The kernel of the Magnus representation of the automorphism group of a free group is not finitely generated

0.18 MB·English

by  Takao Satoh

THE KERNEL OF THE MAGNUS REPRESENTATION OF THE
AUTOMORPHISM GROUP OF A FREE GROUP IS NOT FINITELY
GENERATED
1
Takao Satoh
1
0
Department of Mathematics, Graduate School of Science, Kyoto University,
2
Kitashirakawaoiwake-cho,Sakyo-ku, Kyoto city 606-8502,Japan
n
a
J
Abstract. Inthispaper,weshowthattheabelianizationofthekerneloftheMagnus
4
representation of the automorphism group of a free group is not finitely generated.
2

]
T 1. Introduction
A
Let F be a free group of rank n ≥ 2, and AutF the automorphism group of F . Let
n n n
.
h denoteρ : AutF → AutH thenaturalhomomorphism induced fromtheabelianization
n
t
a F → H. The kernel of ρ is called the IA-automorphism group of F , denoted by IA .
n n n
m
The subgroup IA reflects much richness and complexity of the structure of AutF ,
n n
[
and plays important roles on various studies of AutF .
n

2
Although the study of the IA-automorphism group has a long history since its finitely
v
6 many generators were obtained by Magnus [14] in 1935, a presentation for IA is still
n
8
unknownforn ≥ 3. Nielsen[19]showedthatIA coincideswiththeinnerautomorphism
3 2
0 group, hence, is a free group of rank 2. For n ≥ 3, however, IAn is much larger than
0. the inner automorphism group InnFn. Krsti´c and McCool [11] showed that IA3 is not
1 finitely presentable. For n ≥ 4, it is not known whether IA is finitely presentable or
n
9
not.
0
:
v In general, one of the most standard ways to study a group is to consider its represen-
i
X tations. If a representation of the group is given, it is important to determine whether
r it is faithful or not. Furthermore, if not, it is also worth studying how far it is from
a
faithful, namely, to determine its kernel. In this paper, we consider these matters for
the Magnus representation
r : IA → GL(n,Z[H])
M n
of IA . (See Subsection 2.4.)
n
Classically, the Magnus representation was used to study certain subgroups of IA .
n
One of the most famous subgroup is the pure braid group P . The restriction of r to
n M
P is called the Gassner representation. (For a basic materials for the braid group and
n
the pure braid group, see [4] for example.) It is known due to Magnus and Peluso [15]
that r | is faithful for n = 3. Although the faithfulness of r | has been studied
M Pn M Pn
for a long time by many authors, it seems, however, still open problem to determine it
for n ≥ 4.
Another important subgroup is the Torelli I subgroup of the mapping class group
g,1
of a surface. By a classical work of Dehn and Nielsen, the Torelli group I can be
g,1
1
considered as a subgroup of IA . Suzuki [23] showed that the restriction r | to I
2g M Ig,1 g,1
is not faithful for g ≥ 2. Furthermore, by a recent remarkable work of Church and Farb
[5], it is known that the abelianization of the kernel of r | is not finitely generated
M Ig,1
for g ≥ 2.
Fromthefactsasmentionedabove, itisimmediatelyseenthatr itselfisnotfaithful.
M
There are, however, few results for the abelianization of the kernel K of r . Let FM
n M n
be the quotient group of F by the second derived subgroup [[F ,F ],[F ,F ]] of F .
n n n n n n
The group FM is called the free metabelian group of rank n. The metabelianization
n
F → FM naturally induces a homomorphism ν : AutF → AutFM. Its image
n n n n n
of IA is contained in IAM, the IA-automorphism group of FM. Then it is known
n n n
due to Bachmuth [1] that the Magnus representation r factors through IAM, and
M n
that the induced homomorphism r′ : IAM → GL(n,Z[H]) is faithful. Hence the
M n
metabelianization of F induces the injectivization of the Magnus representation r .
n M
Therefore we see that K coincides with the kernel of ν . From this point of view, in
n n
our previous paper [22], we showed that the abelianization Kab of K contains a certain
n n
free abelian group of finite rank by using the Johnson homomorphism of AutFM.
n
In this paper, we show that r is far from faithful. That is,
M
Theorem 1. (= Theorem 4.1 and Proposition 6.2.) For n ≥ 2, Kab is not finitely
n
generated.
