en K ullen
dimen sioa 1da.
Izan bedi a∈m(m),A ala 2.2ko no azioa ja ai uz, oga hone an zeha Wa
moduan izenda uko dugu W˜sa∈Q[[X]](l)be e u a se ie o malen upla, zein-
nahi W∈R[[X]](l)be e u a se ie o malen upla a ako. Be eziki, Fbada S-
en alde e agike a o mala, o duan Fa=F˜sa alde e agike a o mala da (ikusi
Ko ola ioa 2.8). Izan bedi Tezke ansbe sal ba S- en za G-n, e a suposa
dezagun 1∈Tdela. Hemendik au e a S- ik e a o i ako a lasaz balia uko ga a,
ho s, {( S, φ )} ∈Tnon φ (x) := φ( −1x)den (konpa a u A ala 1.3).
Lema 2.10 e abili a, de ini u L:= nN(d) alde Q-es anda a, zeina en alde
e agike a Fa alde e agike a o malak de ini zen duen, e a H:= T×L alde Q-
anali ikoa (2.3)ko e agike a ekin, zeina L- en gain aldea bali z bezala ikus dai e-
keen. O oi u H- en egi u a Q-anali ikoa {( L, ψ )} ∈T,non ψ ( , l) = l, a lasak
ema en duela.
A al hau buka u a e inka dezagun ( 1, . . . , k)∈T(k) upla, e a demagun,
(4.2) dela e a, edozein l∈Nzenbaki a ako wlhi z un zioa 1S×···× kSmul zo
i ekian Wlbe e u a se ie o malen uplak emanda dagoela, hau da, w{G} ⊆ S
dela ain za ha u a,
φwl(x1, . . . , xk)=Wl(φ 1(x1), . . . , φ k(xk)) ∀xj∈ jS
dugu (no azioa a in zea en explizi uki ida ziko ez den a en, kon uan eduki Wl
be e u a se iea 1, . . . , kbalioen menpekoa e e badela).
Izan bedi wl:H(k)→Hhi z un zioa ∗ae agike a ekin H-n. Lema 2.7 e a
Oha a 2.11 di ela e a,
ψ1wl(x1, . . . , xk)=Wl
a(ψ 1(x1), . . . , ψ k(xk)) ∀xj∈ jL
250
Hemendik au e a [74]ko no azioa ja ai uz, R-Lie e e ikulu ba heina ini uko
R-modulu askea den R-Lie aljeb a ba da. Go pu zen gainean Ado en Teo ema en
oga ani z daude (ikusi, esa e ba e ako, [41, Kapi ulua VI, 2. A ala]), e a ho i-
e ako gehiene a ik ondo ioz a u dai eke e aiki zen den adie azpena en maila soilik
dimKL,aljeb a en K-espazio bek o ial dimen sioa en, menpekoa dela. Zeha zago
esanda, izan bedi
deg L:= min{deg φ|φL- en adie azpen leiala da},
o duan [13] e a [60]ko a gudioe an oina i u a, ikus dai eke Rze o ka ak e is-
ikako go pu za e a = dimKLdi enean:
deg L≤α2
√ (A.1)
dela, α∈Rba en za (ikusi [58, 1.1.2 A ala] oga zeha z ba e ako). Alabaina,
badaude go pu zen gainean deg Laz e zen du en bes e hainba lan e e; aipa u
beha ekoak di a [8], [31] edo a [63] lanak.
Haa ik, RIND o oko ba denean, aipa u ako bi oge a ik ez da zuzenean
ondo ioz a zen deg Lzenbaki osoa soilik k L,e e ikulua en heina en, menpekoa
denik. A eago, [74, P oposizioa 3.4]n ge o a adie azpena en maila izanen dena
ini ua da Re az un noe he da a iza eaga ik ideal segida ba geldiko a delako,
baina ezin da zehaz u zenba idealek osa zen du en segida ho i.
Apendize hone an, R-Lie e e ikulu ba en adie azpen leial ba e aiki zeko modu
kuan i a ibo ba au kez uko dugu, [8] e a [63]ko ideie an oina i uz. Zeha zago:
Teo ema A.4. Izan bi ez Rze o ka ak e is ikako INDa e a L heinako R-Lie
e e ikulua. O duan,
deg L≤ + + 1
4
da.
O oba , aipa u ka ak e is ika posi iboko go pu ze a ako Ado en Teo ema en
pa ekoa be e zen dela, ho i da Iwasawa en Teo ema [40] hain zuzen e e. A e
gehiago, Teo ema A.2 en be sio o oko a i, koe izien een go pu za i bes e inolako
baldin za ik eza i gabe, Ado-Iwasawa en Teoe ema dei zen zaio. Ka ak e is ika
posi iboan emai za o oko asun askoz gehiago ekin eman dai eke:
Teo ema A.5 (c . [19, Teo ema 3]).Izan bi ez Rka ak e is ika posi iboko e az un
ukako a e a Lheina ini uko R-Lie e e ikulua. O duan, exis i zen di a Wheina
ini uko R-modulu askea e a φ:L→EndR(W)R-Lie aljeb a monomo ismoa.
257

Ho i oga zeko nahikoa da oga o iginala hi zez hi z e epika zea, e a, ho -
enbes ez, go pu zen gainean lo zen den bo ne be a lo zen da R-Lie e e ikuluen
maila en za . Ho s,
deg L≤n k3L,
n= cha Rizanik (ikusi [5, A ala 6.2.4]).
Oha ak. F oge an zeha R-modulu askeen (RINDa izanik) ingu uko zenbai
p opie a e e abiliko di a. Hona x gogo a u beha ekoak:
(i) M R-modulu aske ba en azpimoduluak askeak di a, e a gehienez k(M)
heina du e.
