Pub
.
Ma
.
UAB
Nó 17,
Feb e ,
1980
associa eddimensio
n
subg
ó~
POWERS OF
THE
AUGMENTATION
IDEAL
Summa y
o
alk
gi en
a
Uni e si a
Au ónoma
de Ba celona,
19
Ap il,
1978,
by B
.
Ha ley
.
Le
Gbe a
g oup,
R
a
commu a i e
ing
wi h
1
(we
will
in
ac
only
be
conce ned
wi h
he
cases
when
R
is
a
ield
o
ZL
)
.
The
g oup
ing
RG is
he se
o
no mal
ini e
linea
combina ions
E

a g
g

(ag
ER), wi h
he
de ini ions

(Ex
g
g)
+

(
EU
g
g)

=
E(a
g+
u g
)g
and
gE
G
(Ea
9)(EU
h)
=
E
(
E
au)k
.
The
augmen a ion
E
:
RG->
R
gi en
by
s(Ea
g
g) =
Ea
gis a
ing
homomo phism
and
i s
ke nel
o(R,
G)
is
he
au~-
men a ion
ideal
o RG
.
We
a e
in e es ed
in he
powe s
A
(R,
G)
and he
0n(R, G)
=
Gn(1
+
o
n
(R, G))
.
No e ha
Dn(R, G) is
he
ke nel
o
he
na u al
map
o G
in o
he
g oup
o
uni s
U(RG/p
n
(R, G))
o RG/p
n
(R, G)
.
This
is
no
in ended
o
be
a
de ailedsu eyo
his
a ea,
bu
sim-
ply
a
discussion
o
some
sample
esul s
.
I
.
BACKGROUND
Fi s
we
men ion
some
connec ions
be ween
his
subjec
and
o he
p oblems
.
1
.
Conn
e
c ion
wi h
ma ix
ep esen a ions
I is
o en
use ul
o
know ha
ce ain
g oups
can
be
ep esen ed
ai h ully
by ma iceso e
some well
beha ed
ing
.
Theo em
.
I
G
is
a
ini ely
gene a ednilpo en
g oup,
hen
G
can
be
embedded
in
some
GL(n,
71)
(much
mo e
gene al
esul s
a e
ue)
.
Indica ion
o
p oo
.
We
easily educe
o
he
case
when
G
is
o sion-
ee
.
Le
G
ha e
nilpo ency
class
c
.
Then
Dc+1(
2Z
,
G)
=1
(qui e
a
di i-
cul
esul )
.
Hence
G
can
be
embedded
in
he
g oup
o
uni s
o
a
G/oc+1
(TL
,
G)
= A
.
The
addi i e
g oup
o
A
is
ini ely
gene a ed,
and
G
ope a es ai h ully
on his addi i e
g oup
by
igh
mul iplica ion
.
The
addi i e
o sion
subg oup
T
o
Ais
ini e,
and
i
is
easy
o
see
ha
G
ope a es
ai h ully
on
A/T
.
See
:P
.
Hall "The
Edmon on
No es
on
Nilpo en
G oups"
(Queen
Ma y
College,
London)
o he
de ails
.
2
.
Connec ion
wi h
esidual
n
i
lpo encyo g oups
Theo em
(Qaumslag,
Passi, see
J
.
Pu e
App
.
Algeb a
6
(1975))
.
Le
F be a
non-abelian
ee
g oup,
R<1
F
.
Then
F/R' is
esidually
nilpo en
i
and
only i

n

A
(1
,
G) = 0,
whe e
G =
F/R
.
n=1
II
.
DIMENSION
SUBGROUPS
Fo
any
G,

le
G
1
=

G
and
Gn+1

=

[
G
n
,
G],

so

ha
G
1
>
.G2

>

.
.
.

is
he
lowe
cen al
se ies o
G
.
Then
Lemma
2
.1
.
Gn
<
D
n
(R, G)
This
ollows
.b
y
induc ion
om
he
iden i y
1
-
X
-1Y-1
xY
=
X
-1
Y
-1
(
(Y-1)(X-1)-(X-1)(Y-1))
.
Le

/Go be
he
o sion
subg oup
o
G/G
n
.
Theo em
2
.2
.
I K
is
a
ield
o
cha ac e is ic
ze o,
hen
D
n (K,
G)
_
_
N/--G
n'
This
is
due
o
Jennings
;
a
p oo
can
be
ound
in P
.
Hall's Edmon on
No es,
o
D
.S
.
Passman
"The
Algeb aicS uc u e
o
G oup
Rings"
(In e s-
cience)
.
Theo em
2
.3
.
(Laza d)
I
K
is a
ield
o
cha ac e is ic
p >
0,
hen
D
n
(K,
G)
=