Recently, we have heard from Thomas Church, one of the authors of [5], that their
method to show the infinite generation of the abelianization of the kernel of r | can
M Ig,1
be applied to show that of r . Our method, however, is different from theirs. In this
M
paper, in order to prove the theorem for n ≥ 3, we consider some finitely generated
normal subgroups W of F , and embeddings of K into the IA-automorphism group
n,d n n
of W . Then, taking advantage of the first Johnson homomorphisms of AutW , we
n,d n,d
detect infinitely many linearly independent elements in Kab. On the other hand, for
n
n = 2, we show
Proposition 1. K = [[F ,F ],[F ,F ]]
2 2 2 2 2
by using a usual and classical argument in combinatorial group theory.
This paper consists of seven sections. In Section 2, we recall the IA-automorphism
group and the Magnus representation of the automorphism group of a free group. In
Section 3, we consider some embeddings of the kernel of the Magnus representation into
the IA-automorphism group of W . Then, in Section 4, we prove the main theorem
n,d
for n ≥ 3. In Section 5, we discuss how much of Kab can be detected in IA(W )ab.
n n,d
In particular, we show that there exsists a non-trivial element in Kab which can not
n
be detected by each of IA(W )ab for n ≥ 4. In Section 6, we consider the case where
n,d
n = 2.
Contents
1. Introduction 1
2. Preliminaries 3
2.1. Notation and conventions 3
2.2. IA-automorphism group 3
2
2.3. Johnson homomorphisms 5
2.4. Magnus representation 5
3. Embeddings of the IA-automorphism group 6
3.1. Subgroups W of F 6
n,d n
3.2. Action of IA on W 7
n n,d
4. Lower bounds on Kab 8
n
5. On the detectivity of elements of Kab in IA(W )ab 11
n n,d
6. The case where n = 2 12
7. Acknowledgments 14
References 14
2. Preliminaries
In this section, after fixing notation and conventions, we recall the IA-automorphism
group and the Magnus representation of AutF .
n
2.1. Notation and conventions.
Throughout the paper, we use the following notation and conventions. Let G be a
group and N a normal subgroup of G.
• The abelianization of G is denoted by Gab.
• For any group G, we denote by Z[G] the integral group ring of G.
• The group AutG of G acts on G from the right. For any σ ∈ AutG and x ∈ G,
the action of σ on x is denoted by xσ.
• For an element g ∈ G, we also denote the coset class of g by g ∈ G/N if there
is no confusion.
• For elements x and y of G, the commutator bracket [x,y] of x and y is defined
to be [x,y] := xyx−1y−1.
2.2. IA-automorphism group.
Inthispaper, wefixabasisx ,...,x ofF . Let H := Fab betheabelianizationofF
1 n n n n
and ρ : AutF → AutH the natural homomorphism induced from the abelianization
n
of F . In the following, we identify AutH with the general linear group GL(n,Z) by
n
fixing the basis of H induced from the basis x ,...,x of F . The kernel IA of ρ is
1 n n n
called the IA-automorphism group of F . It is clear that the inner automorphism group
n
InnF of F is contained in IA . In general, however, IA for n ≥ 3 is much larger
n n n n
than InnF . In fact, Magnus [14] showed that for any n ≥ 3, IA is finitely generated
n n
by automorphisms
x −1x x , t = i,
j i j
K : x 7→
ij t
x , t 6= i
( t
for distinct i, j ∈ {1,2,...,n} and
x [x ,x ], t = i,
i j l
K : x 7→
ijl t
x , t 6= i
( t
3
for distinct i, j, l ∈ {1,2,...,n} such that j < l. In this paper, for the convenience,
we also consider an automorphism K for j ≥ l defined as above. We remark that
ijl
K = 1 and K = K−1.
ijj ilj ijl
Recently, Cohen-Pakianathan[6,7]CFarb[8]andKawazumi[10]independentlyshowed
(1) IAanb ∼= H∗ ⊗Z Λ2H
as a GL(n,Z)-module where H∗ := HomZ(H,Z) is the Z-linear dual group of H.
