Kg is not finitely generated Daniel Biss* and Benson Farby May 20, 2004 1 Introduction Let g be a closed orientable surface of genus g. The mapping class group Mod g of g is defined to be the group of isotopy classes of orientation- preserving diffeomorphisms g ! g. Recall that an essential simple closed curve fl in g is called a bounding curve, or separating curve, if it is null- homologous in g or, equivalently, if fl separates g into two connected com- ponents. Let K gdenote the subgroup of Mod g generated by the (infinite) collection of Dehn twists about bounding curves in g. Note that K 1is trivial. It has been a long-standing problem in the combinatorial topology of surfaces to determine whether or not the group K gis finitely generated for g 2. For a discussion of this problem, see, e.g., [Jo1 , Jo3, Bi, Mo1 , Mo3 , Ak ]. McCullough-Miller [MM ] proved that K 2is not finitely generated; Mess then proved that K2 is in fact an infinite rank free group. Akita proved in [Ak* * ] that for all g 2, the rational homology H*(K g; Q) is infinite-dimensional as a vector space over Q. Note that as K g admits a free action on the Teichmuller space of g, which is contractible and finite-dimensional, K ghas finite cohomological dimension. For some time it was not known if K g was equal to, or perhaps a finite index subgroup of, the Torelli group Ig, which is the subgroup of elements ______________________________ *This research was conducted during the period the first author served as a * *Clay Math- ematics Institute Long-Term Prize Fellow. ySupported in part by NSF grants DMS-9704640 and DMS-0244542. 1 of Mod g which act trivially on H1( g; Z). Powell [Po ] proved that K 2 = I2. Johnson proved in [Jo2 ] that for g 3, K g has infinite index in Ig by constructing what is now called the Johnson homomorphism, which is the quotient map in a short exact sequence 1 -! Kg -! Ig -! ^3H=H -! 1 where H = H1( g; Z). Johnson then proved in [Jo3 ] that Ig is finitely gen- erated for all g 3. Our main result is the following. Theorem 1.1. The group K gis not finitely generated for any g 2. We will also prove along the way that that the once-punctured analogue of K gis not finitely generated. Theorem 1.1 answers Problem 10 of [Mo2 ], Problem 2.2(i) of [Mo3 ], and the question/conjecture on page 24 of [Bi]. We would still, however, like to know the answer to the following question, asked by Morita (see [Mo3 ], Problem 2.2(ii)). Question 1.2. Is H1(K g; Z) finitely generated for g 3? Note that Birman-Craggs-Johnson (see, e.g. [BC , Jo1]) and Morita [Mo4 ] have found large abelian quotients of K g. We would also like to remark that Morita has discovered (see, e.g., [Mo4 , Mo2 , Mo3 ]) a strong connection between the algebraic structure of K gand the Casson invariant for homology 3-spheres. For example, Morita proved in [Mo4 ] that every integral homology 3-sphere can be obtained by gluing two handlebodies along their boundaries via a map in K g; further, he has been able to express the Casson invariant as a homomorphism K g- ! Z (see, e.g., [Mo1 ]). Rough outline of the proof. Our proof owes a great intellectual debt to the paper [MM ] by D. McCullough and A. Miller, where the theorem is demonstrated in the genus 2 case; indeed we follow the same outline as their proof. First, we find an action of K g on the first homology of an abelian cover Y of g with Galois group Z2g-2. While H1(Y ; Z) is infinitely generated, it is finitely generated as a module over the group-ring