Group and Lie algebra filtrations and homotopy groups of spheres
aa r X i v : . [ m a t h . G R ] M a r HOMOTOPY GROUPS OF SPHERES AND DIMENSIONQUOTIENTS
LAURENT BARTHOLDI AND ROMAN MIKHAILOV
To the memory of John R. Stallings, 1935–2008
Abstract.
We solve the “dimension problem” for groups by exhibiting, forevery prime p , a group with p -torsion in one of its dimension quotients.This p -torsion derives from the Serre element of order p in the homotopygroup π p ( S ).We also obtain an analogous result in the context of Lie algebras. Introduction
This paper solves the classical dimension subgroup problem with the help of ho-motopy groups of spheres: a deep topological phenomenon is translated to algebraicform and used in pure group theory.Consider a group G naturally embedded in its integral group ring Z G . Forthe natural filtrations ( γ n ( G )) of G by its lower central series and ( ̟ n ) of Z G bypowers of its augmentation ideal, we have an induced map G/γ n ( G ) → Z G/̟ n ,whose injectivity is known as the dimension problem for G .Set δ n ( G ) := G ∩ (1 + ̟ n ) = ker( G → Z G/̟ n ), the n th dimension subgroup ;then γ n ( G ) ≤ δ n ( G ), and the dimension problem asks to understand the dimensionquotient δ n ( G ) /γ n ( G ). If it is trivial for all n , one says that G has the dimensionproperty .The study of dimension subgroups was initiated in 1935 by Wilhelm Magnusin [18]; he showed that the dimension property holds for free groups, see [19, 34]. Ifone replaces the ring Z by a field, then there is an elegant, purely group-theoreticaldescription of the corresponding dimension subgroup, depending only on the field’scharacteristic. However, the “universal” case Z is still not well understood. For asmall subset of the literature we refer to [9, 10, 20, 25, 33]. Our main result is: Main Theorem (See Theorem 1.1) . For every prime p , there exists a group G one of whose dimension quotients δ n ( G ) /γ n ( G ) contains Z/p as a subgroup.
It was claimed on numerous occasions [5, 17] that δ n ( G ) = γ n ( G ) holds for all n and all groups G . It is relatively easy to prove δ n ( G ) = γ n ( G ) for n ≤
3, but acounterexample was found by Ilya Rips [26] when n = 4.The quotients δ n ( G ) /γ n ( G ) are abelian, and Jan Sjogren proved in [31] thatthey have finite exponent, bounded by a function of n only: there exists an explicit Date : March 24, 2020.The first author is supported by the “@raction” grant ANR-14-ACHN-0018-01.The second author is supported by the grant of the Government of the Russian Federationfor the state support of scientific research carried out under the supervision of leading scientists,agreement 14.W03.31.0030 dated 15.02.2018. s ( n ) ∈ N (roughly ( n !) n ) with δ n ( G ) s ( n ) ⊆ γ n ( G ) for all groups G . Inder BirPassi [24] gave s (4) = 2.This can be improved in the case of metabelian groups: then Narain Guptaproved in [11] that s ( n ) may be chosen to be a power of 2, so γ n = δ n if G is ametabelian p -group.He claimed that the same bound holds for all groups (a proof is published in [13])and even that it may be improved to s ( n ) = 2, see [12]. However his argumentscontain many unintelligible parts, and we shall see that his claims cannot hold.Our main result is the construction, for every prime p , of a group G and anindex n such that δ n ( G ) /γ n ( G ) contains p -torsion; see Theorem 1.1 below. ThusSjogren’s function cannot be bounded, or even constrained to a finite collection ofprimes.An analogous question may be asked for Lie rings; namely, Lie algebras over Z . Every Lie ring A embeds in its universal enveloping algebra U ( A ), which alsoadmits an augmentation ideal. The dimension subrings are defined analogously by δ n ( A ) = A ∩ ̟ n , see [1]. Again δ n ( A ) = γ n ( A ) when n ≤
3, and there is a Liering A with δ n ( A ) /γ n ( A ) = Z /
2. Sjogren’s bound also holds for Lie rings [30], andmany details are simpler in the category of Lie rings.Even though we are not aware of any direct construction of a group from aLie ring or vice versa that preserves dimension quotients, it often happens thata presentation involving only powers and commutators, which may therefore beinterpreted either as group or Lie algebra presentation, yields isomorphic dimensionquotients.To give a quick taste of dimension quotients in Lie rings, we reproduce firstan example due to Pierre Cartier of a Lie algebra over a commutative ring k not embedding in its universal envelope [4]: consider k = F [ x , x , x ] / ( x , x , x ), and A = h e , e , e | x e + x e + x e = 0 i qua k -Lie algebra . Then α := x x [ e , e ] + x x [ e , e ] + x x [ e , e ] is non-trivial in A , but in anyassociative algebra it