BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS

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1 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS F. R. COHEN, J. PAKIANATHAN, V. VERSHININ, AND J. WU 1. Introduction Let π be a discrete group with Aut(π) the automorphism group of π. Consider the free group F n generated by n letters x 1, x 2,, x n }. The kernel of the natural map Aut(F n ) GL(n, Z) is denoted IA n. Nielsen, and Magnus gave automorphisms which generate IA n as a group [19, 17]. These automorphisms are named as follows: χ k,i for i k with 1 i, k n, and θ(k; [s, t]) for k, s, t distinct integers with 1 k, s, t n and s < t. The definition of the map χ k,i is given by the formula xj if k j, χ k,i (x j ) = (x 1 i )(x k )(x i ) if k = j. Thus the map χ 1 k,i satisfies the formula χ 1 k,i (x xj if k j, j) = (x i )(x k )(x 1 i ) if k = j. The map θ(k; [s, t]) is defined by the formula xj if k j, θ(k; [s, t])(x j ) = (x k ) ([x s, x t ]) if k = j. for which the commutator is given by [a, b] = a 1 b 1 a b. Thus the map θ(k; [s, t]) 1 satisfies the formula θ(k; [s, t]) 1 xj if k j, (x j ) = (x k ) ([x s, x t ] 1 ) if k = j. Date: May 30,

2 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS 2 McCool gave a presentation for the subgroup of IA n generated by the χ k,i [18], the group of basis conjugating automorphisms of a free group. This group is denoted P Σ n in [12] where it is also known as the group of loops. McCool s presentation is listed in Theorem 1.1 below. The subgroup of P Σ n generated by the χ k,i for i < k is denoted P Σ n + here and is called the upper triangular McCool group in [3]. The purpose of this article is to determine the natural Lie algebra obtained from the descending central series for P Σ n + together with related information for P Σ n as well as the structure of the cohomology ring of P Σ n +. One motivation for the work here is that the groups P Σ n and P Σ n + are natural as well as accessible cases arising as analogues of work in [13, 19, 9, 14, 11, 12, 22, 5, 21, 10] where the Johnson filtration is used frequently rather than the descending central series. The techniques here for addressing these Lie algebras are due to T. Kohno [11] and M. Falk/R. Randell [6]. The cohomology of P Σ n was computed by C. Jensen, J. McCammond, and J. Meier [12]. N. Kawazumi [14], T. Sakasai [22], T. Satoh [23] and A. Pettet [21] have given related cohomological information for IA n. The integral cohomology of the natural direct limit of the groups Aut(F n ) is given in work of S. Galatius [8]. The main results here arise from McCool s presentation which is stated next. Theorem 1.1. A presentation of P Σ n is given by generators χ k,j together with the following relations. (1) χ i,j χ k,j χ i,k = χ i,k χ i,j χ k,j for i, j, k distinct. (2) [χ k,j, χ s,t ] = 1 if j, k} s, t} = φ. (3) [χ i,j, χ k,j ] = 1 for i, j, k distinct. (4) [χ i,j χ k,j, χ i,k ] = 1 for i, j, k distinct (redundantly). In what follows below, gr (π) denotes the associated graded Lie algebra obtained from the descending central series of a discrete group π. Work of T. Kohno [11], as well as M. Falk, and R. Randell [6] provide an important description of these Lie algebras for many groups π, one of which is the pure braid group on n strands P n. The Lie algebra gr (P n ), basic in Kohno s work, gave an important ingredient in his analysis of Vassiliev invariants of pure braids in terms of iterated integrals [11]. A presentation for this Lie algebra is given by the quotient of the free Lie algebra L[B i,j 1 i < j k] generated by elements B i,j with 1 i < j k modulo the infinitesimal braid relations or horizontal 4T relations given by the following three relations: (1) If i, j} s, t} = φ, then [B i,j, B s,t ] = 0. (2) If i < j < k, then [B i,j, B i,k + B j,k ] = 0.