In Section 4, we use the following lemmas.
Lemma 2.1. For n ≥ 4, and distinct integers 1 ≤ i,j,l,m ≤ n, let ω ∈ IA be an
n
automorphism defined by
x x x−1x x−1x x−1, t = i,
x 7→ i j l m j l m
t
x , t 6= i.
( t
Then, we have
ω = K K K K K−1 ∈ IA ,
il iml ijm ilj il n
and in particular,
ω = K +K +K ∈ IAab.
iml ijm ilj n
Proof. Since K K K K K−1 fix x for t 6= i, it suffices to consider the image of
il iml ijm ilj il t
x . Then we see
i
x −K−→il x−1x x −K−i−m→l x−1x [x ,x ]x = x−1x x x x−1 −K−i−j→m x−1x [x ,x ]x x x−1
i l i l l i m l l l i m l m l i j m m l m
= x−1x x x x−1x x−1 −K−i→lj x−1x [x ,x ]x x x−1x x−1
l i j m j l m l i l j j m j l m
K−1
= x−1x x x x−1x x−1x x−1 −−i→l x x x−1x x−1x x−1.
l i l j l m j l m i j l m j l m
This completes the proof of Lemma 2.1.
Similarly, we obtain the following lemmas. Since the proofs are done by a straight-
forward calculation as above, we leave them to the reader as exercises.
Lemma 2.2. For n ≥ 5, and distinct integers 1 ≤ i,j,l,m,p ≤ n, let ω ∈ IA be an
n
automorphism defined by
x x x−1x x x−1x−1x x x−1x−1, t = i,
x 7→ i j l p m p j l p m p
t
x , t 6= i.
( t
Then, we have
ω = K K K K K K−1K−1 ∈ IA ,
mp il iml ijm ilj il mp n
and in particular,
ω = K +K +K ∈ IAab.
iml ijm ilj n
Lemma 2.3. For n ≥ 5, and distinct integers 1 ≤ i,j,l,m,p ≤ n, let ω ∈ IA be an
n
automorphism defined by
x x x x−1x−1x x x−1x x−1x−1x x x−1x−1, t = i,
x 7→ i p j p l p m p p j p l p m p
t
x , t 6= i.
( t
4
Then, we have
ω = K K K K K K K−1K−1K−1 ∈ IA ,
jp mp il iml ijm ilj il mp jp n
and in particular,
ω = K +K +K ∈ IAab.
iml ijm ilj n
2.3. Johnson homomorphisms.
Here we recall the definition of the Johnson homomorphisms of AutF . (For a
n
basic materials for the Johnson homomorphisms, see [16], Kawazumi [10] and [21] for
example.)
Let Γ (1) ⊃ Γ (2) ⊃ ··· be the lower central series of a free group F defined by the
n n n
rule
Γ (1) := F , Γ (k) := [Γ (k −1),F ], k ≥ 2.
n n n n n
We denote by L (k) := Γ (k)/Γ (k+1) the graded quotient of the lower central series
n n n
ofF . Foreach k ≥ 1, theactionofAutF onthenilpotent quotient groupF /Γ (k+1)
n n n n
of F induces a homomorphism AutF → Aut(F /Γ (k+1)). We denote its kernel by
n n n n
A (k). Then the groups A (k) define a descending central filtration
n n
IA = A (1) ⊃ A (2) ⊃ ···
n n n
of IA . This filtration is called the Johnson filtration of AutF .
n n
The k-th Johnson homomorphism
τk : An(k)/An(k +1) → HomZ(H,Ln(k +1)) = H∗ ⊗Z Ln(k +1)
of AutF is defined by the rule
n
σ 7→ (x 7→ x−1xσ), x ∈ H.