maps to ( x e + x e + x e ) = 0.Rips’ example, or rather its Lie algebra variant [1, Theorem 4.7], is of a similarspirit. In k = Z one can of course not choose x i nilpotent; but one may choose x i a large power of 2 and impose relations that guarantee that elements with large2-valuation are mapped far in the lower central series: set x i = 2 i and consider A = h e , e , e , · · · | i +2 e i ∈ γ for all i ∈ { , , } ,x j x k e i ± x i x k e j ∈ k +2 γ + γ for all { i, j, k } = { , , }i (1)with the element α = P ≤ i We write iterated commutatorsas left-normed: [ x , x , . . . , x d ] = [[ · · · [ x , x ] , . . . ] , x d ]. In a group or Lie algebrapresentation, we introduce the following notation: for d ∈ N , when we write agenerator x ( d ) i of degree d we mean a list of generators x i, , . . . , x i,d ; when x i appearsin a relator, it is a shorthand for the iterated commutator x i := [ x i, , . . . , x i,d ] ofthe generators x i, , . . . , x i,d . Theorem 1.1. For every prime p there are integers ℓ , n and c , . . . , c p − suchthat, in the following group G : h x ( ℓ )0 , . . . , x ( ℓ )2 p − , y ( ℓ + c )0 , . . . , y ( ℓ + c p − )2 p − | x · · · x p − = 1 , x p ci i = y i for i = 0 , . . . , p − i there is an element of order p in δ n ( G ) /γ n ( G ) . Theorem 1.2. For every prime p there are integers n and c , . . . , c p − such that,in the following Lie algebra A over Z : h x , . . . , x p − , y (1+ c )0 , . . . , y (1+ c p − )2 p − | x + · · · + x p − = 0 , p c i x i = y i for i = 0 , . . . , p − i there is an element of order p in δ n ( A ) /γ n ( A ) . The constants c i are explicit, if a bit unwieldy: c i = (2 p − p − (2 p − i ispossible. We have determined tighter values for p = 2 and p = 3, see § π p ( S ) contain an element of p -torsion due to Jean-Pierre Serre [29]. On the otherhand, π p ( S ) may be expressed as a quotient of normal subgroups in a free group F = h x , . . . , x p − | x · · · x p − = 1 i , following Wu [35]. We write the p -torsionelement as a free group element, and derive some of its properties. In particular, itis a product e α p of commutators of weight 2 p , it does not belong to the symmetriccommutator of the normal subgroups h x i F , . . . , h x p − i F , but e α p − x − Z F, . . . , ( x p − − Z F . The relations x p ci i = y i allow the p c + ··· + c p − th power of e α p to be “pushed down” the dimensionand lower central series; but only symmetric products/commutators (namely, thoseinvolving each h x i i F a single time) may be pushed down maximally. Thus themaximal depth achievable by e α p in the lower central series is strictly less that thedepth achievable by e α p − p = 2 and p = 3 to obtain smaller examples, in particular for p = 2 we obtainstraightforward constructions, for arbitrary n ≥ 4, of Lie algebras in which δ n /γ n contains 2-torsion, and for p = 3 we obtain a Lie algebra and a group in which δ /γ contains 3-torsion. These examples have be checked using computer algebraprograms.Stallings already recognized the value of homological arguments (in particularthe Curtis spectral sequence) towards studying dimension quotients [32, p. 117], ina programme carried out by Sjogren [31]. To the extend of our knowledge, however,Theorems 1.1 and 1.2 are the first instances of the solution of an open problem in LAURENT BARTHOLDI AND ROMAN MIKHAILOV abstract algebra by using classical homotopy theory, to wit the homotopy groupsof spheres. 2. Homotopy groups of spheres We give in this section an explicit generator of the p -torsion in π p ( S ) discoveredby Serre, using the group-theoretic formulation of this homotopy group. It will beused for the construction of the required element in Theorems 1.2 and 1.1.2.1. Groups. Fix an integer n ≥ F = h x , . . . , x n | x · · · x n i be a freegroup of rank n . Consider its normal subgroups R i := h x i i F for i = 0 , . . . , n. Note that F is the fundamental group of a 2-sphere with n + 1 punctures, and R i contains the conjugacy class of a loop around the i th puncture; the operation offilling-in the i th puncture induces the map F → F/R i on fundamental groups.Denote by Σ n +1 the symmetric group on { , . . . , n } , and define the symmetriccommutator product of the above subgroups by[ R , . . . , R n ] Σ := Y ρ ∈ Σ n +1 [ R ρ (0) , . . . , R ρ ( n ) ] . Here and below the iterated commutators are assumed to be left-normalized, namely[ R , R , R ] = [[ R , R ] , R ] etc.We view the circle S as a simplicial set. Milnor’s F construction produces agroup complex, having in degree n a free group on the degree- n objects of S subjectto a single relation ( s n ( ∗ ) = 1) and the same boundaries and degeneneracies as S .According to a formula due