3 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS 3 (3) If i < j < k, then [B i,k, B i,j + B j,k ] = 0. The results below use the methods of Kohno, and Falk-Randell to obtain information about P Σ n, and P Σ n + as well. One feature is that the Lie algebras given by gr (P Σ n ) and gr (P Σ n + ) satisfy two of the horizontal 4T relations. The next theorem is technical, but provides the foundation required to prove the main results here; the proof is given in section 5. Theorem 1.2. There exist homomorphisms defined by π(χ k,i ) = π : P Σ n P Σ n 1 χk,i if i < n, and k < n, 1 if i = n or k = n. The homomorphism π : P Σ n P Σ n 1 is an epimorphism. The kernel of π denoted K n is generated by the elements χ n,i and χ j,n for 1 i, j n 1. Furthermore, this extension is split and the conjugation action of P Σ n 1 on H 1 (K n ) is trivial. In addition, the homomorphism π : P Σ n P Σ n 1 restricts to a homomorphism π P Σ + n : P Σ + n P Σ + n 1 which is an epimorphism. The kernel of π P Σ + n denoted K + n is a free group generated by the elements χ n,i for 1 i n 1. Furthermore, this extension is split and the conjugation action of P Σ + n 1 on H 1 (K + n ) is trivial. Provided that it is clear from the context, the notation χ k,i is used ambiguously to denote both the element χ k,i in P Σ n, or in P Σ n + when defined, as well as the equivalence class of χ k,i in gr 1 (P Σ n ) = H 1 (P Σ n ) or in H 1 (P Σ n + ) when defined. Partial information concerning the Lie algebra gr (P Σ n ) is given next. Theorem 1.3. There is a split short exact sequence of Lie algebras 0 gr (K n ) gr (P Σ n ) gr (P Σ n 1 ) 0. The relations (1)-(4) are satisfied on the level of Lie algebras: (1) If i, j} s, t} = φ, then [χ j,i, χ s,t ] = 0. (2) If i, j, k are distinct, then [χ i,k, χ i,j + χ k,j ] = 0. (3) If i, j, k are distinct, the element [χ k,i, χ j,i + χ j,k ] is non-zero. (4) If i, j, k are distinct, [χ i,j, χ k,j ] = 0.

4 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS 4 Furthermore, there is a split epimorphism γ : K n n 1 Z with kernel denoted Λ n together with a split short exact sequence of Lie algebras 0 L n gr (K n ) where L n is the Lie algebra kernel of gr (γ). gr (γ) n 1 Z 0 The same methods give a complete description for the cohomology algebra of P Σ n + as well as the Lie algebra gr (P Σ n + ). A further application to be given later is a substantial contribution to the cohomology of each of the Johnson filtrations of IA n. To express the answers, the notation χ k,i is used to denote the dual basis element to χ k,i, namely 1 if k = s and i = t χ k,i(χ s,t ) = 0 otherwise. Theorem 1.4. The cohomology algebra of P Σ n + satisfies the following properties. (1) The cohomology algebra is a finitely generated torsion free abelian group in each degree. (2) If 1 k n, a basis for H k P Σ n + is given by χ i 1,j 1 χ i 2,j 2 χ i k,j k where 2 i 1 < i 2 < < i k n 1, and 1 j t < i t for all t. (3) A complete set of relations ( assuming graded commutativity, and associativity ) is given by χ i,k2 = 0 for all i > k, and χ i,j[χ i,k χ j,k ] = 0 for k < j < i. The Lie algebra obtained from the descending central series for P Σ n +, gr (P Σ n + ), is additively isomorphic to a direct sum of free sub-lie algebras with [χ k,j, χ s,t ] = 0 if j, k} s, t} = φ, [χ k,j, χ s,j ] = 0 if s, k} j} = φ and [χ i,k, χ i,j + χ k,j ] = 0 for j < k < i. 2 k n L[χ k,1, χ k,2,, χ k,k 1 ], Remark: Theorem 1.4 does not rule out the possibility that P Σ n + is isomorphic to the pure braid group P n. Notice that P Σ n + is isomorphic to Z F [χ 3,1, χ 3,2 ] and thus P 3 where

5 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS 5 a generator of Z is given by χ 2,1 χ 3,1. In fact, Theorem 1.4 implies that after suspending the classifying spaces of both P Σ n + and P n exactly once, these suspended classifying spaces are homotopy equivalent. Let U[L] denote the universal enveloping algebra of a Lie algebra L. Since the Euler- Poincare series for U[L[χ k,1, χ k,2,, χ k,k 1 ]] is 1/(1 (k 1)t), the next corollary follows at once. Corollary 1.5. The Euler-Poincare series for U[P Σ n + ] is equal to 1/(1 kt). 1 k n 1 Definition 1.6. Let M n denote the smallest subalgebra of H (IA n ; Z) such that (1) M n surjects to H (P Σ + n ) (that such a surjection exists follows from Theorem 1.7) and (2) M n is closed respect to the conjugation action of GL(n, Z) on H (IA n ; Z). Theorem 1.7. The natural inclusion j : P Σ n + IA n composed with the abelianization map A : IA n IA n /[IA n, IA n ] = n( Z given by n 2) P Σ + n j IA n A IA n /[IA n, IA n ] = H 1 (IA n ) induces a split epimorphism in integral cohomology. Thus the integral cohomology of P Σ + n is a direct summand of the integer cohomology of IA n [ a summand which is not invariant under the action of GL(n, Z) ]. Furthermore, the image of A : H (IA n /[IA n, IA n ]) H (IA n ) contains M n. In addition, the suspension of the classifying space BP Σ + n, Σ(BP Σ + n ), is a retract of Σ(BIA n ) and there is an induced map θ : BIA n ΩΣ(BP Σ + n ) which factors the Freudenthal suspension E : BP Σ + n ΩΣ(BP Σ + n ) given by the composite BP Σ + n BIA n ΩΣ(BP Σ + n ). Remarks: Properties of the image of A are addressed in work of N. Kawazumi [14], T. Sakasai [22], T. Satoh [23] and A. Pettet [21]. An analogous map BIA n ΩΣ(BP Σ n ) is constructed in work of C. Jensen, J. McCammond and J. Meier [12].