of the Galois group of the cover. We view this group-ring as the ring Lg of integral Laurent series 2 in 2g - 2 variables. This action a priori gives a rather complicated high- dimensional representation of K g. We first project to a Laurent series ring L in just one variable and then are able to find and quotient out a codimension two fixed submodule. This reduction to a 2-dimensional representation is crucial for what follows. We then analyze this representation æ : K g-! SL2(L) The ring L comes equipped with a discrete valuation, and so SL2(L) can be realized via Bruhat-Tits theory as a group of automorphisms of a certain simplicial tree. The Bass-Serre theory of graphs of groups_equivalently, of groups acting on trees_is especially suited to understanding whether or not such a group is finitely generated; one such criterion is proven in [MM ]. To complete the proof, we compute enough about the image of æ to apply this criterion to show that K gis not finitely generated. 2 Representing Kg on an abelian cover Consider a standard symplectic basis {a1, . .,.ag, b1, . .,.bg} for H1( g; Z), where ai . bj = ffii,jand ai . aj = bi . bj = 0. Here and throughout this article, the symbol . is used to denote the algebraic intersection number of simple closed curves (or homology classes). By abuse of notation, we will also sometimes view the ai and bi as elements of ß1( g), considered as relative to a fixed basepoint. 2.1 The abelian cover Consider the free abelian group Z2g-2 with generators {s2, . .,.sg, t2, . .,.tg} and the surjection _ : H1( g; Z) ! Z2g-2 defined by _(a1) = _(b1) = 0 _(ai) = si, i 2 _(bi) = ti, i 2 Composing with the Hurewicz map ß1( g) ! H1( g; Z) gives a surjection ' : ß1( g) ! Z2g-2; we denote the kernel of ' by K. 3 Let p : Y ! g denote the covering corresponding to the subgroup K ß1( g). The group Z2g-2 then acts on Y by deck transformations. This action induces an action of Z2g-2 on H1(Y ; Z), which is consequently a 1 1 Z s2 , . .,.sg1, t2 , . .,.tg1-module. We denote this Laurent series ring by Lg. It is rather easy to construct the cover Y explicitly. To this end, con- sider the decomposition of g into two subsurfaces g-1,1and 1,1of genus g - 1 and 1, respectively, obtained by cutting along the bounding curve rep- resenting the homotopy class [a1, b1]. Note that the subspace H1( 1,1; Z) H1( g; Z) is the span of {a1, b1}. Let Y 0denote the universal abelian cover of g-1,1, that is, the cover corresponding to the commutator subgroup of ß1( g-1,1). Since the boundary of g-1,1is null-homologous, it lifts to a col- lection of simple closed boundary curves in Y 0, indexed by the set Z2g-2. We then obtain Y by gluing 1,1to each of these curves along its boundary. The Lg-module structure of H1(Y ; Z) can now be read off from this geo- metric description of Y . Proposition 2.1. The homology group H1(Y ; Z) is a free Lg-module of rank 2g-2 2 + 1 on the following generators: a1, b1, [ai, aj] and [bi, bj] for 2 i < j g, and [ai, bj] for 2 i, j g, with [ag, bg] excepted. Denote by W the 2g-2 free submodule of rank 2 - 1 obtained by omitting the generators a1 and b1. We then have p*(a1) = a1, p*(b1) = b1, and p*(c) = 0 for any c 2 W. Proof.It is a standard fact that the homology of the surface obtained by sewing in discs along the boundary circles of Y 0is a free Lg-module on the generators [ai, aj], [bi, bj], and [ai, bj] (for the sake of normalization, we * *choose a single connected fundamental domain X for the action of K on Y and demand that all these generators be supported in X). Note that the resulting space is just the universal abelian cover of g-1. The element [ag, bg] is omitted because the relation [a2, b2] . . ...