Izan bi ez M R-modulua e a N≤Mazpimodulua. N- en isola zailea M-n
IsoM(N) = {x∈M| ∃ ∈R {0}non x ∈N}
azpimodulua da, e a N M-n isola ua dela diogu IsoM(N) = Ndenean.
(ii) M/ Iso(N) o sio ik gabeko R-modulua da.
(iii) M R-modulu askea, N≤Mazpimodulu isola ua e a M/N R-modulu ini-
uki so ua badi a, M/N R-modulu askea da, e a
k(M) = k(N) + k(M/N)
da. Kasu hone an, k(M/N)zenbakia i N- en koheina de i zogu.
(i ) Mheina ini uko R-modulu askea e a Nazpimodulu isola ua badi a, N-k
osaga ia dauka M-n, hau da, exis i zen da L R-modulu askea non M=
N⊕Lden.
A.2 Adie azpen adjun ua e a e egula a
Au kez di zagun zein-nahi Lie aljeb a en bi adie azpen. Alde ba e ik, x∈Lele-
men uak adx:L→L, y 7→ [x, y]aplikazio lineala de ini zen du. Jacobi en iden i-
a ea dela e a, esleipen ho ek L- en k Lmailako adie azpen adjun ua de ini zen
du, hau da,
Ad: L→EndR(L), x 7→ adx.
Hale e, adie azpen ho i o oha ez da leila, ha en nukleoa L- en zen oa bai a, hau
da,
Z(L) := {x∈L|[x, y] = 0 ∀y∈L}.
258
Ho enbes ez, LR-aljeb a semisinplea bada –ho s, ez badu ideal abelda ez-
ibialik–, adie azpen adjun ua leiala da e a deg L≤ k Lda.
Biga en adie azpena au kez eko ingu a ze aljeb a unibe sala de ini u beha
da.
De inizioa A.6 (c . [41, Kapi ulua V, Teo ema 1.1]).Izan bedi LR-Lie aljeb a.
L- en R- en so e aljeb a
TR(L) = R⊕L1⊕L2⊕···⊕Li⊕. . .
da, Li=L⊗(i)
. . . ⊗L en sio R-modulua dela ik. Ho en R-modulu egi u a bide -
ke a en so iala ena da, e a bide ke a
(x1⊗···⊗xi)⊗(y1⊗···⊗yi) = x1⊗···⊗xi⊗y1⊗···⊗yi
a auak de ini zen du. Izan bedi R
[u, ]−(u⊗ − ⊗u), u, ∈L
elemen uek so u ako TR(L)- en ideala. O duan, L- en ingu a ze aljeb a unibe -
sala
UR(L) := TR(L)
R
iden i a edun R-aljeb a elka ko a da.
Gaine a, Le a L1elka ekin iden i ika uz ge o, ι:L→ UR(L)homomo ismoa
dugu. Ikus dai eke L ini uki so ua e a RINDa di enean, ιinjek iboa dela (ikusi
[74, Teo ema 3.2]), e a, be az, L⊆ UR(L)dela asumi dezakegu. Halabe , no azioa
sinpli ika zea en, x1⊗···⊗xkelemen ua x1. . . xkmonomioa bezala ida ziko dugu.
Aljeb a unibe sala Poinca é-Bi kho -Wi en eo emak desk iba zen du:
Teo ema A.7 (c . [74, Teo ema 3.2]).Izan bi ez L heinako R-Lie e e ikulua
e a {x1, . . . , x }ha en oina ia. O duan, UR(L)R-modulu askea da, e a
{xα1
1. . . xα
|αi∈N0}(A.2)
monomioek oina i ba osa zen du e.
Bi monomio emanda, haien bide ke a (A.2) moduko monomioen konbinazio lin-
eal gisa adie az dai eke, hu enez hu en xjxi=xixj−[xi, xj]iden i a ea e abiliz
259
inde e mina uak o dena zeko.
Edozein LR-Lie e e ikuluk UR(L)- en gainean e agi en du ezke bide ke az.
Ho s, x∈Lbakoi ze ako `x:UR(L)→ UR(L), u 7→ xu R-endomo ismoa dugu.
Alde ba e ik, edozein x, y ∈L- a ako
[x, y] = xy −yx
da UR(L)-n e a, be az, L:L→EndR(UR(L)), x 7→ `xR-Lie aljeb a homomo -
ismoa da. Halabe , ingu a ze aljeb a unibe sala iden i a eduna denez x6=y
denean
`x(1) = x6=y=`y(1)
da. Ondo ioz, Ladie azpen leiala da, (ezke ) adie azpen e egula dei uko duguna.
Al a, UR(L)heina in ini ukoa denez, adie azpen ho i ez da ini ua. Baina ingu a ze
aljeb a unibe sala ka ak e iza zen duen p opie a e hau dela e a, Lie aljeb a adie-
azpen guz iek UR(L)- en za idu a ba en gainean eki en du e.
Teo ema A.8 (P opie a e unibe sala, c . [11, Kapi ulua I, § 2.1, P oposizioa
1]).Izan bi ez LR-Lie e e ikulua, A R-aljeb a elka ko a, [a, b] = ab −ba Lie en
ko xea (a, b ∈Aguz ie a ako) e a ψ:L→(A, [,]) R-Lie aljeb a homomo ismoa.
O duan, exis i zen da ψ∗:UR(L)→A R-aljeb a homomo ismo baka a non ψ=
ψ∗◦ιden. Alegia, diag ama hau ukako a da:
UR(L)
LA.
ψ∗
ι
ψ
Hain zuzen e e, p opie a e unibe sal ho i da UR(L)- en izena en a azoia. A e
gehiago, oga dai eke p opie a e unibe sala be e zen duen edozein R-aljeb a
UR(L)- i isomo oa dela (ikusi [41, Kapi ulua V, Teo ema 1.1.1]).
A.3 Ado en Teo ema
Ado en Teo ema oga zeko asko an LR-Lie e e ikulu ik LK:= L⊗RK K-
espazio bek o iale a pasako ga a. Oha u LKK-aljeb a ba dela e a ha en Lie en
ko xe ea L- en Lie en ko xe eak en so iza uz induzi u iko K-aplikazio lineala
dela. Kon uan ha u be ehalako p opie a e hauek:
•LKespazio bek o ialak k Ldimen sioa du,
260
•L=hx1, . . . , x iRbada, o duan LK=hx1, . . . , x iKda e a
•I⊴LKideal guz ie a ako, I∩L⊴Lideal isola ua da.
Teo ema hi u u a se an oga uko dugu.
A.3.1 Lie en e e ikulu nilpo en eak
Lehenik e a behin, demagun L heinako R-Lie e e ikulu nilpo en ea dela. Gogo-
a u, LLie e e ikulua en se ie zen al behe ako a e eku siboki
γ1(L) = L, γi(L) = [γi−1(L),L]∀i≥2
moduan de ini zen dela, e a Lnilpo en ea dela exis i zen bada γc+1(L) = {0}
be e zen duen c∈Nzenbaki osoa. Zenbaki oso ho ie an xikiena i, exis i uz
ge o, R-Lie e e ikulua en nilpo en zia klasea de i zo. Bide ke a en so iala lineala
denez, e az ikus dai eke honako emai za hau:
Lema A.9. Izan bi ez LR-Lie e e ikulua e a I,H⊴Lidealak. O duan,
[I⊗RK, H⊗RK] = [I,H]⊗RK
da. Be az, Lcklaseko R-Lie e e ikulu nilpo en ea bada, LKcklaseko K-Lie
aljeb a nilpo en ea da.
O duan,
γ0(LK)> γ1(LK)>··· > γi(LK)>··· > γc+1(LK) = {0}
K-espazio bek o ialen segida he siki behe ako a denez, c≤dim LK= da.
De ini u Li=γi(LK)∩L⊴Lideal isola uak, e a ha u L- en {x1, . . . , x }oina ia
non lehenengo x1, . . . , x 1elemen uak Lc- en oina ia di en, lehenengo x1, . . . , x 2
( 2> 1) elemen uak Lc−1-en oina ia di en, e a ho ela hu enez hu en. Teo ema
A.7 en a abe a,
xα:= xα1
1. . . xα
, α = (α1, . . . , α )∈N( )
0
monomioek UR(L)ingu a ze aljeb a unibe sala en oina ia osa zen du e. Ho i
kon uan ha u a de ini dezagun ω:UR(L)→N0∪ {∞} pisu un zioa ondoko
moduan:
261
ω(xi) = max{m|xi∈Lm}ω(xα) = P
i=1 αiω(xi)
ω(P
i=1 cαxα) = min {ω(xα)|cα6= 0}e a ω(0) = ∞.
Edozein m∈N0- a ako de ini u
Um(L) := {u∈ UR(L)|ω(u)> m}
–Le e ikulua zein den ga bi dagoenean, Umbaino ez dugu ida ziko–. Ikus deza-
gun Um⊴UR(L)ideal isola ua dela:
(i) Oha u 0∈Umdela mguz ie a ako, e a
ω( x) = ω(x)e a ω(x+y)≥min{w(x), w(y)}
di ela x, y ∈ UR(L)e a ∈R {0}guz ie a ako. Gaine a, ω([xi, xj]) ≥
ω(xi) + ω(xj)denez (edozein i, j ∈ {1, . . . , }- a ako),
ω(xy)≥ω(x) + ω(y)(A.3)
da. Ondo ioz, Umideala da.
(ii) Izan bedi ∈R {0},o duan
x ∈Um=⇒ω(x) = ω( x)> m =⇒x∈Um,
hau da, Umideal isola ua da.
Bes alde, UR(L)/Um ini uki so ua da,
Bm={xα+Um|ω(xα)≤m}
mul zoak so zen bai u. Ho az, Umkoheina ini uko R-modulu aske isola ua
da, e a, (A.3) en a abe a, x∈Lguz ie a ako `x(Um)⊆Umda. Be az, edozein
m- a ako ezke adie azpen e egula ak
Lm:L→EndR(UR(L)/Um), x 7→ `x
adie azpen ini ua ema en du –izan bedi ∈EndR(UR(L)) non (X)⊆Xden
X⊴UR(L)ideal ba e ako, no azio abusu ba ekin, be i o e abiliko da x+X7→
(x) + Xmoduan de ini u iko ∈EndR(UR(L)/X)endomo ismoa izenda zeko–.
262

Bes e ik, L∩Uc(L) = {0}da. Izan e e, x=P
i=1 αixibada,
ω(x) = ω
X
i=1
αixi!≤max
i=1,..., ω(xi) = c
da.
Ho i dela e a, LcL- en adie azpen leial ini ua da, e a ho en maila bo na zeko,
nahikoa da |Bc|goi ik bo na zea. Alde ba e ik, Bc-n dauden monomio guz iek
gehienez cpisua du e, e a, be az, gehienez cpolinomio maila. Bes e ik, aldagai-
an gehienez cmailako monomio kopu ua +1 aldagai an cmailako monomio ko-
pu ua da (aldagai lagun zaile ba gehi uz homogeneiza u monomio guz iak zeha z-
meha z cmaila izan deza en).
Lema A.10. aldagai an cmailako monomio kopu ua  +c−1
cda.
Be az, bo ne laño hau dugu (konpa a u [31, Ko ola ioa 5.1]):
deg L≤ k UR(L)
Uc(L)≤ +c
c.