6 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS 6 The remainder of this introduction is devoted to the structure of the braid-permutation group BP n introduced by R. Fenn, R. Rimányi and C. Rourke [7]. The group BP n is defined as the subgroup of Aut(F n ) generated by ξ i and σ i, where the action of an element φ in Aut(F n ) is from the right with x i+1 j = i, (x j )ξ i = x i j = i + 1, x j otherwise; (x j )σ i = x i+1 j = i, x 1 i+1 x ix i+1 j = i + 1, x j otherwise. The group BP n is presented by the set of generators ξ i and σ i for 1 i n 1, and by the relations: ξi 2 = 1, (1) ξ i ξ j = ξ j ξ i i j > 1, ξ i ξ i+1 ξ i = ξ i+1 ξ i ξ i+1 ; (2) (3) σ i σ j = σ j σ i i j > 1, σ i σ i+1 σ i = σ i+1 σ i σ i+1 ; ξ i σ j = σ j ξ i i j > 1, ξ i ξ i+1 σ i = σ i+1 ξ i ξ i+1, σ i σ i+1 ξ i = ξ i+1 σ i σ i+1, The group BP n is also characterized [7] as the the subgroup of Aut(F n ) consisting of automorphism φ Aut(F n ) of permutation-conjugacy type which satisfy (4) (x i )φ = w 1 i x λ(i) w i for some word w i F n and permutation λ Σ n the symmetric group on n letters. Theorem 1.8. The group BP n is the semi-direct product of the symmetric group on n-letters Σ n and the group P Σ n with a split extension 1 P Σ n BP n Σ n 1.

7 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS 7 Theorems 1.3, 1.4 and 1.8 provide some information about the cohomology of as well as Lie algebras associated to BP n. The authors take this opportunity to thank Toshitake Kohno, Nariya Kawazumi, Shigeyuki Morita, Dai Tamaki as well as other friends for this very enjoyable opportunity to participate in this conference. The authors would also like to thank Benson Farb for his interest in this problem. The first author is especially grateful for this mathematical opportunity to see friends as well as to learn and to work on mathematics with them at this conference. Consider the map defined by the formula 2. Projection maps P Σ n P Σ n 1 p(x j ) = p : F n F n 1 xj if j n 1, 1 if j = n. In addition, let i : P Σ + n P Σ n denote the natural inclusion. Theorem 2.1. The projection maps p : F n F n 1 induce homomorphisms given by π(χ k,i ) = π : P Σ n P Σ n 1 Furthermore, these homomorphisms restrict to together with a commutative diagram χk,i if i < n, and k < n, 1 if i = n or k = n. π : P Σ + n P Σ + n 1 P Σ n + i P Σ n π P Σ n 1 + i π P Σ n 1. Proof. Consider the following three commutative diagrams: F n p χ n,j Fn p F n 1 1 F n 1,

8 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS 8 F n p χ j,n Fn p F n 1 1 F n 1. If k, j < n, then the following diagram commutes. F n p χ j,k Fn p F n 1 χ j,k Fn 1. Thus if either i, k < n, i = n, or k = n, the functions χ k,i restrict to isomorphisms of F n 1. The restriction is evidently compatible with composition of isomorphisms. Hence there is an induced homomorphism given by π(χ k,i ) = π : P Σ n P Σ n 1 χk,i if i < n, and k < n, 1 if i = n or k = n. These homomorphisms are compatible with the restriction maps i : P Σ n + theorem follows. P Σ n, and the 3. On automorphisms of P Σ n, and P Σ + n The conjugation action of Aut(F n ) on itself restricted to certain natural subgroups of Aut(F n ) has P Σ n, and P Σ n + as characteristic subgroups. The purpose of this section is to give two such natural subgroups. One of these subgroups is used below to determine the relations in the cohomology algebra for P Σ n +. A choice of generators for Aut(F n ) is listed next. Let σ denote an element in the symmetric group on n letters Σ n which acts naturally on F n by permutation of coordinates with σ i given