to Jie Wu [7, 35], considered in the standard basis ofMilnor’s F [ S ]-construction, homotopy groups of the sphere S can be presentedin the following way: π n +1 ( S ) ≃ R ∩ · · · ∩ R n [ R , . . . , R n ] Σ . Consider now for i = 0 , . . . , n the ideals r i := ( R i − Z [ F ] in the free group ring Z [ F ], and their symmetric product( r , . . . , r n ) Σ := X ρ ∈ Σ n +1 r ρ (0) · · · r ρ ( n ) which is also an ideal in Z [ F ]. Proposition 2.1. For n ≥ we have R ∩ · · · ∩ R n ≤ F ∩ (1 + ( r , · · · , r n ) Σ ) whenconsidered in Z [ F ] .Proof. It is shown in [21] that the quotient r ∩···∩ r n ( r ,..., r n ) Σ can be viewed as the n thhomotopy group of the simplicial abelian group Z [ F [ S ]], and the map F → Z [ F ]given by f f − R ∩ · · · ∩ R n [ R , . . . , R n ] Σ r ∩ · · · ∩ r n ( r , . . . , r n ) Σ π n +1 ( S ) = π n (Ω S ) H n (Ω S ) . OMOTOPY GROUPS OF SPHERES AND DIMENSION QUOTIENTS 5 The lower map is the n th Hurewicz homomorphism for the loop space Ω S . Sinceall homotopy groups π n (Ω S ) are finite for n ≥ 3, but all homology groups H n (Ω S )are infinite cyclic ( H ∗ (Ω S ) is the tensor algebra generated by the homology of S in dimension one), we conclude that, for n ≥ 3, the map in the above diagram iszero. (cid:3) Lie algebras. One obtains an analogous picture in the case of Lie algebrasover Z . The homotopy groups of the simplicial Lie algebra L [ S ] := M i γ i ( F [ S ]) /γ i +1 ( F [ S ])are equal to the direct sum of terms in rows of the E -term of the Curtis spectralsequence E i,j := π j (cid:0) γ i ( F [ S ]) /γ i +1 ( F [ S ]) (cid:1) = ⇒ π j +1 ( S ) . The mod- p -lower central series spectral sequence is well-studied, see for example,the foundational paper [3]. The integral case which we consider here has similarproperties, see [2, 14]. Here we will only need elementary properties of this spectralsequence and will consider essentially the (pre)image of the Serre elements.Observe that the E -page of the above spectral sequence consists of derivedfunctors L j in the sense of Dold-Puppe, applied to Lie functors: if L i denotes the i th Lie functor in the category of abelian groups, then π j (cid:0) γ i ( F [ S ]) /γ i +1 ( F [ S ]) (cid:1) = L j L i ( Z , . Recall the definition of derived functors. Let B be an abelian group, and let F bean endofunctor on the category of abelian groups. For every i, n ≥ F in the sense of Dold-Puppe [6] are defined by L i F ( B, n ) = π i ( F KP ∗ [ n ])where P ∗ → B is a projective resolution of B , and K is the Dold-Kan trans-form, inverse to the Moore normalization functor from simplicial abelian groupsto chain complexes. We denote by L F ( B, n ) the object F K ( P ∗ [ n ]) in the ho-motopy category of simplicial abelian groups determined by F K ( P ∗ [ n ]), so that L i F ( B, n ) = π i ( L F ( B, n )).Consider a free Lie algebra L over Z with generators x , . . . , x n and relation x + · · · + x n = 0, and the Lie ideals I i := h x i i L for i = 0 , . . . , n. Define their symmetric product by[ I , . . . , I n ] Σ = X ρ ∈ Σ n +1 [ I ρ (0) , . . . , I ρ ( n ) ] . The same arguments as in the group case imply the Lie analog of Wu’s formula: I ∩ · · · ∩ I n [ I , . . . , I n ] Σ ≃ M i ≥ E i,n = M i ≥ L n L i ( Z , . Consider the universal enveloping algebra U ( L ), the corresponding ideals i i := x i U ( L ) in U ( L ), and their symmetric product:( i , . . . , i n ) Σ = X ρ ∈ Σ n +1 i ρ (0) · · · i ρ ( n ) . LAURENT BARTHOLDI AND ROMAN MIKHAILOV Proposition 2.2. For n ≥ we have I ∩· · ·∩ I n ≤ L ∩ ( i , . . . , i n ) Σ when consideredin the universal enveloping algebra.Proof. Similarly to the group case, the natural map L → U ( L ) induces I ∩ · · · ∩ I n [ I , . . . , I n ] Σ i ∩ · · · ∩ i n ( i , . . . , i n ) Σ M i ≥ E i,n H n ( U ( L [ S ])) . By [27] the E i,j -terms of the lower central series spectral sequence for S are finitefor all j ≥ 3, while the universal enveloping simplicial algebra U ( L [ S ]) has infinitecyclic homology groups in all dimensions. It follows that the map is 0. (cid:3) Homotopy groups of S . Let p be a prime. In this subsection we describeexplicitly a copy of Z /p in π p ( S ) due to Serre [29], by computing its (pre)image α p in the E -term of the lower central spectral sequence associated to F [ S ]. Thereis a single ( Z /p )-term in dimension 2 p − p-torsion (cid:18) I ∩ · · · ∩ I p − [ I , . . . , I p − ] Σ (cid:19) = L p − L p ( Z , 1) = Z /p, and α p will be a generator of this subgroup. Theorem 2.3. Let x i for i = 0 , . . . , p − be free generators of a free Lie algebra,and consider the following element α p = X ρ ∈ Σ p − a p − -shuffle ρ (1) <ρ (3) < ··· <ρ (2 p − ( − ρ [[ x ρ (0) , x p − ] , [ x ρ (1) , x p − ] , [ x ρ (2) , x ρ (3) ] , . . . , [ x ρ (2 p − , x ρ (2 p − ]]; the sum is taken over all permutations ( ρ (0) , . . . , ρ (2 p − ∈ Σ p − satisfying ρ (0) < ρ (1) , . . . , ρ (2 p − < ρ (2 p − as well as ρ (1) < ρ (3) < · · · < ρ (2 p − .Then α p represents a generator of the p -torsion in L p − L p ( Z , .Proof. Consider the free abelian simplicial group K ( Z , σ in degree 2, and its other generators may be chosen to be all iterated degeneraciesof σ . We will use the dual notation for generators: for k > K ( Z , k is generated by ordered sequences of two elements( i i ) := s k − · · · c s i · · · c s i · · · s ( σ )with 0 ≤ i < i < k . For example, K ( Z , has generators(0 1) := s s s ( σ ) , (0 2) := s s s ( σ ) , (0 3) := s s s ( σ ) , (0 4) := s s s ( σ ) , (1 2) := s s s ( σ ) , (1 3) := s s s ( σ ) , (1 4) := s s s ( σ ) , (2 3) := s s s ( σ ) , (2 4) := s s s ( σ ) , (3 4) := s s s ( σ ) . For n ≥ 1, define the functor J n as the metabelianization of the n th Lie functor L n . For a group A , there is a natural epimorphism L p ( A ) ։ J p ( A )with kernel generated by Lie brackets of the form [[ ∗ , ∗ ] , [ ∗ , ∗ ]]. The elements of J p can also be written as linear combinations of Lie brackets, namely as elements of OMOTOPY GROUPS OF SPHERES AND DIMENSION QUOTIENTS 7 the Lie functor L p , but there is additional rule which holds in J p but not hold in L p in general: [ a , a , . . . , a p ] = [ a , a , a ρ (3) , . . . , a ρ ( p ) ]for arbitrary a i and permutation ( ρ (3) , . . . , ρ ( p )) of { , . . . , p } . For p = 3, thefunctors L and J are equal.For n ≥ 1, denote by S n the n th symmetric power functor S n : Abelian groups → Abelian groups . For a free abelian group A , there is a natural short exact sequence [27, Proposition3.2](2) 0 → J n ( A ) → S n − ( A ) ⊗ A → S n ( A ) → , where the left-hand map is given by(3) [ b , . . . , b n ] b b . . . b n ⊗ b − b b . . . b n ⊗ b for b i ∈ A. Applying the functors J p ֒ → S p − ⊗ id ։ S p to the simplicial abelian group K ( Z , n ), and taking the homotopy groups, we get the long exact sequence π pn (cid:0) S p − K ( Z , n ) ⊗ K ( Z , n ) (cid:1) → L pn S p ( Z , n ) → L pn − J p ( Z , n ) →→ π pn − (cid:0) S p − K ( Z , n ) ⊗ K ( Z , n ) (cid:1) . It follows from [6, p. 307] that the above sequence has the following form: π pn (cid:0) S p − K ( Z , n ) ⊗ K ( Z , n ) (cid:1) L pn S p ( Z , n ) L pn − J p ( Z , n ) Z Z Z /p. p By [27, Proposition 4.7], the natural epimorphism L p ։ J p gives a natural iso-morphism of derived functors L pn − L p ( Z , n ) ≃ −→ L pn − J p ( Z , n ) ≃ Z /p. Let us first find a simplicial generator of L p S p ( Z , S p ֒ → ⊗ p induces anisomorphism of derived functors L p S p ( Z , → L p ⊗ p ( Z , . A simplicial generator of L p ⊗ p ( Z , 2) can be given by the Eilenberg-Zilber shuffle-product theorem. Using interchangeably the notation ρ ( i ) and ρ i , this is the element X ρ ∈ Σ p a 2 p -shuffle ( − ρ ( ρ ρ ) ⊗ ( ρ ρ ) ⊗ · · · ⊗ ( ρ p − ρ p − ) . It follows immediately from the definition of 2 p -shuffles that the symmetric groupΣ p , acting by permutation on blocks { i, i + 1 } of size 2, acts on 2 p -shuffles. Agenerator of L p S p ( Z , 2) can be chosen by keeping only a single element per Σ p -orbit, and replacing tensor products by symmetric products: β := X ρ ∈ Σ p a 2 p -shuffle ρ (1) <ρ (3) < ··· <ρ (2 p − ( − ρ ( ρ ρ ) · ( ρ ρ ) · · · ( ρ p − ρ p − ) . LAURENT BARTHOLDI AND ROMAN MIKHAILOV The conditions imply ρ (2 p − 1) = 2 p − 1. For example, for p = 3 we get the element(0 1)(2 3)(4 5) − (0 1)(2 4)(3 5) + (0 1)(3 4)(2 5) − (0 2)(3 4)(1 5) − (0 2)(1 3)(4 5)+ (0 3)(2 4)(1 5) − (0 3)(1 4)(2 5) + (1 2)(0 3)(4 5) − (2 3)(0 4)(1 5) − (1 2)(0 4)(3 5)+ (1 2)(3 4)(0 5) − (1 3)(2 4)(0 5) + (2 3)(1 4)(0 5) + (0 2)(1 4)(3 5) + (1 3)(0 4)(2 5) . Now we lift the element from S p K ( Z , p to ( S p − K ( Z , ⊗ K ( Z , p in a stan-dard way:˜ β := X ρ ∈ Σ p a 2 p -shuffle ρ (1) <ρ (3) < ··· <ρ (2 p − ( − ρ ( ρ ρ ) · · · ( ρ p − ρ p − ) ⊗ ( ρ p − ρ p − ) . Observe that we have d j ( i i ) = ( i i ) if i < j, ( i i − 1) if i < j ≤ i , ( i − i − 1) if j ≤ i with the understanding that ( i i ) = 0, that we use the same notation ( i i ) forelements of varying degree, and that d (0 i ) = 0 and d j ( i i ) = 0 if deg( i i ) = j = i − 1. Thus e.g. d (0 4) = d (0 4) = 0 and d (0 4) = d (0 4) = d (0 4) = d (0 4) = (0 3) while d (2 3) = d (2 3) = d (2 3) = (1 2) and d (2 3) = 0 and d (2 3) = d (2 3) = (2 3).Clearly d ( ˜ β ) = d p − ( ˜ β ) = 0. If j < p − 2, then we express ˜ β as a sum overall possible values of r := ρ (2 p − 2) (remembering ρ (2 p − 1) = 2 p − 1) and obtain d j ( ˜ β ) = p − X r =0 ( − r (cid:0) d j ( · · · ) ⊗ ( r p − 1) + ( · · · ) ⊗ d j ( r p − (cid:1) . Now the sum in ( · · · ) is a symmetric product similar to β , but with p − p factors, so ( · · · ) is exact. The second terms telescope, so we get d j ( ˜ β ) = 0 when j < p − 2. However, ˜ β is not a cycle in S p − ( Z , ⊗ K ( Z , d p − ( ˜ β ) isnot zero: we compute d p − ( ˜ β ) = X ρ ∈ Σ p a 2 p -shuffle ρ (1) < ··· <ρ (2 p − p − >ρ (2 p − ( − ρ ( ρ ρ ) . . . ( ρ p − ρ p − ) ⊗ ( ρ p − p − . We use the long exact sequence associated with (2) to obtain a cycle in J p ( Z , p − .The ascending 2 p -shuffles ( ρ (0) , . . . , ρ (2 p − p − -shuffles ( ρ (0) , ρ (1) , . . . , ρ (2 p − , ρ (2 p − , ρ (2 p − , ρ (2 p − ρ (0) , ρ (1) , . . . , ρ (2 p − , ρ (2 p − , ρ (2 p − , ρ (2 p − ρ (2 p − < ρ (2 p − 4) or not, and in all cases completed by the values (2 p − , p − X ρ ∈ Σ p − a 2 p − -shuffle ρ (1) < ··· <ρ (2 p − ( − ρ [( ρ p − p − , ( ρ p − p − , ( ρ ρ ) , . . . , ( ρ p − ρ p − )] . We now consider the simplicial map K ( Z , → L K ( Z , σ [ s ( σ ′ ) , s ( σ ′ )] , where σ ′ is the generator of K ( Z , ; it is a homotopy equivalence of complexes. OMOTOPY GROUPS OF SPHERES AND DIMENSION QUOTIENTS 9 The abelian group K ( Z , k is k -dimensional, with generators x i := s k · · · b s i · · · s ( σ ′ )for all 0 ≤ i < k , and we have ( i i ) [ x i , x i ] under this homotopy equivalence.Thus L ∗ ( Z , p − is a free Lie algebra on 2 p − L p K ( Z , → L p ◦ L K ( Z , → L p K ( Z , X ρ ∈ Σ p − a 2 p − -shuffle ρ (1) < ··· <ρ (2 p − ( − ρ [[ x ρ (2 p − , x p − ] , [ x ρ (2 p − , x p − ] , [ x ρ (0) , x ρ (1) ] , . . . , [ x ρ (2 p − , x ρ (2 p − ]] . Up to sign and renumbering, this is exactly our element α p . (cid:3) Note that we considered, in the beginning of this section, a free Lie algebraof rank 2 p − p generators x , . . . , x p − subject to the relation P x i =0. Any choice of 2 p − p generators yields a free Lie algebra on2 p − α p . The point being made is that every suchexpression involves one of the generators (here x p − ) twice, and omits another(here x p − ).We summarize as follows the properties of the element α p that will be useful tous: Proposition 2.4. For every prime p there is an element e α p in the free group h x , . . . , x p − | x · · · x p − = 1 i with the properties: • e α p − ∈ ( r , . . . , r p − ) Σ ; • e α p [ R , . . . , R p − ] Σ ; • e α pp ∈ [ R , . . . , R p − ] Σ .Furthermore, e α p − ∈ ([ r , r ] , . . . , [ r p − , r p − ]) Σ , namely in the sum of all p -foldassociative products of brackets of r i in any of the (2 p )! orderings.Proof. The first claim follows from Proposition 2.1, since α p represents an elementof π p ( S ). The second claim holds because this element is non-trivial in π p ( S ).The third claim holds because it has order p in π p ( S ). The last claim followsfrom general facts: L i L n ( Z , 1) = 0 for odd n , and L i L n ( Z , 1) = L i L n ( Z , (cid:3) The same statement holds for Lie algebras; we omit the proof. Proposition 2.5. For every prime p there is an element α p in the free Lie algebra h x , . . . , x p − | x + · · · + x p − = 0 i with the properties: • α p ∈ ( I , . . . , I p − ) Σ ; • α p [ I , . . . , I p − ] Σ ; • pα p ∈ [ I , . . . , I p − ] Σ .Furthermore, α p ∈ ([ R , R ] , . . . , [ R p − , R p − ]) Σ . (cid:3) Example 2.6. Here is an explicit generator of π ( S ) = Z / 2. If we consider p = 2in Theorem 2.3, we have only one [2]-shuffle and the element α is [[ x , x ] , [ x , x ]].Reintroducing x = − x − x − x , we can easily check that α ∈ ( I , I , I , I ) Σ : α := [[ x , x ] , [ x , x ]]= [ x , x ] · [ x , x ] + [ x , x ] · [ x , x ] − [ x , x ] · [ x , x ] . (5) Applying to it the Dynkin idempotent u · v [ u, v ] gives then 2 α ∈ [ I , I , I , I ] Σ .It is only slightly harder to write a generator of π ( S ) in the language of groups.We may lift α to e α ∈ F , the free group h x , x , x , x | x · · · x i , as e α = [[ x , x ] , [ x x , x ]] , since then the Hall-Witt identities give e α = [[ x , x ] , [ x − , x ] x − ] = [[ x , x ] , [ x , x ] x [ x , x ]] =[[ x , x ] , [ x , x ]] · [[ x , x ] , [[ x , x ] , x x ]] so e α ∈ R ∩ · · · ∩ R . We have thus pro-duced a non-trivial cycle e α ∈ ( R ∩ R ∩ R ∩ R ) / [ R , R , R , R ] Σ . Example 2.7. Here is a generator of the 3-torsion in π ( S ). For p = 3, we havesix [2 , , , , 3) with sign = 1 , (0 , , , 3) with sign = − , (0 , , , 2) with sign = 1 , (2 , , , 1) with sign = 1 , (1 , , , 2) with sign = − , (1 , , , 3) with sign = 1 . The element α representing 3-torsion in π ( S ) is α := [[ x , x ] , [ x , x ] , [ x , x ]] − [[ x , x ] , [ x , x ] , [ x , x ]]+ [[ x , x ] , [ x , x ] , [ x , x ]] + [[ x , x ] , [ x , x ] , [ x , x ]] − [[ x , x ] , [ x , x ] , [ x , x ]] + [[ x , x ] , [ x , x ] , [ x , x ]] . It may be expressed as a sum of 30 associative products of the form ± [ x a , x b ] · [ x c , x d ] · [ x e , x f ] with { a, b, c, d, e, f } = { , , , , , } .Again it is possible (but now with considerably more effort) to lift α to agenerator of π ( S ) in terms of free groups. We return to the notation of simplicialfree groups: we consider the free group F = h z , . . . , z i and normal subgroups R = h z i F , R i = h z − i − z i i F for i ∈ { , . . . , } and R = h z i F . In other words,we set z i := x · · · x i . Here is a lift of α to F which defines a simplicial cycle,i.e., which lies in the intersection R ∩ · · · ∩ R : it is the product of the followingfourteen elements e α = [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] − · [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] · [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] − · [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] · [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] · [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] − · [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] · [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] − · [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] · [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] − · [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] − · [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] − · [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] · [[ z , z ] , [ z , z ] , [ z , z ] [ z ,z ] ] . One can directly check that e α defines a simplicial cycle and that modulo theseventh term of the lower central series it represents exactly the element α . OMOTOPY GROUPS OF SPHERES AND DIMENSION QUOTIENTS 11 Remark 2.8. We have π ( S ) = Z / × Z / 4, and it is also possible to give anexplicit generator of the 4-torsion. In the same notation as above, it is e α = [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] · [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] · [[[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] , [[ z , z ] , [ z , z ]]] · [[[ z , z ] , [ z , z ]] , [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]]] · [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] · [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] · [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] · [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] · [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] · [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] · [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] · [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] · [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] · [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] · [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] . Note that, contrary to prime torsion, this element will not (at least, easily) leadto 4-torsion in a dimension quotient. Indeed e α is, up to the symmetric commutator[ R , . . . , R ] Σ , equal to(6) [[[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] , [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]]]and this element does not have the form “one letter repeats, all the others appearonce”.Here is a brief explanation of the origin of e α . The elements of the E -page of thespectral sequence can be coded by generators of lambda-algebra. Serre elements,which we study, correspond to the elements λ . The element e α corresponds to λ λ of the lambda-algebra. The E ∞∗ , column of S has the following non-trivial terms: E ∞ , = Z / λ λ ), E ∞ , = Z / λ for p = 3), E ∞ , = Z / λ ). The 4-torsion in π ( S ) is glued from two terms in E ∞ : λ and λ λ . A representative of λ is the bracket (6), see for example [7]. To show that e α represents the 4-torsion, we observe first that it is a cycle, namely that it lies in R ∩ · · · ∩ R , and secondly we show that, modulo γ γ , it represents the element λ λ of the simplicial Lie algebra, given as a sum[[[ z , z ] , [ z , z ]] + [[ z , z ] , [ z , z ]] + [[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]] + [[ z , z ] , [ z , z ]] + [[ z , z ] , [ z , z ]]]= − [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] − [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]]+ [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]]+ [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]]+ [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]]+ [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]]+ [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]]+ [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]]+ [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]]+ [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]]+ [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]]+ [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]]+ [[[ z , z ] , [ z , z ]] , [[ z , z ] , [ z , z ]]] . Proof of Theorems 1.2 and 1.1 We will prove our main theorems by using the Lie algebra element α p contructedin Theorem 2.3, respectively a lift e α p to a free group. Recall that α p belongs to I ∩· · ·∩ I p − \ [ I , . . . , I p − ] Σ , and e α p belongs to R ∩· · ·∩ R p − \ [ R , . . . , R p − ] Σ . Lemma 3.1. For every d there are integers c , . . . , c d such that n c + · · · + n d c d = c + · · · + c d has only n = · · · = n d = 1 as solutions in N .Proof. Set c i = d d +1 − d i . Then, looking at the equation n c + · · · + n d c d = c + · · · + c d modulo d , one sees n ≡ d ). Furthermore, c > c > · · · > c d so n ≤ ( c + · · · + c d ) /c < d + 1 and therefore n = 1. Continue then with theremaining terms, n ( c /d ) + · · · + n d ( c d /d ) = ( c + · · · + c d ) /d which has the sameform but one term fewer. (cid:3) (It seems that the c i can be improved to c i = 2 d +1 − i but no better.)3.1. The Lie algebra case. Determine weights c i from Lemma 3.1 with d = 2 p − ω := p c + ··· + c p − α p of the Lie ring A constructed in thestatement of Theorem 1.2, using the element α p given in Proposition 2.5. Set n = 2 p + c + · · · + c p − .Firstly, ω ∈ δ n ( A ). Indeed, α p is a linear combination of associative products of x , . . . , x p − in some order (with possible repetition), so ω is a linear combinationof associative products of p c x , . . . , p c p − x p − , namely of y , . . . , y p − of totalweight (1 + c ) + · · · + (1 + c p − ) = n .Secondly, pω ∈ γ n ( A ). Indeed pα p is a linear combination of left-normed bracketsof x , . . . , x p − in some order (with possible repetition), so by the same argument pω is a linear combination of brackets of y , . . . , y p − of total weight n .We finally show ω γ n ( A ). Note that our element ω is in the subring generatedby h x , . . . , x p − i which is free of rank 2 p − 1. We perform calculations in the free OMOTOPY GROUPS OF SPHERES AND DIMENSION QUOTIENTS 13 Lie ring L generated by S := { x , . . . , x p − , y , . . . , y p − } . It is a graded ring, inwhich we give weight 1 to the each generator x i and weight 1 + c i to generator y i .By the last claim of Proposition 2.5, we may present ω as a linear combinationof brackets of length 2 p , and furthermore assume each such bracket to be of theform [ z , . . . , z p − ] with each z i ∈ S .Suppose that after applying the relations p c i x i = y i the element ω can be made tolie in γ n ( A ). Since ω belongs to a free, graded ring, each such bracket [ z , . . . , z p − ]must have weight at least n . Write z i ∈ { x σ ( i ) , y σ ( i ) } for a map σ : { , . . . , p − } →{ , . . . , p − } , and let n i be the number of occurrences of y i among the z σ − ( i ) .We know that n i of the original x i letters in the summand of ω under consider-ation have been substituted into y i . Each such substitution divides the coefficientby p c i and brings the bracket down c i steps along the central series. In orderto make [ z , . . . , z p − ] of degree at least n , we therefore need 2 p + n c + · · · + n p − c p − ≥ n ; and at the same time n c + · · · + n p − c p − ≤ c + · · · + c p − ,since this is the number of available p -powers. Combining these two equations,we get n c + · · · + n p − c p − = c + · · · + c p − , and then from Lemma 3.1we get ( n , . . . , n p − ) = (1 , . . . , x i occur once in[ z , . . . , z p − ] and σ is a permutation; so α p ∈ [ I , . . . , I p − ] Σ , a contradiction.3.2. The group case. The difference with the Lie algebra case is that we giveweight ℓ to each generator x i and weight ℓ + c i to each generator y i . Let us write ℓ := c + · · · + c p − and n := 2 pℓ + ℓ , and note that n = ( ℓ + c ) + · · · + ( ℓ + c p − )is the total weight of the y i generators. We construct the element ω as above as e α p ℓ p , with e α p given by Proposition 2.4. To show that we have ω ∈ δ n ( G ), we usethe following classical identity: if k is an integer and g ∈ δ s ( G ) then g k − g − k − k ( g − 1) + (cid:18) k (cid:19) ( g − + · · · ≡ k ( g − 1) mod ̟ s . We thus have ω − e α p ℓ p − ≡ p ℓ ( e α p − 1) mod ̟ n since e α p ∈ δ pℓ ( G ) and 2 · pℓ ≥ n ∈ p ℓ (( x − , . . . , ( x p − − Σ by Proposition 2.4. We then continue with a typical summand p ℓ [( x σ (0) − , . . . , ( x σ (2 p − − p c σ (0) ( x σ (0) − , . . . , p c σ (2 p − ( x σ (2 p − − ≡ [ x p cσ (0) σ (0) − , . . . , x p cσ (2 p − σ (2 p − − 1] mod ̟ n since p c i ( x i − ≡ x p ci i − ̟ ℓ ), all other terms of the commutator belongto ̟ ℓ , and (2 p − ℓ + 2 ℓ = n = [ y