It is known that each of τ is a GL(n,Z)-equivariant injective homomorphism. Since
k
the target of the Johnson homomorphism τ is a free abelian group of finite rank, we
k
can detect non-trivial elements in the abelianization A (k)ab of A (k) by τ .
n n k
In particular, we remark that the isomorphism (1) is induced from the first Johnson
homomorphism
τ1 : IAn → H∗ ⊗Z Λ2H,
and that (the coset classes of) the Magnus generators K s and K s form a basis of
ij ijk
IAab as a free abelian group.
n
2.4. Magnus representation.
In this subsection we recall the Magnus representation of AutF . (For details, see
n
[4].) For each 1 ≤ i ≤ n, let
∂
: Z[F ] → Z[F ]
n n
∂x
i
be the Fox derivation defined by
r
∂ (w) = ǫ δ xǫ1 ···x12(ǫj−1) ∈ Z[F ]
∂x j µj,i µ1 µj n
i
j=1
X
for any reduced word w = xǫ1 ···xǫr ∈ F , ǫ = ±1. Let a : F → H be the abelianiza-
µ1 µr n j n
tion of F . We also denote by a the ring homomorphism Z[F ] → Z[H] induced from a.
n n
5
For any matrix A = (a ) ∈ GL(n,Z[F ]), set Aa = (aa ) ∈ GL(n,Z[H]). The Magnus
ij n ij
representation r : AutF → GL(n,Z[H]) of AutF is defined by
M n n
a
∂x σ
i
σ 7→
∂x
(cid:18) j (cid:19)
for any σ ∈ AutF . This map is not a homomorphism but a crossed homomorphism.
n
Namely,
r (στ) = r (σ)τ∗ ·r (τ)
M M M
where r (σ)τ∗ denotes the matrix obtained from r (σ) by applying the automorphism
M M
τ∗ : Z[H] → Z[H] induced from ρ(τ) ∈ Aut(H) on each entry. Hence by restricting r
M
to IA , we obtain a homomorphism IA → GL(n,Z[H]), also denote it by r .
n n M
Let K be the kernel of the homomorphism r : IA → GL(n,Z[H]). The main pur-
n M n
pose of the paper is to show that the abelianization Kab of K is not finitely generated.
n n
More precisely, we prove that it contains a free abelian group of infinite rank.
Here we consider the group K from the view point of the metabelianization of F .
n n
Let FM be the quotient group of F by the second derived subgroup [[F ,F ],[F ,F ]]
n n n n n n
of F . The group FM is called the free metabelian group of rank n. The abelianization
n n
ofFM iscanonicallyisomorphictoH = Fab. Letµ : AutF → AutFM betheinduced
n n n n n
homomorphism from the action of AutF on FM. It is known that µ is surjective for
n n n
n ≥ 2 except for n = 3. (See [1] for n = 2, and see [3] for n ≥ 4.) Let IAM be the
n
IA-automorphism group of FM. Namely, the group IAM consists of automorphisms of
n n
FM which act on the abelianization of FM trivially. Then the restriction of µ induces
n n n
a homomorphism IA → IAM, also denoted by µ .
n n n
Now, in [1], Bachmuth constructed a faithful representation
r′ : IAM → GL(n,Z[H])
M n
using the Magnus representation of FM. Then we can easily see that r = r′ ◦µ by
n M M n
observing the image of the Magnus generators of IA . (For details, see [1].) Therefore,
n
the faithfulness of r′ induces K = Ker(µ ). In particular, we have
M n n
(2) K = {σ ∈ AutF |x−1xσ ∈ [[F ,F ],[F ,F ]], x ∈ F } ⊂ IA .
n n n n n n n n
3. Embeddings of the IA-automorphism group
In this section, we consider certain finitely generated subgroups of the free group F ,
n
and its automorphism group.
3.1. Subgroups W of F .
n,d n
For any integer d ≥ 2, let C be the cyclic group of order d generated by s. Let
d
ι : F → C be a homomorphism defined by
n,d n d
s, i = 1,
xιn,d :=
i
1, i 6= 1.