9 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS 9 by the transposition (i, i + 1). Thus x j if j} i, i + 1} = φ, σ i (x j ) = x i+1 if j = i and x i if j = i + 1. Next consider τ i : F n F n which sends x i to x 1 i, and fixes x j for j i. Thus xj if j i and τ i (x j ) = x 1 i if j = i. The elements τ i, and σ j for 1 i, j n generate the signed permutation group in Aut(F n ); this group, also known as the wreath product Σ n Z/2Z, embeds in GL(n, Z) via the natural map Aut(F n ) GL(n, Z). In this wreath product, a Coxeter group of type B n, the elements τ i can be expressed in terms of τ 1 and σ i. Let δ denote the automorphism of F n which sends x 1 to x 1 x 2 while fixing x i for i > 1. It follows from [17] that Aut(F n ) is generated by the elements σ i for 1 i n, τ i for 1 i n and δ. It is natural to consider the action of some of these elements on P Σ n, and P Σ n + conjugation. Proposition 3.1. Subgroups of the automorphism groups of P Σ + n, and P Σ n are listed as follows. (1) Conjugation by the elements τ i for 1 i n leaves P Σ + n invariant. Thus n Z/2Z is isomorphic to a subgroup of Aut(P Σ + n ) with induced monomorphisms θ : n Z/2Z Aut(P Σ + n ) obtained by conjugating an element in P Σ n + by the τ i. The conjugation action of τ i on the elements χ s,t is specified by the formulas χ s,t if i} s, t} = φ, τ i (χ s,t )τ 1 i = χ 1 s,i if t = i, if s = i χ i,t by

10 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS10 for s > t. (2) Conjugation by the elements τ i and σ i for 1 i n leaves P Σ n invariant. Thus Σ n Z/2Z is isomorphic to a subgroup of Aut(P Σ n ) with induced monomorphisms θ : Σ n Z/2Z Aut(P Σ n ) obtained by conjugating an element in P Σ n by the τ i and σ i. The conjugation action of τ i and σ i on the χ s,t is specified by the formulas χ s,t if i} s, t} = φ, τ i (χ s,t )τ 1 i = χ 1 s,i if t = i, χ i,t if s = i, and for all s t. σ i (χ s,t )σ 1 i = χ s,t if i, i + 1} s, t} = φ, χ i+1,t if s = i and t i + 1, χ i+1,i if s = i, and t = i + 1, χ i,t if s = i + 1, and t i, χ i,i+1 if s = i + 1, and t = i, χ s,i+1 if s i + 1, and t = i, χ s,i if s i, and t = i + 1 Proof. The proof of this proposition is a direct computation with details omitted. 4. On certain subgroups of IA n The purpose of this section is to consider the subgroups (1) G n of P Σ n generated by the elements χ n,i and χ j,n for 1 i, j n 1 and (2) G + n of P Σ + n generated by the elements χ n,i for 1 i n 1. Proposition 4.1. (1) The following relations are satisfied in P Σ n. i: χ 1 i,j χ n,j χ i,j = χ n,j with i, j < n. ii: χ 1 i,k χ n,j χ i,k = χ n,j with i, k} n, j} = φ. iii: χ 1 j,k χ n,j χ j,k = χ n,k χ n,j χ 1 n,k with k, j < n. iv: χ 1 i,k χ j,n χ i,k = χ j,n with i, k} n, j} = φ. v: χ 1 j,i χ j,n χ j,i = χ n,i χ j,n χ 1 n,i with i, j < n. vi: χ 1 i,j χ j,n χ i,j = (χ n,j χ 1 i,n χ 1 n,j ) (χ i,n χ j,n ) with i, j < n.

11 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS11 (2) The group G n is a normal subgroup of P Σ n and is the kernel of the projection π : P Σ n P Σ n 1. Thus G n = K n as given in Theorem 2.1. (3) The group G + n is a normal subgroup of P Σ + n and is the kernel of the projection π P Σ + n : P Σ + n P Σ + n 1. Thus G + n = K + n as given in Theorem 2.1. Proof. Consider the elements χ 1 i,j χ t,n χ i,j and χ 1 i,j χ n,t χ i,j for various values of t together with McCool s relations as listed in Theorem 1.1: (1) χ i,j χ k,j χ i,k = χ i,k χ i,j χ k,j for i, j, k distinct. (2) [χ k,j, χ s,t ] = 1 if j, k} s, t} = φ. (3) [χ i,j, χ k,j ] = 1 for i, j, k distinct. The verification of part (1) of the proposition breaks apart into six natural cases where the first three are given by the conjugation action on χ n,j while the second three are given by the conjugation action on χ j,n. i: χ 1 i,j χ n,j χ i,j = χ n,j by formula (3) with i, j < n. ii: χ 1 i,k χ n,j χ i,k = χ n,j by formula (2) with i, k} n, j} = φ. iii: χ 1 j,k χ n,j χ j,k = χ n,k χ n,j χ 1 n,k by formulas (1) and (3) with k, j < n. iv: χ 1 i,k χ j,n χ i,k = χ j,n by formula (2) with i, k} n, j} = φ. v: χ 1 j,i χ j,n χ j,i = χ n,i χ j,n χ 1 n,i by formulas (1) and (3) with i, j < n. vi: χ 1 i,j χ j,n χ i,j = (χ n,j χ 1 i,n χ 1 n,j ) (χ i,n χ j,n ) by formulas (1) and (3) with i, j < n. A sketch of formula (iv) is listed next for convenience of the reader. Assume that i, j, n are distinct. a: χ i,n χ j,n χ i,j = χ i,j χ i,n χ j,n. b: χ j,n χ i,j = χ 1 i,n χ i,j χ i,n χ j,n. c: χ 1 i,j χ j,n χ i,j = (χ 1 i,j χ 1 i,n χ i,j) (χ i,n χ j,n ). d: By formula (v) above, χ 1 i,j χ i,n χ i,j = χ n,j χ i,n χ 1 Thus e: Thus χ 1 i,j χ j,n χ i,j = (χ n,j χ i,n χ 1 (1) G n is a normal subgroup of P Σ n and (2) G + n is a normal subgroup of P Σ + n n,j. n,j ) (χ i,n χ j,n ) and formula (vi) follows.