[ag, bg] in ß1( g-1) implies that,* * in the homology of the cover, [ag, bg] is in the span of the [ai, bi] for 2 i * *g-1. It is then apparent that the remainder of H1(Y ; Z) is free on the generators a1 and b1. Finally, the identification of the images of the generators under p* follows by definition. We will need to compute the algebraic intersection numbers of certain curves in Y . To ease the exposition of the next result, it will be convenient * *to 4 introduce another piece of notation. We denote the set {a2, . .,.ag, b2, . .,.b* *g} by {c1, . .,.c2g-2} and {s2, . .,.sg, t2, . .,.tg} by {u1, . .,.u2g-2}. Thus, H* *1(Y ) is a free Lg-module on the generators a1, b1, and [ci, cj] for 1 i < j 2g -* * 2 except i = g and j = 2g. Proposition 2.2. Suppose i, j, i0, j0 2 {1, . .,.2g - 2} with i 6= j and i06= j* *0. Assume first that {i, j} \ {i0, j0} = ;. Then there exists ffli,j,i0,j02 {1, 0,* * -1} such that r r2g-2 [ci, cj] . u11. .u.2g-2[ci0, cj0]= 8 < (-1)ri+rj+ri0+rj0ffli,j,i0,j0rk 2 {ffii,k, ffij,k, -ffii0,k, -ffij0,k} f* *or all k = (1) : 0 otherwise. Now, assume that i = i0. Then there exists ffli,j,j02 {1, 0, -1} such that r r2g-2 [ci, cj] . u11. .u.2g-2[ci, cj0]= 8 < (-1)ri+rj+rj0ffli,j,j0rk 2 {ffii,k, ffij,k, -ffii,k, -ffij0,k} for a* *ll k = (2) : 0 otherwise. Lastly, r r2g-2 r r2g-2 [ci, cj] . u11. .u.2g-2a1 = [ci, cj] . u11. .u.2g-2b1 = 0(3) regardless of the integers rk. Proof.Equation (3) is clear since the curves in question are apaprently dis- joint. To prove equation (1), notice that the curve representing the cycle [ci, cj] is a kind of quadrilateral beginning at some basepoint y in the funda- mental domain X, then passing to uiy, followed by uiujy, then ujy, and then back to the original basepoint y. The curve [ci0, cj0] thus intersects [ci, cj]* * only once, at y, but this intersection is not necessarily transverse, so we cannot determine the value of ffli,j,i0,j0= [ci, cj] . [ci0, cj0] aside from observing* * that it lies in the set {1, 0, -1}. Now, the curve ur11. .u.r2g-22g-2[ci0, cj0] cannot possibly meet [ci, cj] un* *less rk 2 {ffii,k, ffij,k, -ffii0,k, -ffij0,k} for all k. On the other hand, if the * *two curves do meet, then by symmetry, their intersection numbers are determined by [ci, cj] . [ci0, cj0] as indicated in the statement of the proposition. 5 The verification of equation (2) proceeds in much the same way. The only subtlety comes in checking the cases ri = 0, in which the curves in question actually have an entire segment in common. But one can perturb one of the curves so that they only meet at one endpoint of the segment; the computation then follows from the usual symmetry. 2.2 The representation It will be useful for us to consider pointed versions of Ig and K g. We work with respect to the basepoint x = p(y) 2 g. Denote by I g,*the group of components of the group of basepoint-preserving diffeomorphisms of g which act trivially on H1( g; Z). öF rgetting the basepoint" clearly gives a surjective homomorphism Ig,*! Ig. Denote by Kg,*the pullback of the diagram Ig,* | " fflfflfflffl|| Kg Ø____//Ig that is, the subgroup of Ig,*generated by twists about bounding curves which avoid the basepoint. Again, the operation of forgetting the basepoint induces a surjection Kg,*! Kg. Recall that K = ß1(Y ). Note that since K is not a