(
We denote by W the kernel of ι . Then, W is also a free group, and its rank is
n,d n,d n,d
given by d(n − 1) + 1 since the index of W in F is d. Furthermore, applying the
n,d n
6
Reidemeister-Schreier method to the following data:
X := {x ,x ,...,x }; the set of generators of F ,
1 2 n n
T := {1,x ,...,xd−1}; a Schreier-transversal of W of F ,
1 1 n,d n
we see that a set
(3) {(t,x) ∈ W |t ∈ T, x ∈ X, (t,x) 6= 1}
n,d
form a basis of W , where (t,x) = tx(tx)−1 and a map¯: F → T is defined by the
n,d n
condition that
W y = W y ⊂ F
n,d n,d n
for any y ∈ F . (Fordetails for the Reidemeister-Schreier method, see [12] forexample.)
n
Then the set (3) is written down as
xd,
1
x , x x x−1, ..., xd−1x x−(d−1),
2 1 2 1 1 2 1
(4)
.
.
.
x , x x x−1, ..., xd−1x x−(d−1).
n 1 n 1 1 n 1
In the following, we fix this set as a basis of W .
n,d
3.2. Action of IA on W .
n n,d
SinceeachofW containsthecommutatorsubgroup[F ,F ]ofF ,theIA-automorphism
n,d n n n
group
IA = {σ ∈ AutF |x−1xσ ∈ [F ,F ], x ∈ F }
n n n n n
naturally acts on them. Let ρ = ρ : IA → AutW be the homomorphism induced
n,d n n,d
from the action. Then we have
Lemma 3.1. For any n ≥ 2 and d ≥ 2, the homomorphism ρ is injective.
Proof. For an element σ ∈ IA , assume ρ(σ) = 1. Since xσ = x for 2 ≤ i ≤ n, in
n i i
order to see σ = 1, it suffices to show xσ = x .
1 1
Set xσ = x v for some irreducible word v. Then we have
1 1
(5) xd = (xd)σ = (xσ)d = x vx v···x v,
1 1 1 1 1 1
In the right hand side of the equation above, no cancellation of words happen. In
fact, if v is of type x−1w for some irreducible word w, then we have xd = wd. By a
1 1
usual argument in the Combinatorial group theory, there exists some z ∈ F such that
n
x = za and w = zb for some a,b ∈ Z. (See proposition 2.17. in [12].) From the former
1
equation, we have z = x and a = 1. Hence
1
xdw−d = xd−bd = 1.
1 1
Since F is torsion free, b = 1. This shows that w = x . This is a contradiction to the
n 1
irreducibility of v = x−1w. On the other hand, If v is of type wx−1 for some irreducible
1 1
word w, then we have
xd = x−1xdx = x−1(x wdx−1)x = wd.
1 1 1 1 1 1 1 1
By an argument similar to the above, we see that this is also contradiction.
7
Thus, observing the word length of both side of (5), we obtain that v = 1. This
shows that σ = 1.
By this lemma, we can consider IA as a subgroup of each of AutW . In the
n n,d
following, we identify IA with the image of it by ρ.
n
Now,letU betheabelianizationofW . WedenotebyIA(W )theIA-automorphism
n,d n,d n,d
group of W . Namely, the group IA(W ) consists of automorphisms of W which
n,d n,d n,d
act on U trivially. Let us define an linear order among (4) by
n,d
xd < x x x−1 < ... < xd−1x x−(d−1) < ... < x x x−1 < ... < xd−1x x−(d−1).
1 1 2 1 1 2 1 1 n 1 1 n 1
Then, by a result of Magnus [14] as mentioned in Subsection 2.2, the IA-automorphism
group IA(W ) is finitely generated by automorphisms
n,d
K : α 7→ α−1βα, α < β,
α,β
(6)
K : α 7→ α[β,γ], α 6= β < γ 6= α
α,β,γ
where α, β, γ are elements of the basis (4).
On the other hand, from (1), we have
IA(Wn,d)ab ∼= Un∗,d ⊗Z Λ2Un,d
where Un∗,d denotes the Z linear dual group HomZ(Un,d,Z) of Un,d. This isomorphism
is induced from the first Johnson homomorphism
τ1 : IA(Wn,d) → Un∗,d ⊗Z Λ2Un,d.