12 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS12 by inspection of the previous relations (i-vi). Next notice that G n is in the kernel of the projection π : P Σ n P Σ n 1. In addition, every element W in P Σ n is equal to a product W n 1 X n where W n 1 is in the image of the section σ applied to P Σ n 1 and X n is in G n by inspection of the relations relations (i-vi). Thus, the kernel of π is generated by all conjugates of the elements χ n,i and χ j,n for 1 i, j n 1 which coincides with G n. A similar assertion and proof applies to G + n using relations (iv, v, vi). The following result [11, 6] will be used below to derive the structure of certain Lie algebras in this article. Theorem 4.2. Let 1 A B C 1 be a split short exact sequence of groups for which conjugation by C induces the trivial action on H 1 (A). Then there is a split short exact sequence of Lie algebras 0 gr (A) gr (B) gr (C) 0. To apply Theorem 4.2, features of the local coefficient system in homology for the projection maps π : P Σ n P Σ n 1 and π P Σ + n : P Σ + n P Σ + n 1 are obtained next. Proposition 4.3. (1) The natural conjugation action of P Σ n 1 on H 1 (K n ) is trivial. Thus there is a split short exact sequence of Lie algebras 0 gr (K n ) gr (P Σ n ) gr (P Σ n 1 ) 0. (2) The natural conjugation action of P Σ + n 1 on H 1 (K + n ) is trivial. Thus there is a split short exact sequence of Lie algebras 0 gr (K + n ) gr (P Σ + n ) gr (P Σ + n 1) 0. Proof. As before, consider the elements χ 1 i,j χ n,t χ i,j and χ 1 i,j χ t,n χ i,j together with McCool s relations as given in Proposition 4.1 to obtain the following formulas. i: χ 1 i,j χ n,j χ i,j = χ n,j with i, j < n. ii: χ 1 i,k χ n,j χ i,k = χ n,j with i, k} n, j} = φ. iii: χ 1 j,k χ n,j χ j,k = χ n,j (χ 1 n,j χ n,k χ n,j χ 1 n,k ) with k, j < n. Thus χ 1 j,k χ n,j χ j,k = χ n,j ([χ 1 n,j, χ n,k]). iv: χ 1 i,k χ j,n χ i,k = χ j,n with i, k} n, j} = φ. v: χ 1 j,i χ j,n χ j,i = χ n,i χ j,n χ 1 n,i = χ j,n χ 1 j,n χ n,i χ j,n χ 1 n,i with i, j < n. Thus χ 1 j,i χ j,n χ j,i = χ j,n [χ 1 j,n, χ n,i].

13 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS13 vi: χ 1 i,j χ j,n χ i,j = (χ n,j χ 1 i,n χ 1 n,j ) (χ i,n χ j,n ) = [χ n,j, χ 1 i,n ] χ j,n with i, j < n. It then follows that the conjugation action of χ s,t on the class of either χ n,j or χ j,n in H 1 (K n ) fixes that class (in H 1 (K n )). Thus there is a short exact sequence of Lie algebras 0 gr (K n ) gr (P Σ n ) gr (P Σ n 1 ) 0 by Theorem 4.2 A similar assertion and proof follows for H 1 (K n + ) by inspection of formulas (iv,v,vi). The proposition follows. 5. Proof of Theorem 1.2 The first part of Theorem 1.2 that the projection map π : P Σ n P Σ n 1 is an epimorphism with kernel generated by χ n,i and χ j,n for 1 i, j n 1 follows from Proposition 4.1. Furthermore, this extension is split by the section σ : P Σ n 1 P Σ n with σ(χ j,i ) = χ j,i. That the local coefficient system in homology is trivial follows from Proposition 4.3. Similar properties are satisfied for π P Σ + n : P Σ + n P Σ + n 1 by Propositions 4.1 and 4.3. Consider the free group F n 1 with generators x 1,, x n 1. There is a homomorphism obtained by defining Φ n 1 : F n 1 K + n Φ n 1 (x i ) = χ n,i. This homomorphism is evidently a surjection. To check that the subgroup K n + is a free group generated by the elements χ n,i for 1 i n 1, it suffices to check that Φ n 1 is a monomorphism. Observe that if W = x ɛ 1 i1 x ɛ 2 i2 x ɛr i r for ɛ i = ±1 is a word in F n 1, then Φ n 1 (W ) is an automorphism of F n with Φ n 1 (W )(x n ) = W x n W 1. Thus if W is in the kernel of Φ n 1, then W x n W 1 = x n. Furthermore, if W is in F n 1, then W = 1. Theorem 1.2 follows. 6. Proof of Theorem 1.3 Theorem 1.3 states that there a split short exact sequence of Lie algebras 0 gr (K n ) gr (P Σ n ) gr (P Σ n 1 ) 0. This follows from Proposition 4.3. The next assertion is that certain relations are satisfied in the Lie algebra gr (P Σ n ): If i, j} s, t} = φ, then [χ j,i, χ s,t ] = 0 by one of McCool s relations in Theorem 1.1.