A al hau buka zeko deg L- en bo ne ez oso zo o z ba emanen dugu, baina
soilik k L- en menpe. Izan e e, c∈ {1, . . . , }da, hau da, c=α da α∈
{1/ , . . . , −1/ ,1}izanik. Bes e ik, S i lingen hu bilke a o mula en a abe a,
√2π ( /e) ≤ !≤√2π ( /e) e1
12 ∀ ∈N.
da. Ondo ioz,
 +α
α ≤p2π( +α )( +α ) +α
√2πα (α )α √2π e1
12(1+α)
≤e1/12(1 + α)
√2π
√1 + α
√α(1 + α)1+α
αα
.
Gaine a, bi ezbe din ze an ezke eko e a eskuineko e minoak asin o ikoki balio-
kideak di a, hau da, haien za idu a 1e a doa in ini u a joan ahala. Halabe ,
α∈[1/ ,1] denez,
√1 + α
√α≤√ + 1 e a (1 + α)1+α
αα≤4
263
di a. Ho s,
e1/12(1 + α)
√2π
1
√
√1 + α
√α(1 + α)1+α
αα
≤ + 1
4 .
Ho enbes ez,
 +c
c≤ + 1
4 ∀c∈ {1, . . . , }.(A.4)
A.3.2 Lie en e e ikulu bananga iak
Biga en u a sean R-Lie aljeb a adie azpen egoki ba emanen dugu R-Lie e e i-
kulu bananga i dei uko di ugune a ako.
Ideal nilpo en een ba u a be i o ideal nilpo en ea da (konpa a u [41, Kapi u-
lua I, P oposizioa 7.6]), be az, ini uki so u ako LR-Lie aljeb a guz iek badu e
ideal nilpo en e o o ba uan duen ideal nilpo en e ba , L- en e adikal nilpo en e
dei uko duguna e a Rn(L)adie aziko dena. Be eziki, Z(L)≤Rn(L)da, e a Rn(L)
ideal isola ua da, Rn(L) = Rn(LK)∩Lda e a.
Ho ela, Lheina ini uko R-Lie e e ikulua bananga ia dela diogu, exis i zen
bada S≤LR-Lie azpialjeb a ba non L=Rn(L)⊕Sden, hau da, LLie aljeb a
S- en bide ke a e dizuzena da Rn(L)ideala ekin. Ka ego ia eo ia en ikuspegi ik
baliokidea da esa ea
0→Rn(L)→L→L/Rn(L)→0
segida zeha z labu a bana u egi en dela R-Lie aljeb en ka ego ian.
Kasu bananga i ako, au eko adie azpen e egula a e a de ibazioe a ik e a o-
i ako adie azpenak konbina dai ezke:
De inizioa A.11. Izan bedi A R-aljeb a. O duan, D∈EndR(A)R-modulu
endomo imoa de ibazioa da Leibnizen iden i a ea be e zen badu, hau da,
D(ab) = aD(b) + D(a)b∀a, b ∈A.
De ibazio guz ien mul zoa De R(A)izenda zen da.
Esa e ba e ako, Jacobi en iden i a ea en e aginez, edozein x∈L- a ako adx∈
De R(L)da. Bes alde, LR-Lie e e ikulua en Dde ibazio ik abia u a UR(L)- en
264
de ibazio ba e aiki dai eke, D∗dei uko duguna, Leibnizen iden i a ea be e zea
inposa uz. Ho s,
D∗(u1. . . u ) =
X
i=1
u1. . . ui−1D(ui)ui+1 . . . u
a aua en hedapen lineala ha u e a D∗(1) = 0 de ini u (iden i a edun aljeb a guz-
ie a ako be e beha bai u azken ho ek). Hedadu a ho i p opie a e unibe sala en
kasu pa ikula a baino ez da (konpa a u [41, Kapi ulua V, Teo ema 1.1(7)]).
Lema A.12. Izan bi ez LR-Lie e e ikulu nilpo en ea e a D∈De R(L).O duan,
D∗(Um(L)) ⊆Um(L)da m∈Nguz ie a ako.
F oga. Demagun ωpisu un zioa L- en {x1, . . . , x }R-oina ia ekiko de ini u dela
(konpa a u Azpia ala A.3.1). O duan, D R-Lie aljeb a homomo ismoa denez,
D(Li)⊆Lida i∈ {1, . . . , c}guz ie a ako, e a, be az, ω(D(xi)) ≥ω(xi)da i∈
{1, . . . , }guz ie a ako, hau da, oina iko elemen u guz ie a ako. Ho enbes ez,
xi1. . . xi ∈Umbada, (A.3) dela e a,
ω(D∗(xi1. . . xi )) = min
j=1,..., {ω(xi1. . . D(xj). . . xi )} ≥ ω(xi1. . . xi )> m.
Teo ema A.13 (Zassenhaussen hedapena, c . [11, Kapi ulua I, § 7.3, Teo ema
1] e a [41, Kapi ulua VI, Teo ema 2.1]).Izan bedi LR-Lie e e ikulu bananga ia
e a izan bedi czenbaki osoa Rn(L)- en nilpo en zia klasea. O duan, exis i zen da
L- en
Φ: L→EndRUR(Rn(L))
Uc(Rn(L)) 
adie azpena zeina injek iboa den Rn(L)-n. Be eziki, deg Φ soilik k Rn(L)- en
menpekoa da.
F oga. Izenda u N:= Rn(L).O duan, L=N⊕Sda, S≤LR-Lie azpialjeb a
ba e ako, e a Lema A.12 dela e a, ad∗
x(Uc(N)) ⊆Uc(N)da x∈Lguz ie a ako.