In particular, (the coset classes of) the elements K s and K s form a basis of
α,β α,β,γ
IA(W )ab as a free abelian group.
n,d
In general, the induced action of IA on U by ρ is not trivial. However, its restric-
n n,d
tion to the kernel of the Magnus representation K is trivial by (2). Namely, ρ induces
n
an embedding
ρ| : K ֒→ IA(W ).
Kn n n,d
By this embedding, we consider K as a subgroup of IA(W ) for any d ≥ 2.
n n,d
4. Lower bounds on Kab
n
In the following, we always assume n ≥ 3 and d ≥ 3. For each 2 ≤ m ≤ d−1, let
σ ∈ AutF be an automorphism defined by
m n
x [[x ,x ],[xm,x ]], i = 2,
x 7→ 2 1 3 1 3
i
x , i 6= 2.
( i
Clearly, we see σ ∈ K . We show that σ ,...,σ are linearly independent in Kab.
m n 2 d−1 n
To do this, we consider the images of σ by the natural homomorphism
m
π : K ֒→ IA(W ) → IA(W )ab.
n,d n n,d n,d
Tobeginwith, inordertodescribeσ inIA(W )ab withK andK , weconsider
m n,d α,β α,β,γ
the action of σ on the basis (4) of W . Observing
m n,d
xσm = x ·x x x−1 ·x−1 ·xmx x−m ·x x−1x−1 ·x ·xmx−1x−m,
2 2 1 3 1 3 1 3 1 1 3 1 3 1 3 1
8
we see
(xd)σm = xd,
1 1
(xkx x−k)σm = xkx x−k, i 6= 2,
1 i 1 1 i 1
(xkx x−k)σm = xkx x−k ·xk+1x x−(k+1) ·xkx−1x−k ·xm+kx x−(m+k)
1 2 1 1 2 1 1 3 1 1 3 1 1 3 1
·xk+1x−1x−(k+1) ·xkx x−k ·xm+kx−1x−(m+k).
1 3 1 1 3 1 1 3 1
Hence, for any 0 ≤ k ≤ d−1, if we denote by ω the automorphism of W defined
m,k n,d
by
xkx x−k 7→ xkx x−k ·xk+1x x−(k+1) ·xkx−1x−k ·xm+kx x−(m+k)
1 2 1 1 2 1 1 3 1 1 3 1 1 3 1
(7)
·xk+1x−1x−(k+1) ·xkx x−k ·xm+kx−1x−(m+k),
1 3 1 1 3 1 1 3 1
then
σ = ω ω ···ω .
m m,0 m,2 m,d−1
Here we remark that for any 0 ≤ k,l ≤ d − 1, the automorphisms ω and ω are
m,k m,l
commutative in AutW .
n,d
Let us describe ω in IA(W )ab with K and K for each 0 ≤ k ≤ d−1.
m,k n,d α,β α,β,γ
Case 1. The case of 0 ≤ k ≤ d−1−m.
In this case, we have k +1 ≤ d−1 and m+k ≤ d−1. Thus, by (7) and Lemma 2.1,
we see
ωm,k = Kxk1x2x−1k,xk1x3x−1kKxk1x2x−1k,xm1+kx3x−1(m+k),xk1x3x−1k
·K
xk1x2x−1k,xk1+1x3x−1(k+1),xm1+kx3x−1(m+k)
·K K−1
xk1x2x−1k,xk1x3x−1k,xk1+1x3x−1(k+1) xk1x2x−1k,xk1x3x−1k
in IA(W ), and hence,
n,d
ω = K +K
m,k xk1x2x−1k,xm1+kx3x−1(m+k),xk1x3x−1k xk1x2x−1k,xk1+1x3x−1(k+1),xm1+kx3x−1(m+k)
+K .
xk1x2x−1k,xk1x3x−1k,xk1+1x3x−1(k+1)
in IA(W )ab.
n,d
Case 2. The case of d−m ≤ k ≤ d−2.