14 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS14 Next notice that one of the horizontal 4T relations follows directly from McCool s identity in Theorem 1.1. Since [χ i,j χ k,j, χ i,k ] = 1 for i, j, k distinct, it follows that [χ i,k, χ i,j + χ k,j ] = 0 on the level of Lie algebras. In addition, [χ k,j, χ s,j ] = 0 on the level of Lie algebras if s, k} j} = φ by inspection of Theorem 1.1. That [χ k,i, χ j,i + χ j,k ] is non-zero follows from [4]. The details are omitted: they are a direct computation using the Johnson homomorphism together with structure for the Lie algebra of derivations of a free Lie algebra. Next notice that H 1 (P Σ n ) = ( n 2) Z. Project to the summand with basis χ i,n for 1 i n 1. This composite restricted to γ : K n n 1 Z is evidently a split epimorphism as [χ i,n, χ j,n ] = 1. The remaining properties follow by inspection. Theorem 1.3 follows. 7. Proof of Theorem 1.4 Proposition 4.3 gives that the action of P Σ n 1 + on H 1 (K n + ) is trivial. There are two consequences of this fact. The first consequence is that there is a split short exact sequence of Lie algebras 0 gr (K n + ) gr (P Σ n + ) gr (P Σ n 1) + 0 by Proposition 4.3 or [11, 6]. Notice that one of the horizontal 4T relations follows directly from McCool s identity in Theorem 1.1. As checked in the proof of Theorem 1.3, the relation [χ i,k, χ i,j + χ k,j ] = 0 is satisfied on the level of Lie algebras. The remaining two relations [χ j,k, χ s,t ] = 0 if j, k} s, t} = φ, and [χ j,k, χ s,j ] = 0 if s, k} j} = φ follow by inspection of McCool s relations. Thus the asserted structure of Lie algebra follows. The second consequence is that the local coefficient system for the Leray spectral sequence of the extension 1 K + n P Σ + n P Σ + n 1 1 has trivial local coefficients in cohomology. Since K n + is a free group, it has torsion free cohomology which is concentrated in degrees at most 1. Thus the E 2 -term of the spectral sequence splits as a tensor product H (P Σ + n 1) Z H (K + n ).

15 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS15 Since the extension is split, all differentials are zero, and the spectral sequence collapses at the E 2 -term. Note that P Σ + 2 is isomorphic to the integers. Thus the cohomology of P Σ + n is torsion free and a basis for the cohomology is given as stated in the theorem by inspection of the E 2 -term of the spectral sequence by induction on n. It thus remains to work out the product structure in cohomology which is asserted to be χ k,i2 = 0 for all i > k, and χ i,j[χ i,k χ j,k ] = 0 for k < j < i. for which χ i,k is the basis element in cohomology dual to χ i,k. Notice that χ i,k2 = 0. It suffices to work out the relation χ 3,2 [χ 3,1 χ 2,1] = 0 in case n = 3 by the natural projection maps. Consider the natural split epimorphism obtained by restriction to π P Σ + 3 : P Σ+ 3 P Σ + 2. The kernel is a free group on two letters χ 3,1, χ 3,2 and the spectral sequence of the extension collapses. On the level of cohomology, it suffices to work out the value of χ 3,1 χ 3,2. Thus the product χ 3,1 χ 3,2 is equal to the linear combination Aχ 2,1 χ 3,1 + Bχ 2,1 χ 3,2 for scalars A, and B. Next, consider McCool s relations ( as stated in 1.1 ) which gives χ 3,1 χ 2,1 χ 3,2 = χ 3,2 χ 3,1 χ 2,1. Thus the commutator [χ 3,1 χ 2,1, χ 3,2 ] is 1 in P Σ 3. Since this commutator is trivial, there is an induced homomorphism defined by the equation ρ((a, b)) = ρ : Z Z P Σ + 3 χ3,2 if (a, b) = (1, 0) and χ 3,1 χ 2,1 if (a, b) = (0, 1). Let (1, 0) and (0, 1) denote the two associated natural classes in H 1 (Z Z; Z) which are dual to the natural homology basis. Notice that ρ (χ 2,1) = (0, 1), ρ (χ 3,2) = (1, 0), and ρ (χ 3,1) = (0, 1).