n

Gp
J
ip
3 >
n
whe e
i H is a
g oup,
H
m
=
<h
m
:
h
c
H>
.
Ano he
desc ip ion
o
D
n
(K, G)
was
gi en
ea lie
by
Jennings
.
Theo em
2
.3
can
also
be
ound
in
Passman's
book
.
No e ha
in
Theo em
2 .3, i G has
exponen
p,
hen
Dn
(K,
G)
=
Gn
o
all n
.
Theo em
2
.
.2
and
2
.3
show ha o e
ields,
he
dimensionsubg oups
can
be
comple ely
desc ibed
in
g oup
heo e ic
e ms
.
Fo he
case
R =
71
he
si ua ion
is
mo e
complica ed
.
-I
was
a
one
ime
conjec u ed
ha
D
n
(71
,
G)
=
Go
o
all
G
(Dimen-
sion
subg oup
conjec u e)
and
se e al
alse
p oo s
ha e
been
gi en
.
Theo em
2
.4
.
(i)
I
F is
ee,
hen
D
n
(F)
=
F
n
o
all
n
.
(Magnus,
1937)
(ii)
I G
is
any g oup,
hen
Dn(G) = G
n
o
n =
1,2,3
(I
don'
know
who
i s
p o ed
hese
esul s
;
qui e
a
.
numbe
o
p oo s
a e now
a aila-
ble,
o
example
A
.H
.M
.
Hoa e,
J
.
LondonMa h
.
Soc
.)
iii) I G is
a
ini e p-g oup,
hen
Dn(G)
=
G
n
o
n
< p
(Mo an
;
P oc
.
Camb idge
Philos
.
Soc
.)
I
ha e
w i en
D n
(G)
o
D
n
(7L
,
G)
.
E en ually
he
Dimension
Subg oupConjec u e
was
e u ed
by
Theo em
2
.5
(Rips,
1972)
The e
exis s
a
ini e
2-g oup
G
o
class
3
such
ha
ID4(G)1
=
2
.
(Is ael
J
.
Ma h
.)
The
bes
esul
o
da e
is
con ained
in a
ecen
and
di icul
pape
o
Sjog en
(J
.
Pu e Appd
.
Algeb a
(1979))
.
Theo em
2
.6
(Sjog en)
The e
exis s
an
(explici ly
gi en)
unc ion
cn
such
ha
i
G
is
any
g oup, hen
D
n
(G)/G
n
has
exponen
di iding
cn
.
The
p oo
uses some
elemen a y
spec al sequence
ideas
a id
some com-
plica ed
Lie- heo e ic
me hods
.
I
ha e
a
simpli ied
e sion
o
he
p oo
.
We
ha e
c l
= c2
=
c
3
=
1,
c 4
=
2,
so 2
.4(ii)
ollows
om
Sjog en's
wo k,
and
Rips'
example
shows
ha
c
4
is
bes possible
.
Also by
examining
he

unc ion

c n
one can
eplace
he
es ic ion
n
<

p in 2
.4(iii)

by
n
< p+1
.
Some
open p oblems
P oblem
1
.
Dóes
he eexis
an
in ege
alued unc ion
(c)
such ha
i G
.is
a
nilpo en
g oup
o
class
.c,
hen
D (c)(G)=l?
Rips'
example
shows
ha
(c)=
c+1
will
no
do, bu
he e
is no
coun e example
known
o
ule
ou
he
possibili y
(c)
=
c+2
.
P oblem
2
.
(a
weake
e sion
o
P oblem
1)
.'I
G is
nilpo en ,
is
i
ue
ha

n
1
D
n(G) = 1?
P oblem3
.
Does
he eexis
a
ini e
p-g oup
(p
~
2)
such
ha
D n
(G)
:~
G
n
o
some
n?
The
smalles
possibili y
is
p = 3, n
=
5
.
III
.
THE
.LIERING
AND
THE
GRADEDRING
Le
G
be
a
g oup,
wi h
lowe
cen al
se ies
We
w i e
Gn
/G
n+1
addi i ely
and
o o
he
abelian
g oup
and
de ine
L(G)

6)
G
n
/G
n+1
n=1
( es ic ed
di ec
sum),
and
de ine
an
ope a ion
( ,
on
L(G)by
(xGn+1>
.yGM+l)
=
[x,
y]Gn+m+1
(x e
Gn
,
y
c
G
m
)
and
ex ending
by
addi i i y
.
This
u ns
L(G) in o
a
Lie
ing
;
he
Jacobi
iden i y
ollows
om
he
Wi
iden i y
.
Now
w i ing
o
=
A(a,
G),
we
o o
he
abelian
g oup
G(G)
=
®
on/on+1
,
n=0
(a+on+1)(,+o o+1)
=as+,n+m+l
.
We ex end
his
ope a ion
by
addi i i y
again,
and
ob ain
his
ime
an
associa i e
ing,
called
he
associa ed
g aded
ing
o
71
G
.
We
can
hink
o
G(G)
as
a
Lie
ing
unde
usual
u - u
ope a ion,
and
hen
he
map
xGn+1
;
(x-1)

+
o
n+l

(x c

Gn
)