In this case, we have k +1 ≤ d−1 < m+k. Hence (7) is written as
xkx x−k 7→ xkx x−k ·xk+1x x−(k+1) ·xkx−1x−k ·xd ·xm+k−dx x−(m+k−d) ·x−d
1 2 1 1 2 1 1 3 1 1 3 1 1 1 3 1 1
·xk+1x−1x−(k+1) ·xkx x−k ·xd ·xm+k−dx−1x−(m+k−d) ·x−d.
1 3 1 1 3 1 1 1 3 1 1
Then using Lemma 2.2, we obtain
ω = K +K
m,k xk1x2x−1k,x1m+k−dx3x−1(m+k−d),xk1x3x−1k xk1x2x−1k,xk1+1x3x−1(k+1),x1m+k−dx3x−1(m+k−d)
+K .
xk1x2x−1k,xk1x3x−1k,xk1+1x3x−1(k+1)
in IA(W )ab.
n,d
9
Case 3. The case of k = d−1.
In this case, we have k +1 = d and d ≤ m+k. Hence (7) is written as
xd−1x x−(d−1) 7→ xd−1x x−(d−1)
1 2 1 1 2 1
·xd ·x ·x−d ·xd−1x−1x−(d−1) ·xd ·xm−1x x−(m−1) ·x−d
1 3 1 1 3 1 1 1 3 1 1
·xd ·x−1 ·x−d ·xd−1x x−(d−1) ·xd ·xm−1x−1x−(m−1) ·x−d.
1 3 1 1 3 1 1 1 3 1 1
Then using Lemma 2.3, we obtain
ω = K +K
m,d−1 xd1−1x2x−1(d−1),xm1−1x3x1−(m−1),x1d−1x3x−1(d−1) x1d−1x2x−1(d−1),x3,xm1−1x3x1−(m−1)
+K .
xd1−1x2x−1(d−1),xd1−1x3x−1(d−1),x3
in IA(W )ab.
n,d
Therefore for 2 ≤ m ≤ d−1, we obtain
σm = {Kxk1x2x−1k,xm1+kx3x−1(m+k),xk1x3x−1k
0≤k≤d−1−m
X
(8) +Kxk1x2x−1k,xk1+1x3x−1(k+1),xm1+kx3x−1(m+k) +Kxk1x2x−1k,xk1x3x−1k,xk1+1x3x−1(k+1)}
+ {Kxk1x2x−1k,x1m+k−dx3x−1(m+k−d),xk1x3x−1k
d−m≤k≤d−1
X
+Kxk1x2x−1k,xk1+1x3x−1(k+1),x1m+k−dx3x−1(m+k−d) +Kxk1x2x−1k,xk1x3x−1k,xk1+1x3x−1(k+1)}
in IA(W )ab. Using this, we obtain
n,d
Lemma 4.1. For any n ≥ 3 and d ≥ 3, the elements σ ,...,σ are linearly indepen-
2 d−1
dent in IA(W )ab.
n,d
Proof. Suppose
a σ +···+a σ = 0 ∈ IA(W )ab.
2 2 d d n,d
Then, observing the coefficients of K in the left hand side of (8), we see
x2,∗,∗
am{Kx2,xm1 x3x−1m,x3 +Kx2,x1x3x−11,xm1 x3x−1m +Kx2,x3,x1x3x−11} = 0.
2≤m≤d−1
X
Hence, we have a = ··· = a = 0.
2 d−1
This Lemma induces our main theorem.
Theorem 4.1. For n ≥ 3, Kab is not finitely generated.
n
Proof. For any d ≥ 3, the image of the induced map
Kab → IA(W )ab
n n,d
from the natural homomorphism π : K → IA(W )ab contains a free abelian group
n,d n n,d
of rank d−2. Since we can take d ≥ 3 arbitrarily, we obtain the required result. This
completes the proof of Theorem 4.1.
As a Corollary, we have
Corollary 4.1. For n ≥ 3, K is not finitely generated.
n
10