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