characteristic subgroup of ß1( g), an arbitrary mapping class need not lift to Y . In fact, there are even elements of Ig which don't lift to Y . However, we have the following. Proposition 2.3. Each element of Kg,*has a lift to a basepoint-preserving diffeomorphism of Y which is unique up to basepoint-preserving isotopy. Proof.The uniqueness is clear. Moreover, by the universal lifting property for covering maps, the collection of basepoint-preserving mapping classes that admit such a lift constitutes a subgroup. Thus, we need only verify the result for Dehn twists about bounding curves, as these generate Kg,*. So, let C be a bounding curve on g, and denote by tC the twist about C. Since p is an abelian cover, C lifts to a simple closed curve in Y . Consider the map etC, which is a simultaneous Dehn twist about all the lifts of C. This obviously constitutes a lift of tC . 6 These observations are enough to give us our main tool. Henceforth C will denote an arbitrary bounding curve in g, and eCwill denote a lift of C to Y . The homology class of eCwill be written X c + mp2,...,pg,q2,...,qgsp22. .s.pggtq22. .t.qgga1 + np2,...,pg,q2,...qgsp22* *. .s.pggtq22. .t.qggb1 where c 2 W (recall W was defined in the statement of Proposition 2.1), the sum is taken over all integers p2, . .,.pg, q2, . .,.qg, and the m's and n's are integral coefficients, all but finitely many of which vanish. To simplify the notation, we will use underlined symbols to refer to (g - 1)-tuples of objects indexed by the set {2, . .,.g}. For example, p_will stand for p2, . .,.pg, the symbol s_will stand for s2, . .,.sg and, crucially, binary operations on under- lined quantities will be performed componentwise, so that s_p_= sp22. .s.pgg. We are now ready to lift the action of K g,*. Proposition 2.4. The operation which associates to an element of Kg,*, the action of its lift to Y on H1(Y ; Z) gives rise to a representation eæ: Kg,*! GL 1+(2g-2(Lg) 2 ) Proof.We must check that eætakes composition to multiplication and that its image respects the Lg-action on H1(Y ; Z). The former condition follows from the uniqueness up to isotopy of lifts; the latter holds because for any bounding curve C in g, the set of all lifts of C to Y is Lg-invariant. 2.3 Reducing dimension The representation eæ is high-dimensional; we instead would like to work with a 2-dimensional representation. We will achieve this by proving that eæ contains a large subrepresentation, namely W , that we will be able to ignore. In order to do this we first need to analyze the image under eæof a twist about a bounding curve. Proposition 2.5. Let C be a bounding curve on g and 1 i < j 2g - 2. Then ~æ(tC )([ci, cj]) = [ci, cj] + d where d can be written as a sum of terms each of which is divisible by (uk - 1)(ul- 1) for some 1 k < l 2g - 2. 7 Proof.We first assume that C~ = [ci0, cj0]. Recall now that if fi = {fik} is a family of mutually disjoint and nonisotopic simple closed curves on a surface, and if ff is another simple closed curve, then the homology class of the twist tfi(ff) of ff about fi is X [tfi(ff)] = [ff] + (ff . fik)[fik] (4) k Now, if {i, j} = {i0, j0}, then of course ~æ(tC )([ci, cj]) = [ci, cj]. If, * *instead, {i, j} \ {i0, j0} = ;, then equation (1) tells us that ~æ(tC )([ci, cj]) = [ci, cj] + ffli,j,i0,j0(ui- 1)(uj - 1)(u-1i0- 1)(u-1j0- 1* *)[ci0, cj0] which is of the desired form if we set k = i and l = j. Lastly, suppose that {i, j} \ {i0, j0} contains a single element, say without loss of generality