16 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS16 Let (1, 1) denote the fundamental cycle of H 2 (Z Z) given by the cup product (1, 0) (0, 1). Consider (1, 1) = ρ (χ 3,1 χ 3,2) = ρ (Aχ 2,1 χ 3,1 + Bχ 2,1 χ 3,2) = B(1, 1). Thus B = 1. Next, consider the automorphisms of P Σ 3 +. Recall that there are automorphisms of P Σ n + specified by the conjugation action of τ i on the elements χ s,t given by the formulas: χ s,t if i} s, t} = φ, τ i (χ s,t )τ 1 i = χ 1 s,i if t = i, χ i,t if s = i. Notice that τ 1 leaves P Σ 3 + invariant and is thus an automorphism of P Σ 3 +. Thus on the level of H 1 (P Σ 3 + ), conjugation by τ 1 denoted φ 1, is given by the formula (1) φ 1 (χ 2,1 ) = χ 2,1, (2) φ 1 (χ 3,1 ) = χ 3,1 and (3) φ 1 (χ 3,2 ) = χ 3,2. Thus on the level of cohomology, (1) φ 1(χ 2,1) = χ 2,1, (2) φ 1(χ 3,1) = χ 3,1 and (3) φ 1(χ 3,2) = χ 3,2. Apply the automorphism φ 1 to the equation χ 3,1 χ 3,2 = Aχ 2,1 χ 3,1 + Bχ 2,1 χ 3,2 where B = 1 to obtain the following. χ 3,1 χ 3,2 = Aχ 2,1 χ 3,1 Bχ 2,1 χ 3,2. Aχ 2,1 χ 3,1 Bχ 2,1 χ 3,2 = [Aχ 2,1 χ 3,1 + Bχ 2,1 χ 3,2] in a free abelian group of rank two with basis χ 2,1 χ 3,1, χ 2,1 χ 3,2}. Hence A = 0 and χ 3,2 [χ 3,1 χ 2,1] = 0. It follows that χ i,j [χ i,k χ j,k ] = 0 for k < j < i by a similar argument. The Theorem follows. 8. Proof of Theorem 1.7 Recall that H 1 (IA n ) is a free abelian group of rank n ( n 2) with basis given by the equivalence classes of χ i,k for i k with 1 i, k n, and θ(k; [s, t]) for k, s, t distinct positive natural numbers with s < t [14, 5, 4, 23]. Thus the natural composite P Σ + n IA n IA n /[IA n, IA n ]

17 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS17 is a split monomorphism on the level of the first homology group with image spanned by χ k,j for k > j. The cohomology algebra of P Σ n + is generated as an algebra by elements of degree 1 given by χ k,j, for k > j in the dual basis for H 1 (P Σ + n ) by Theorem 1.4. Thus the composite P Σ + n IA n IA n /[IA n, IA n ] is a split surjection in integral cohomology. That the image is not GL(n, Z)-invariant follows by an inspection. Since the composite H (IA n /[IA n, IA n ]) A H (IA n ) H (P Σ + n ) is an epimorphism, and A is GL(n, Z)-equivariant, the image of A contains M n. Notice that the classifying space BIA n /[IA n, IA n ] is homotopy equivalent to a product of circles (S 1 ) n(n 2). Furthermore, the composite P Σ + n IA n IA n /[IA n, IA n ] induces a split epimorphism on integral cohomology. That the map BP Σ + n (S 1 ) n(n 2) is split after suspending once follows directly from the next standard property of maps into products of spheres. Lemma 8.1. Let f : X Y be a continuous map which satisfies the following properties. Then (1) The space X is homotopy equivalent to a CW-complex, (2) Y is a finite product of spheres of dimension at least 1 and (3) the map f induces a split epimorphism on integral cohomology. (1) the suspension of X, Σ(X), is a retract of Σ(Y ) (2) Σ(X) is homotopy equivalent to a bouquet of spheres and (3) the Freudenthal suspension E : X ΩΣ(X) factors through a map Y ΩΣ(X). Hence there is a map Θ : BIA n ΩΣ(P Σ + n ) which gives a factorization of the Freudenthal suspension. The Theorem follows. 9. Proof of Theorem 1.8 The purpose of this section is to give the proof of Theorem 1.8 along with other information. There is a homomorphism ρ n : BP n Σ n

18 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS18 defined by ρ(σ i ) = ξ i = ρ(ξ i ). One way to see that such a homomorphism exists is to consider the pullback diagram π Ξ n Σ n i Aut n GL(n, Z) and to observe that BP n is a subgroup of Ξ n. Recall that the group BP n is characterized as the subgroup of Aut(F n ) consisting of automorphisms φ Aut(F n ) of permutation-conjugacy type which satisfy (5) (x i )φ = w 1 i x λ(i) w i for some word w i F n and permutation λ Σ n the symmetric group on n letters ( where the action of an element φ is Aut(F n ) is from the right ) [7]. Thus (6) ((x i )φλ 1 ) = ((w 1 i )λ 1 )(x i )((w i )λ 1 ). Furthermore σ i ξ i = χ i+1,i and so σ i = χ i+1,i ξ 1 i. The group BP n is thus generated by (1) Σ n and (2) P Σ n. Thus the kernel of the natural map ρ n : BP n Σ n is P Σ n. Theorem 1.8 follows. 10. Problems (1) Is P Σ + n isomorphic to P n? If n = 2, 3, the answer is clearly yes. (2) Let K n denote the subgroup of IA n generated by all of the elements θ(k; [s, t]). Identify the structure of gr (K n ). (3) Are the natural maps and/or gr (P Σ + n ) gr (IA n ), gr (P Σ n ) gr (IA n ) gr (K n ) gr (IA n ) monomorphisms? (4) Give the Euler-Poincaré series for U[gr (IA n ) Q] and U[gr (P Σ n ) Q] where U[L] denotes the universal enveloping algebra of a Lie algebra L. (5) Give the Euler-Poincaré series for M n.