Ho az, de ini u
Φ: L=N⊕S→EndR(UR(N)/Uc(N)), n +s7→ `n+ ad∗
s.
un zioa. Ikus dezagun ΦR-Lie aljeb a homomo ismoa dela. Ho e a ako nahikoa
da
Φ ([s, n]) = [Φ(s),Φ(n)] = [ad∗
s, `n]
265
dela ikus ea n∈Ne a s∈Sguz ie a ako. Alde ba e ik, oha u n∈Ne a
D∈De R(UR(N)) guz ie a ako
[D, `n](u) = D◦`n(u)−`n◦D(u) = D(n)u=`D(n)(u)∀u∈N
dela. Bes alde, Nideala denez, [s, n]∈Nda, e a, ho enbes ez,
Φ ([s, n]) = `[s,n]=`ad∗
s(n)= [ad∗
s, `n] = [Φ(s),Φ(n)] .
Be az, ΦR-Lie aljeb a adie azpena da, e a ha en nukleoak ebakidu a ibiala du
e adikal nilpo en ea ekin, Φ|N=Lce a Lcadie azpen leiala bai i a.
Buka zeko, iden i a ea (A.4) en a abe a,
deg Φ ≤s k Rn(L)+1
k Rn(L)·4 k Rn(L)(A.5)
da.
A.3.3 Mu gilke a eo ema
Nilpo en zia bezala, R-Lie aljeb e an ebazga i asuna de ini dai eke: LR-Lie al-
jeb a en se ie de iba ua e eku siboki
L(1) := L,L(i):= L(i−1),L(i−1)∀i≥2
moduan de ini zen da, e a Lebazga ia da exis i zen bada `∈Nzenbaki osoa non
L(ℓ)={0}den. Ho i be e zen du en zenbaki osoe an xikiena da, hain zuzen e e,
L- en de ibazio luze a. A e gehiago, ideal ebazga ien ba u a be i o e e ebazga -
ia da (konpa a u [41, Kapi ulua I, P oposizioa 7.4]), e a, ho enbes ez, ini uki
so u ako LLie aljeb a en e adikal ebazga ia de ini dai eke, hau da, bes e ideal
ebazga i o o ba uan duen Rs(L)ideal ebazga ia. Noski, aljeb a nilpo en e guz-
iak ebazga iak di a e a, ondo ioz, Rn(L)≤Rs(L)da.
Le i en Teo ema en a abe a (ikusi [41, Kapi ulua III, A ala 9]), Rze o ka ak-
e is ikako go pu za denean, LR-Lie aljeb a o o Rs(L)⊕Smoduan deskon-
posa zen da, S≤LR-Lie azpialjeb a semisinple ba dela ik, Le i en osaga i
dei ua. Eskua ki deskonposizio ho ek ga an zia handia du go pu zen gainean
Ado en Teo ema oga ze akoan. Izan e e, lehenbiziko Rs(L)- en adie azpen leial
ini u ba e aiki ohi da, e a ondo ik adie azpen ho i L- a heda u, Zassenhaussen
hedapena en bidez (konpa a u Teo ema A.13).
266
E e e en ziak
[1] Abe comb ie, A. G.: Subg oups and sub ings o p o ini e ings, Ma h. P oc.
Camb . Phil. Soc. 116, 209-222 (1994).
[2] Ado, I. D.: The ep esen a ion o Lie algeb as by ma ices, Ame . Ma h. Soc.
T ansla ions 21(2) (1949); hemendik i zulia: Usehi Ma em. Nauk (NS) 6(22),
159-173 (1947).
[3] A iyah, M., F. e a Macdonald, I., G.: In oduc ion o Commu a i e Algeb a,
Addison-Wesley, Reading (1969).
[4] Bae , R.: Nilpo en g oups and hei gene aliza ion, T ans. Ame . Ma h. Soc.
47, 393-434 (1940).
[5] Bah u in, Y.: Iden i ical ela ions in Lie algeb as, De G uy e Exposi ions in
Ma hema ics 68, De G uy e , Be lin-Bos on (2021).
[6] Ba nea, Y. e a Shale , A.: Hausdo dimension, p o-pg oups and Kac-
Moody algeb as, T ans. Ame . Ma h. Soc. 349, 5073-5091 (1997).
[7] Ba nea, Y., Shale , A. e a Zelmano , E.I.: G aded subalgeb as o affine Kac-
Moody algeb as, Is ael J. Ma h. 104, 321-334 (1998).
[8] Bi kho , G.: Rep esen a ibili y o Lie algeb as and Lie g oups by ma ices,
Ann. o Ma h. 38, 526-532 (1937).
[9] Bishmu h, R.: Ra ional o mal g oup laws, B. Sc. Tesia, McMas e Uni e si y,
Hamil on, On a io (1976).
[10] Bou baki, N.: Commu a i e Algeb a, Chap e s 1-7, Sp inge -Ve lag, Be lin-
Heidelbe g (1989).
[11] Bou baki, N.: Lie G oups e a Lie Algeb as, Chap e s 1-3, Sp inge -Ve lag,
Be lin-Heidelbe g (1989).
273

[12] B euilla d, E. e a Gelande , T.: A opological Ti s al e na i e, Ann. o Ma h.
166, 427-474 (2007).
[13] Bu de, D.: On a e inemen o Ado’s Theo em, A ch. Ma h. 70 118-127,
(1998).
[14] Camina, R. e a Du Sau oy, M.: Linea i y o Zp[[ ]]-Pe ec G oups, Geom.
Dedica a 107, 1-16 (2004).
[15] Ca e , R., W.: Simple g oups o Lie ype, John Wiley & Sons, Lond es-New
Yo k (1989).
[16] Casals-Ruiz, M. e a Zozaya, A.: Linea i y o compac R-analy ic g oups,
submi ed.