i =* * i0. Then equation (2) gives us -1 -1 ~æ(tC )([ci, cj])=[ci, cj] + ffli,j,,j0-ui + 1 - ui(uj - 1) uj0 - 1 [ci0, cj0] -1 = [ci, cj] + ffli,j,,j0u-1j0ui - 1 + ui (uj - 1)(uj0- 1)[ci0, * *cj0] which again gives us what we want, with k = j and l = j0. The general case follows from this calculation by the linearity present in equation (4) along with the vanishing of equation (3). We now use Proposition 2.5 to find a substantially smaller representation of Kg,*. Denote by L the Laurent series ring Z[t, t-1], and define : Lg -! L by (si) = 1 for 2 i g and ( t if i = 2 (ti) = 1 if 3 i g The homomorphism induces a homomorphism ^ : GL n(Lg) -! GL n(L) for any n 1. Now define ^æ: Kg,*! GL 1+(2g-2(L) by 2 ) æ^= ^ O eæ This is of course the same as the representation obtained by tensoring H1(Y ; Z) over Lg with L. Recall that W H1(Y ; Z) was defined in the statement of Proposition 2.1. 8 Corollary 2.6. The representation æ^ becomes trivial upon restriction to W Lg L. Proof.Proposition 2.5 guarantees us that for any bounding curve C and any 1 i < j 2g - 2, we have eæ(tC )([ci, cj]) = [ci, cj] + d, where d is a sum * *of terms each of which is divisible by (uk-1)(ul-1) for some 1 k < l 2g-2. Since k 6= l, at least one of uk and ul is not equal to t2, so it must be the case that each of the summands of d vanishes when we tensor with L. Thus, ^æ(tC )([ci, cj]) = [ci, cj]. The desired result then follows from the fact tha* *t the tC generate Kg,*and the [ci, cj] generate W Lg L. We are now able to define the representation that will actually allow us to prove our result. Since the image of ^æ: Kg,*! GL 1+(2g-2(L) fixes W Lg L, 2 ) we may pass to a quotient representation ~ ~ H1(Y ; Z) Lg L æ~: Kg,*-! Aut L _______________ GL 2(L) W Lg L The first thing we will need to know about ~æis the following. Proposition 2.7. The image of ~æis actually contained in SL2 (L) rather than GL 2(L). Moreover, for a bounding curve C on g, we have _ P P ! 1 + ni_,jmp,qs_p_-i_t_q_-j_- mi_,jmp,qs_p_-i_t_q_-j_ æ~(tC ) = P _ _ _p-i q-j P _ _ _p-iq-j (5) ni_,j_np_,q_s___t___1 - mi_,j_np_,q_s___t___ Furthermore, ~ædescends to a representation æ : Kg -! SL2(L) Proof.Observe that the statement that the image of ~ælies in SL2(L) rather than GL 2(L) follows formally from equation (5), so it suffices to verify that equality. To establish that, we compute before projecting to L via by simply expanding out the summations X (etC)*(a1) a1 + (a1 . s_i_t_j_eC)s_i_t_j_[Ce] (mod W ) and X (etC)*(b1) b1 + (b1 . s_i_t_j_eC)s_i_t_j_[Ce] (mod W ) 9 using the formulas a1 . s_i_t_j_eC= n-i_,-j_ and b1 . s_i_t_j_eC= -m-i_,-j_ To verify the last statement, consider an element j of Kg,*that lies in the kernel of the projection Kg,*! Kg. Denote by ejthe basepoint-preserving lift of j to Y. Since j is isotopic to the identity once we forget basepoints, ejmust be isotopic to a diffeomorphism covering the identity map on g. Thus, we must have an equation _ ! (s_p_t_q_)0 ^æ(j) = pq 0 (s__t__) But in order for this to lie in SL2, it must be the identity matrix, so ^æfacto* *rs through the quotient Kg of Kg,*. 