10
ex e
.nds
o a
Lie
homomo phism
o
L(G)
in o G(G)
.
The ke nel
o
he
es-
ic ion
o 0 o Gn/G
n+l
is
(Dn+1
.
(G)
n
Gn )/Gn+l
,
and
so
in
his
way
we
ob ain
a
connec ion
be ween
dimensionsubg oüps
and
Lie
alneb as
.
The map
0
also
de e mines
a
homomo phism
om
L(G)
o
iz
K

->G(G)
®
2z
K,
i Kis
any
ield
o
cha ac e is ic
ze o,
and
we ha e
he
ollowing
beau-
i ul
esul
o
Quillen
(J
.
Algeb a
10
(1968))
.
Theo em
3
.1
.
Wi h
he
abo e
no a ion,
0
is
injec i e
and
G(G)
o
K
he
uni e sal
en eloping
algeb a
o
L(G)
a K
.
I is in
ac mo e
na u al
o
hese
pu poses
ing om
he
cha ac e is ic
ze o
dimension
se ies
is
Ní~G1
>
_'/-G2
hough
he wo Lie
ings
become
isomo phic-on enso ing
wi h
a
ield
o
cha ac e is ic
ze o
.
Howe e
using
analogy
wi h
cha ac e is ic
p > 0
;
in ol ing
he
es ic ed
uni e sal
su eyo
hese
ma e s,
see
I
.B .S
.
Passi
(J
.
LondonMa h
.
Soc
.
1979)
.
IV
.POWERS
OF
THE
AUGMENTATION
IDEAL
He e we
con ine
ou sel es
o
he
ques ion
:
n
o
n =
0,
whe e
o =
o(7L,
G)?
In
dealing
wi h
his
ques ion,
i
seems
n=1
una oidable
o
conside
also
he
ing ZL/p
m
7L
.
Theo em
4
.1
.

n

o
n
(G,
ZL/p
m
ZZ)
=
0
i
and
n=1
a
nilpo en
p-g oup
o ini e
exponen
.
o
cons uc he
Lie
he
dimension
se ies
gi es
he
igh
he e we
ge
a
esul
like
Theo em
3
.1
en eloping
algeb a
.
Fo
a
mo e
de ailed
When
is i
ue
ha
only
i G is
esidual
ly
1
5
1

i,
:~-~
.ii~

aiki~<
ü
.n1
.11
.'+
.
.i1~ws
:illh
:¢imi i ~d
~J,
IIUC
. ~
~s~
.
. .
il
This
was
p o ed
by B
.
Ha ley
(P oc
.
LondonMa h
.
Soc
.
1969)
and al-
so by K
.41
.
G uenbe g(unpublished)
.
The
case
m=
1
was
done
much
ea lie
by
Mal'ce
.
Theo em
4
.2
.
I G is
esidually
o sion- ee
nilpo en ,
hen
o
n
(7Z
,
G)
=
0
.
n=1
This
is
also
due
o
Ha ley
in
he
abo e
pape
.
The
main pa o
he
p oo
consis so
adap ing
a
cons uc ion
o
P
.
Hall o
ob ain,
o
a
ini ely
gene a ed
o sion- ee
nilpo en
g oup
G,
a 7Z
-basiso
ZZ
G
which
is
closely
ela ed
o he
powe s o
o
.
This
can
now
be
done a he
be e
.
Le
~~
n
(7Z
,
G)/o
n
(7Z
,
G)
deno e
he
addi i e
o sión
subg oup
o
7Z
G/o
n
(TZ
,
G)
.
Theo em
4.3
.
Le
G be a
ini ely
gene a ed
o sion- ee
nilpo en
g oup
.
Then
7Z
G has a
7Z
-basis
B
such
ha ,
o
each
n >
1,
-,/
o
n
(7Z
,
G)
is
spanned
by
a
subse o
B
.
(See
SymposiaMa h
.,
1976)
O he
p oo s
o 4 .2
ha e
since
been
gi en
by A
.L
.
Smel'kin
using
echniques om
Lie
Algeb as
(T ans
.
Moscow
Ma h
.
Soc
.
1973,
and
also
in
a
ecén
issue
o
Uspekhi
Ma
.
Nauk
.)
Finally,
A
.I
.
Lich man
showed ha su icien condi ions
gi en
by
4
.1
and
4
.2
a e
e y nea ly
necessa y
.
I G
is
a
g oup,
hen we
say
ha
G is
disc imina ed
by
nilpo en
p-g oups
o ini eexponen i
and
only
i
o
each ini e
se
o elemen s
x
1
,
. .
.,
xm
e
G,
he e
exis s
a
p ime
p
and
a
homomo phism
e
o
G
in o
a
nilpo en
p-g oup
o
ini e
exponen ,
such ha
6(x
i
)
#
1

(1
<
i<
m)
.
o
n
(IL
,
G)
= 0
i
and only
i
G
is
ei he
esidually
o sion-
n=1
ee
nilpo en
o
disc imina ed
by
nilpo en
p-g oups
o ini e
exponen
.
Theo em
(Is ael
J
.
Ma h
.
1977)