[17] Chang, C. C. e a Keisle , H. J.: Model Theo y, No h Holland, New Yo k
(1990).
[18] Che alley, C.: Publica ions de l’Ins i u Ma héma ique de l’Uni e si é de
Nancago, Ac uali és Scien i iques e Indus ielles 1226, He mann, Pa is
(1968).
[19] Chu kin, V. A.: The ma ix ep esen a ion o Lie algeb as o e ings, Ma h.
No es 10(6), 835-839 (1971); hemendik i zulia: Ma . Zame ki 10, 671-678
(1971).
[20] Clemen , A. E., Majewicz, S. e a Zyman, M.: The heo y o nilpo en g oups,
Bi khäuse , Cham (2017).
[21] Cohen, I. S.: On he s uc u e and ideal heo y o comple e local ings, T ans.
Ame . Ma h. Soc. 59, 54-106 (1946).
[22] De omi, E., Klopsch, B. e a Shumya sky, P.: S ong conciseness in p o ini e
g oups, J. London Ma h. Soc. 102, 977-993 (2020).
[23] De omi, E., Mo igi, M. e a Shumya sky, P.: On conciseness o wo ds in p o i-
ni e g oups, J. Pu e Appl. Algeb a 220, 3010-3015 (2016).
[24] Dixon, J., Du Sau oy, M., Mann, A. e a Segal, D.: Analy ic P o-pG oups,
2nd ed., Camb idge Uni e si y P ess, Camb idge (1999).
274
[25] Eisenbud, D.: Commu a i e algeb a wi h a iew owa d algeb aic geome-
y, G adua e Tex s in Ma hema ics 150, Sp inge -Ve lag, Be lin-Heidelbe g-
New Yo k (1995).
[26] Falcone , K.: F ac al geome y: ma hema ical ounda ions e a applica ions,
3 d edn. Wiley, Chiches e (2014).
[27] Fe nández-Alcobe , G. A., Giannelli, E. e a González-Sánchez, J.: Hausdo
dimension in R-analy ic p o ini e g oups, J. o G oup Theo y 20, 579-587
(2017).
[28] Folland, G. B.: Real Analysis: mode n echniques and hei applica ions,
John Wiley & Sons, New Yo k (1999).
[29] Geo ge, K. M.: Ve bal p ope ies o ce ain g oups, Dok o ego Tesia, Uni e -
si y o Camb idge, Camb idge (1976).
[30] González-Sánchez, J. e a Zozaya, A.: S anda d Hausdo spec um o com-
pac Fp[[ ]]-analy ic g oups, Mona sh Ma h 195(3), 401-419 (2021).
[31] de G aa , W. A.: Cons uc ing ai h ul ma ix ep esen a ion o Lie algeb as,
P oceedings o he 1997 In e na ional Symposium on Symbolic and Algeb aic
Compu a ion, ISSAC’97, pp. 54-59 ACM P ess, New Yo k (1997).
[32] G eco, S. e a Salmon, P.: Topics in m-adic opologies, E gebnisse de Ma he-
ma ik und ih e G enzgebie e 58, Sp inge -Ve lag, Be lin-Heidelbe g (1971).
[33] Hall, P.: Ve bal and ma ginal subg oups, J. Reine Angew. Ma h. 182, 156-
157 (1949).
[34] Ha ish-Chand a: On ep esen a ion o Lie algeb a, Ann. o Ma h. 50, 900-915
(1949).
[35] Hazewinkel, M.: Fo mal G oups e a Applica ions, Academic P ess, New
Yo k-San F an zisko-Lond es (1978).
[36] de las He as, I. e a Zozaya, A.: Hausdo en dimen sioa alde p o ini ue an,
Ekaia 40, 291-314 (2021).
[37] Humph eys, J. E.: Linea Algeb aic G oups, Sp inge -Ve lag, New Yo k-
Be lin-Heidelbe g (1975).
275
[38] Huppe , B.: Endliche G uppen I, Sp inge -Ve lag, Be lin-Heidelbe g-New
Yo k (1967).
[39] I ano , S. V.: P. Hall’s conjec u e on he ini eness o e bal subg oups, So .
Ma h. 33, 71-76 (1989); ansla ion om Iz . Vyssh. Uchebn. Za ed. Ma . 6,
60–70 (1989).
[40] Iwasawa, K.: On he ep esen a ion o Lie algeb as, Jap. J. Ma h. 19, 405-426
(1948).
[41] Jacobson, N.: Lie Algeb as, Do e Publica ions, Inc., New Yo k (1979).
[42] Jaikin-Zapi ain, A.: On linea jus in ini e p o-pg oups, J. Algeb a 255,
392-404 (2002).
[43] Jaikin-Zapi ain, A.: On linea i y o ini ely gene a ed R-analy ic g oups,
Ma h. Z. 253, 333-345 (2006).
[44] Jaikin-Zapi ain, A.: On he e bal wid h o ini ely gene a ed p o-pg oups,
Re . Ma . Ibe oam. 24, 617-630 (2008).
[45] Jaikin-Zapi ain, A. e a Klopsch, B.: Analy ic g oups o e gene al p o-pdo-
mains, J. London Ma h. Soc. 76, 365-383 (2007).
[46] Jo dan, C.: T ai é des sub i u ions e de équa ions algéb iques, Les G ands
Classiques Gau hie -Villa s, Pa is (1870).
[47] Klopsch, B. e a Thillaisunda am, A.: A p o-pg oup wi h in ini e no mal
Hausdo spec a. Paci ic J. Ma h. 303, 569-603 (2019).
[48] Klopsch, B., Thillaisunda am, A. e a Zugadi-Reizabal, A.: Hausdo dimen-
sion in p-adic analy ic g oups. Is ael J. Ma h. 231, 1-23 (2019).