3 Amalgamated products and infinite gener- ation Denote by H the image of the homomorphism æ : Kg -! SL2(L). Our goal is to prove that H is not finitely generated. We now describe how we will do this. Consider the inclusion SL2 (L) SL2(Q[t, t-1]). The field Q(t) obtained by adjoining a free variable t to the rational numbers is equipped with a discrete valuation and contains Q[t, t-1], so one can apply the construc- tion of Bruhat-Tits-Serre to find a (locally infinite) simplicial tree on which SL2(Q[t, t-1]) acts by isometries. The Bass-Serre theory of groups acting on trees can then be applied (see [BM ], x5) to express SL2 (Q[t, t-1]) as an amalgamated product: SL2(Q[t, t-1]) ~=A *U B (6) where A = SL2(Q[t]), _ ! _ ! t-1 0 t 0 B = A 0 1 0 1 10 and U = A \ B. This decomposition allows one to apply the theory of graphs of groups to obtain the following criterion, which is Proposition 5 in [MM ]. Proposition 3.1 (Criterion for infinite generation). Let A *U B be an amalgamated product, and let H be any subgroup. Suppose there exist elements Mk 2 A\U and Nk 2 B\U such that 1. MkNkM-1k2 H; and 2. (H \ A)MkU 6= (H \ A)MlU whenever k 6= l. Then H is not finitely generated. We apply Proposition 3.1 to the situation above, with SL2 (Q[t, t-1]) ~= A *U B and with H = æ(Kg). Our goal now is to find matrices Mk and Nk satisfying the desired criterion. 3.1 The elements Mk and N For a positive integer k, we let _ ! 1 0 Mk = 2 SL2(L) k 1 We also set _ ! 1 t - 2 + t-1 N = 2 SL2(L) 0 1 We now verify that the first hypothesis of Proposition 3.1 holds in our case; here we are taking Nk = N for all k. Proposition 3.2. For each k 1, the matrix MkNM-1klies in H. Proof.First of all, consider the simple closed bounding curve C shown in Figure 1. The figure is drawn so that the homology of the leftmost handle of g is spanned by {a1, b1}. We now lift C to Y ; this is shown in Figure 2. Here, each octogan with a handle coming out of it corresponds to a single fundamental domain for the Z2g-2-action on Y ; we have drawn the two lifts of C that meet the fundamental domain X. In general, of course, the base 11 Figure 1: The curve C Figure 2: The two lifts of C that meet b1. is a 4(g - 1)-gon; the figure corresponds to the case g = 3. It is clear that no lifts of C meet a1, so that æ(tC )(a1) = a1. Moreover, by twisting b1 about the two curves shown in Figure 2, one sees that æ(tC )(b1) = b1 + (t2 - 2 + t-12)a1 = b1 + (t - 2 + t-1)a1 and therefore that æ(tC ) = N. 12 Secondly, since b1 is in the kernel of the map ß1( g) ! Z2g-2, the twist tb1 lifts to Y . Denoting by T the simultaneous twist about all the lifts of b1 to Y , we see that Mk = T*k. Set C0 = tkb1(C). We then have -k MkNM-1k = T*k etC*T* = etC0* the last equality following from the general formula ftaf-1 = tf(a), where f is any mapping class and ta any Dehn twist. Since C0 bounds in g, we see that MkNM-1k= æ(tC0). 3.2 Distinctness of double cosets The rest of this paper is devoted to proving the following. Proposition 3.3. With the notation as above, we have (H \ A)MkU 6= (H \ A)MlU for all k 6= l. Given Proposition 3.3, whose proof we present in the next section, we are now able to establish our main result, Theorem 1.1. Proof of Theorem 1.1. We apply Proposition 3.1 to the subgroup H = æ(Kg) of SL2 (Q[t, t-1]) ~= A *U B, with Mk and Nk = N as above. First observe that Mk 2 A\U since t does not divide k, and N 2 B\U since t - 2 + t-1 62 Q[t]. Therefore, in light of Propositions 3.2 and 3.3, Proposition 3* *.1 implies that H is not finitely generated. As Kg surjects onto H, it is not finitely generated. Note that since Kg,*surjects onto Kg, it follows that Kg,*is also not finitely generated. 4 The proof of Proposition 3.3 In this section we prove Proposition 3.3. In order to do this we will prove that the elements of H = æ(Kg) are of a very special form. To state this precisely, we will need the following. 