19 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS19 (6) Observe that the kernel of the natural map out of the free product P Σ n K n IA n is a surjection with kernel a free group. The natural morphism of Lie algebras ι : gr (P Σ n ) gr (K n ) gr (IA n ) is an epimorphism ( where denotes the free product of Lie algebras). Is the kernel a free Lie algebra? 11. Acknowledgements This work was started during the visit of the first and the third authors to the National University of Singapore. They are very thankful for hospitality to the Mathematical Department of this university and especially to Jon Berrick. The first author was partially supported by National Science Foundation under Grant No and the Institute for Advanced Study. The third author was supported in part by INTAS grant No and the ACI project ACI-NIM Braids and Knots. The fourth author was partially supported by a grant from the National University of Singapore. The first and third authors were supported in part by a joint CNRS-NSF grant No References [1] S. Andreadakis,On automorphisms of free groups and free nilpotent groups, Proc. London Math Soc. (3) 15 (1965), [2] K. Brown, Cohomology of Groups, Graduate Texts in Mathematics 87, Springer-Verlag, [3] D. Cohen, F. R. Cohen and S. Prassidis, Centralizers of Lie Algebras Associated to the Descending Central Series of Certain Poly-Free Groups, math.gr/ [4] F. R. Cohen, and J. Pakianathan, Notes on automorphism groups, on Pakianathan s website. [5] B. Farb, The Johnson homomorphism for Aut(F n ), 2003 (in preparation). [6] M. Falk, and R. Randell, The lower central series of a fiber type arrangement, Invent. Math., 82 (1985), [7] R. Fenn; R. Rimányi; C. Rourke, The braid-permutation group, Topology 36 (1997), no. 1, [8] S. Galatius, to appear. [9] R. Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math.Soc., 10 (1997), [10] Y. Ihara, Galois group and some arithmetic functions, Proceedings of the International Congress of mathematicians, Kyoto, 1990, Springer (1991), [11] T. Kohno, Série de Poincaré-Koszul associée aux groupes de tresses pure, Invent. Math., 82 (1985), [12] C. Jensen, J. McCammond, and J. Meier, The integral cohomology of the group of loops, to appear. [13] D. Johnson, An abelian quotient of the mapping class group I g, Math. Ann. 249 (1980), [14] N. Kawazumi, Cohomological aspects of Magnus expansions, to appear.

20 BASIS-CONJUGATING AUTOMORPHISMS OF A FREE GROUP AND ASSOCIATED LIE ALGEBRAS20 [15] S. Krstić, J. McCool, The non-finite presentability of IA(F 3 ) and GL 2 (Z[t, t 1 ]), Invent. Math. 129 (1997), [16] W. Magnus, Über n-dimensionale Gittertransformationen, Acta Math., Vol. 64 (1934), [17] W. Magnus, A. Karass, D, Solitar, Combinatorial Group Theory, Wiley (1966). [18] J. McCool, On basis-conjugating automorphisms of free groups, Canadian J. Math., vol. 38,12(1986), [19] S. Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke J. Math.,70(1993), [20] J. Nielsen, Über die Isomorphismen unendlicher Gruppen ohne Relation, (German) Math. Ann. 79 (1918), no. 3, [21] A. Pettet, The Johnson homomorphism and the second cohomology of IA n, AGT. [22] T. Sakasai, The Johnson homomorphism and the third rational cohomology group of the Torelli group, to appear. [23] T. Satoh, The abelianization of the congruence IA-automorphism group of a free group, to appear. [24] V. V. Vershinin, On homological properties of singular braids, Trans. Amer. Math. Soc. 350 (1998), no. 6, Department of Mathematics, University of Rochester, Rochester, NY U.S.A. address: cohf@math.rochester.edu Department of Mathematics, University of Rochester, Rochester, NY address: jonpak@math.rochester.edu Département des Sciences Mathématiques, Université Montpellier II, Place Eugéne Bataillon, Montpellier cedex 5, France address: vershini@math.univ-montp2.fr Sobolev Institute of Mathematics, Novosibirsk, , Russia address: versh@math.nsc.ru Department of Mathematics, National University of Singapore, Singapore , Republic of Singapore, address: matwuj@nus.edu.sg