[49] Lang, S.: Algeb a, G adua e Tex s in Ma hema ics 211, Sp inge -Ve lag,
New Yo k (2005).
[50] Laza d, M.: G oupes analy iques p-adiques, Publ. Ma h. I.H.E.S. 26, 5-219
(1965)
[51] Lee, J. M.: In oduc ion o Smoo h Mani olds, 2nd ed., Sp inge , New Yo k
(2013).
276
[52] Lubo zky, A.: G oup p esen a ion, p-adic analy ic g oups and la ices in
SL2(C),Ann. o Ma h. 118, 115-130 (1983).
[53] Lubo zky, A. e a Mann, A.: On g oups o polynomial subg oup g ow h, In-
en . Ma h. 104, 521-533 (1991).
[54] Lubo zky, A. e a Shale , A.: On some Λ-analy ic p o-pg oups, Is ael J.
Ma h. 85, 307-337 (1994).
[55] Mal’ce , A.: On isomo phic ma ix ep esen a ions o in ini e g oups, Rec.
Ma h. N.S. 8(50), 405-422 (1940).
[56] Me zljako , Yu. I.: Algeb aic linea g oups as ull g oups o au omo phisms
and he closeness o hei e bal subg oups, Algeb a i Logika 6, 83-94 (1967)
(e usie az).
[57] Me zljako , Yu. I.: Ve bal e a ma ginal subg oups o linea g oups, Dokl.
Akad. Nauk SSSR 177, 1008-1011 (1967).
[58] Moens, W. A.: Fai h ul ep esen a ions o minimal deg ee o Lie algeb as
wi h an abelian adical, Dok o ego Tesia, Uni e si ä Wein, Viena (2009).
[59] Munk es, J. R.: Topology, 2nd ed., Pea son Educa ion, Ha low (2014).
[60] Ne e in Yu.: A cons uc ion o ini e-dimensional ai h ul ep esen a ion o
Lie algeb as. P oceedings o he 22nd Win e School on “Geome y and
Physics” (S ni, 2002). Rend. Ci c. Ma . Pale mo (2) Suppl. No. 71, 159–
161 (2003).
[61] Neumann, H.: Va ie ies o G oups, Sp inge -Ve lag, New Yo k (1967).
[62] Ol’shanskiĭ, A., Yu.: Geome y o de ining ela ions in g oups, ol. 70, Ma he-
ma ics and Applica ions (So ie Se ies), Kluwe Academic Publishe s G oup,
Do d ech (1991).
[63] Reed, B. E.: Rep esen a ions o sol able Lie algeb as, Michigan Ma h. J. 16,
227-233 (1969).
[64] Ri es, L. e a Zalesskii, P.: P o ini e G oups, 2nd ed., A Se ies o Mode n
Su eys in Ma hema ics 40, Sp inge -Ve lag, Be lin-Heidelbe g (2000).
[65] Robinson, D. J. S.: A cou se in he heo y o g oups, 2nd ed., Sp inge -Ve lag,
New Yo k (1982).
277
[66] Roman’ko , V. A.: Wid h o e bal subg oups in sol able g oups, Algeb a
and Logic 21, 41-49 (1982); hemendik i zulia: Algeb a i Logika 21, 60–72
(1982).
[67] Segal, D.: Wo ds. No es on Ve bal Wid h in G oups, Camb idge Uni e si y
P ess, Camb idge (2009).
[68] Se e, J. P.: Lie Algeb as and Lie G oups, Lec u e No es in Ma hema ics ol.
1500, Sp inge -Ve lag, Be lin-Heidelbe g (2006).
[69] Simons, N. J.: The wid h o e bal subg oups in p o ini e g oups, Dok o ego
Tesia, Uni e si y o Ox o d, Ox o d (2009).
[70] S ade, H.: Simple Lie Algeb as O e Fields o Posi i e Cha ac e is ic. S uc-
u e Theo y, ol. I, De G uy e , Be lin-Bos on (2004).
[71] S oud, P. W.: Topics in he heo y o e bal subg oups, Dok o ego Tesia,
Uni e si y o Camb idge, Camb idge (1966).
[72] Tu ne -Smi h, R. F.: Fini eness condi ions o e bal subg oups, J. London
Ma h. Soc. 41, 166-176 (1966).
[73] Weh i z, B. A. F.: In ini e Linea G oups, Sp inge -Ve lag, Be lin-
Heidelbe g-New Yo k (1973).
[74] Weigel, T.: A Rema k on he Ado-Iwasawa Theo em, J. Algeb a 212, 613-625
(1999).
[75] Za iski, O. e a Samuel, P.: Commu a i e Algeb a, Vol. I e a II, Sp inge -
Ve lag, Be lin-Heidelbe g-New Yo k (1958).
[76] Zozaya, A.: Conciseness in compac R-analy ic g oups, bidali a.
[77] Zozaya, A.: A ema k on Ado’s Theo em o p incipal ideal domains,
p es a zen.