13 Definition 4.1 (Balanced polynomials). Let f 2 Z[u11, . .,.un1] be a Laurent polynomial in n variables over the integers. We say that f is balanced if 1. f(1, 1, . .,.1) = 0; and 2. for all n-tuples (i1, . .,.in) 2 Zn, the coefficients of ui11. .u.innand u-i11. .u.-innin f are equal. Parallel with a crucial observation of McCullough-Miller [MM ], we have the following. Proposition 4.2. Each element of H has the form _ ! 1 + P Q1 Q2 1 - P where P, Q1, and Q2 are balanced. Proof.Recall the map : Lg ! L above. We begin by fixing an element T = æ(tC ) 2 H and writing _ ! (1 + R1) (S1) T = (S2) (1 - R2) where now the Ri and Si lie in Lg. Equation (5) gives us expressions for R1, R2, S1, and S2 in terms of the m and n coefficients. Now, since the twist tC lies in the Torelli group Ig, we have p* etC* (a1) = p*(a1) = a1. But equation (5) tells us that p* etC* (a1) = a1 + R1(1, 1, . .,.1)a1 + S2(1, 1, . .,.1)b1 and so R1(1, 1, . .,.1) = S2(1, 1, . .,.1) = 0. A similar analysis of b1 allows us to conclude that R2(1, 1, . .,.1) = S1(1, 1, . .,.1) = 0. It follows via formal manipulations from equation (5) that S1 and S2 also satisfy the other criterion for balancedness. We now turn our attention to R1 and R2. Notice that for all p_and q_, we have 0 = Ce . s_-p_t_-q_ X X = mi_,j_ni_+p_,j_+q_- ni_,j_mi_+p_,j_+q_ i_,j_ i_,j_ 14 From this, it follows that R1 = R2, from which one can deduce formally that R1 is balanced. Since it is clear that takes balanced polynomials to balanced polyno- mials, we have the desired property for elements of the form æ(tC ). But the set of elements of SL2(L) of the desired form is evidently a subgroup, so the result follows since the tC generate Kg. Following Lemma 7 in [MM ], we will now see how Proposition 3.3 follows rather formally from Proposition 4.2. Proof of Proposition 3.3.Suppose, that the Mk and Mlare in the same dou- ble coset, that is, that we have a matrix equation _ ! _ ! _ ! _ ! 1 0 1 + P Q1 1 0 u v = (7) k 1 Q2 1 - P l 1 wt z with _ ! 1 + P Q1 2 H \ A Q2 1 - P and _ ! u v 2 U wt z By Proposition 4.2, we know that P, Q1, and Q2 are balanced. By the definition of A, they also lie in Q[t]. Thus, they are constant and hence vanish. Therefore, setting t = 0 in equation (7) gives _ ! _ ! _ ! _ ! 1 0 1 0 u(0) v(0) u(0) v(0) = = k 1 l 1 0 z(0) lu(0) lv(0) + z(0) which obviously implies that k = l. References [Ak] T. Akita, Homological infiniteness of Torelli groups, Topology, Vol. 40 (2001), no. 2, 213-221. [BM] S. Bachmuth and H. Mochizuki, IA-automorphisms of the free metabelian group of rank 3, J. Algebra 55 (1978), 106-115. 15 [Bi] J. Birman, Mapping class groups of surfaces, in Braids (Santa Cruz, CA, 1986), 13-43, Contemp. Math., 78, Amer. Math. Soc., 1988. [BC] J. Birman and R. Craggs, The ~-invariant of 3-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2-manifold, Trans. Amer. Math. Soc. 237 (1978), 283-309. [Jo1] D. Johnson, A survey of the Torelli group, Contemp. Math., Vol. 20 (1983), 165-17. [Jo2] D. Johnson, An abelian quotient of the mapping class group Ig, Math. Ann. 249 (1980), no. 3, 225-242. [Jo3] D. 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Morita, Casson's invariant for homology 3-spheres and char- acteristic classes of surface bundles. I. Topology 28 (1989), no. 3, 305-323. [Po] J. Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978), no. 3, 347-350. Dept. of Mathematics, University of Chicago 5734 University Ave. Chicago, Il 60637 E-mail: daniel@math.uchicago.edu, farb@math.uchicago.edu 17