278

Indizea
δ-es alki, 210
p-adiko anali iko, 157
adie azpen
∼adjun u, 258
∼e a o i, 188
∼e egula , 260
∼ ini u, 256
∼leial, 256
∼maila, 256
∼ma izial, 256
lineal, 187
Ado
Teo ema, 256
alde an zizko
∼ o mal, 158
∼ un zio, 157
aljeb a
∼bakun zen al, 236
∼bananga i, 264
∼semisinple, 259
en so e ∼, 259
anali iko
azpiba ie a e ∼, 178
azpimul zo ∼, 180
azpi alde ∼, 181
ba ie a e ∼, 155
dimen sio ∼, 162
un zio ∼, 154, 155
he siki ∼, 155
alde ∼, 157
a las, 154
∼ba e aga i, 154
∼maximal, 154
azpimu gilke a, 172
azpiba ie a e, 178
∼anali iko, 178
∼ahul, 178
azpimul zo mu gildu, 177
azpi ake a, 170
∼ahul, 168
Bai e
Ka ego ia Teo ema, 243
Bake -Hausdo -Campell
∼ o mula, 190
ba ie a e
∼anali iko, 155
∼pu u, 155
be e u a se ie
∼p opie a e unibe sal, 152
∼e az un, 151
∼konbe gen e, 152
bianali iko, 169
bide ke a
∼ un zio, 157
ezke ∼ un zio, 168
bilipschi zia , 210
279
Che alley
alde klasiko, 232
Cohen
Egi u a Teo ema, 150, 174
de ibazio, 264
∼luze a, 266
di e en zial, 165
dimen sio
∼anali iko, 155, 162
∼anali iko lokal, 155
azpiba ie a e ∼, 178
Hausdo en ∼, 210
Hausdo en ∼es anda , 218
ku xa ∼, 210
Minkowski-Bouliganden ∼, 210
disk iminazioa, 198
ebakidu a ini uen p opie a e, 202
ebaluake a
∼epimo ismo, 196
∼ un zio, 152
ebazga i
e adikal ∼, 266
egonko asun
∼ ini u, 211
∼kon aga i, 210
e adikal
∼ebazga i, 248, 266
∼nilpo en e, 264
∼unipo en e, 248
e eduk ibo
alde ∼, 248
eskala e mu izke a, 176
espazio ul ame ikoa, 150
es anda
il azio se ie ∼, 214
alde ∼, 157
ezbe din za iangula gogo , 150
ak o izazio baka eko domeinu
(FBD), 185
il azio se ie, 207
∼es anda , 159, 214
∼no mal, 211
p-be e u a ∼, 208
il o, 201
Hall-Pe escu
o mula, 246
Hall-Wi
iden i a e, 164
Hausdo
∼den si a e, 234
∼dimen sio, 210
∼dimen sio es anda , 209, 218
∼espek o, 208
∼espek o es anda , 209, 218
∼neu i, 210
Hilbe
∼ un zio, 215
∼polinomio, 215
hi z, 239
∼azpi aldea, 240
∼balio, 239
∼baliokide, 239
∼behe zen al, 241
∼de iba u, 241
∼elip ikoa, 243
∼hu s, 239
∼zabale a, 243
∼ un zio, 239
Bu nside en ∼, 241
kommu ado e ∼, 241
hi zezko
∼elip iko asun, 242
labu asun ∼, 242
sendo asun ∼, 242
honda go pu z, 149
280
honda -heina, 170
ideal nagusie ako domeinu (IND),
150
iden i a e osagai, 247
Igo ze Teo ema, 196
ingu a ze aljeb a unibe sal, 259
p opie a e unibe sal, 260
inon kons an e un zio, 252
isola u modulu, 258
isola zaile, 258
Iwasawa
Teo ema, 257
Jacobi
iden i a e, 164, 256
jacobia , 165
ka a, 154
∼ba e aga i, 154
∼e egula , 155
∼molda u, 182
ka ea en e egela, 165
koheina, 181, 258
koo dena u
∼aldake a, 169
∼aldake a un zio, 154
∼sis ema, 156
∼sis ema kanoniko, 156
K ull
Ebakidu a Teo ema, 149
ku xa-dimen sio, 210
∼es anda , 222
behe ∼es anda , 216
behe ∼, 210
goi ∼, 210
labu
hi zez ∼, 242
labu asun, 242
∼gogo a, 252
Leibniz
iden i a e, 264
Le i
∼osagai, 266
Teo ema, 266
Lie algeb a
∼nilpo en ea, 261
Lie aljeb a, 165, 255
∼adie azpen, 256
∼ebazga i, 266
∼g adua ua, 234
∼homomo ismo, 256
∼konk e ua, 256
∼pe ek u, 235
∼zen oa, 258
∼elka ua, 165
Lie e e ikulu, 257
∼be e u a-be e, 190
Lie en ko xe e, 164, 255
lineal
adie azpen ∼, 187
alde ∼, 187
alde ∼be ezi, 231
alde ∼o oko , 156, 228
alde aljeb aiko ∼, 243
alde aljeb aiko ∼, 247
lokal
e az un ∼, 149
e az un homomo ismo ∼, 191
ma jinal
azpi alde ∼, 240
Minkowski-Bouligand
dimen sio, 210
mono ono, 210
mo ismo o mal, 166
mul zo a in, 228
mu gilke a, 170
281
∼ahul, 168
nilpo en e
e adikal ∼, 264
klase, 261
o ogonal be ezi
alde ∼, 232
Poinca é-Bi kho -Wi
Teo ema, 259
p o-pdomeinu, 150
p o ini u espazio, 252
pun u e egula , 181
Schu
Teo ema, 246
se ie
∼de iba u, 266
∼zen al behe ako , 261
sinplek iko
alde ∼, 232
alde e agike a o mal, 158
∼ba uko , 158
∼bide kako , 158
Ti s
al e na iba opologiko, 231
opologia m-adiko, 149
opologikoki nilpo en e, 152
anslazio, 156
ul abe e u a, 203
ul abide ke a, 202
ul a il o, 201
∼ez-nagusi, 201
∼ eo ema, 202
ul ame iko espazio, 150
uni o meki be e u a-be e, 189
uni o miza zaile, 151
unipo en e
azpi alde ∼, 248
e adikal ∼, 248
ma ize ∼, 248
Zassenhauss
hedapen, 265
zen oide, 236
Zo n
Lema, 236
282