σ (0) − , . . . , y σ (2 p − − ∈ ̟ ℓ + σ (0)+ ··· + ℓ + σ (2 p − = ̟ n . The other claims — that ω p ∈ γ n ( G ) and ω γ n ( G ) — are exactly the same as inthe Lie algebra case and need not be repeated. Examples The examples presented above yielded with relatively little computational effortLie algebras and groups with p -torsion in their dimension quotients. Using morecomputational resources, we were able to find p -torsion in lower degree for p = 2and p = 3.A general simplification (see Propositions 2.4 and 2.5) is that we can start byan element α p of degree p an not 2 p , by writing generators x ij in place of [ x i , x j ].Indeed all the computations that express α p as an symmetrized associative productactually take place in L p L ( Z p ) ⊂ L p ( Z p ). In fact, this amounts to working inMilnor’s simplicial construction F [ S ], whose geometric realization is Ω S , and inits Lie analog L [ S ]. Observe that, for higher spheres S n with n > 3, as well as ofMoore spaces, there is a description of homotopy groups as centers of explicitelydefined finitely generated groups [22]. However, these groups are not as easilydefined as in the case of S , when we quotient by the symmetric commutator. Anapplication of homotopy groups of higher spheres in group-theoretical questionssuch as the problems considered here is obviously possible, however, it will involvemore complicated constructions.4.1. p = 2 . The construction given in the proof of Theorem 1.2 has generators x , x , x and x := − x − x − x . The element ω belongs to δ ( A ) \ γ ( A ). Itis possible to be a little bit more economical, by keeping the nilpotency degrees ofthe y i more under control: consider A = h x , x , x , x , y (1)0 , y (2)1 , y (3)2 , y (4)3 | x + x + x + x = 0 , x = 2 y , x = y , x = y , x = y i and the element ω = [[ x , x ] , [ x , x ]]. In that Lie algebra, we have ω ∈ δ ( A ) \ γ ( A ) and 2 ω ∈ γ ( A ). This can been checked by computer using the program lienq by Csaba Schneider [15, 28].Rewriting [ x i , x j ] as x ij and simplifying somewhat, we obtain A = h x , x , x , x , x , x , y (2)01 , y (3)02 , y (4)03 , y (4)12 , y (5)13 , y (6)23 | x + x + x = − x + x + x = − x − x + x = 0 , x = y , x = y , x = y , x = y , x = y , x = y i with 2 [ x , x ] ∈ δ ( A ) \ γ ( A ).There is in fact substantial flexibility in this example: suppose that the element ω is 2 d [ x x ] and we want to show that it belongs to δ ℓ ( A ) \ γ ℓ ( A ). Using theassociative rewriting [ x , x ] = x x + x x − x x from (5), we will have ω ∈ δ ℓ ( A ) as soon as A has relations of the form 2 d x = 2 a y and 2 a x = y with deg( y ) + deg( y ) = ℓ ; and similarly for the other generators. The condition ω γ ℓ ( A ) can be checked by a direct calculation, e.g. using lienq . For instance,we have (eliminating the x i ) A = h x , x , x , y (2)01 , y (3)02 , y (4)03 , y (5)12 , y (6)13 , y (7)23 | x = y , x = y , x = y , ( − x − x ) = 2 y , ( x − x ) = 2 y , ( x + x ) = 2 y i OMOTOPY GROUPS OF SPHERES AND DIMENSION QUOTIENTS 15 with 2 [ x , x ] ∈ δ ( A ) \ γ ( A ).Another modification of the examples often leads to lower values of ℓ : we mayreplace the variables x ij by 2 b ij x ij for well-chosen b ij . This amounts, essentially,to letting the x ij have distinct, negative degrees. For instance, replacing x ij by2 i + j x ij in the previous example, we get A = h x , x , x , y (2)01 , y (2)02 , y (2)03 , y (2)12 , y (2)13 , y (2)23 | x = y , x = y , x = y , ( − x − x ) = 2 y , ( x − x ) = 2 y , ( x + 2 x ) = 2 y i with ω := 2 [ x , x ] ∈ δ ( A ) \ γ ( A ). We have ω ≡ [ x , x ] + 2 [ x , x ] +2 [ x , x ] modulo γ ( A ); note the similarity with Rips’s original example (1). Wemay also choose ℓ ≥ 2, let y i have degree ℓ for all i and in this manner obtainexamples with 2-torsion in δ ℓ +2 ( A ) /γ ℓ +2 ( A ).It is straightforward to convert the example above into a group: it will be G = h x , x , x , y (2)01 , y (2)02 , y (2)03 , y (2)12 , y (2)13 , y (2)23 | x = y , x = y , x = y ,x − x − = y , x x − = y , x x = y i and the element ω = [ e , e ] [ e , e ] [ e , e ] belongs to δ ( G ) \ γ ( G ). Increasingthe degree of the x ij and y ij leads, for every ℓ ≥ 4, to a group G with 2-torsion in δ ℓ ( G ) /γ ℓ ( G ).4.2. p = 3 . As in the p = 2 example, we may construct a Lie algebra with generators x ij as follows: A = h x ij , y ( i + j +1) ij for 0 ≤ i < j ≤ ,x + x + x + x + x = 0 , − x + x + x + x + x = 0 , − x − x + x + x + x = 0 , − x − x − x + x + x = 0 , − x − x − x − x + x = 0 , i + j x ij = y ij for 0 ≤ i < j ≤ i and the element ω = 3 ([ x , x , x ] − [ x , x , x ]+[ x , x , x ]+[ x , x , x ] − [ x , x , x ] + [ x , x , x ]) which belongs to δ ( A ) \ γ ( A ).Again there is substantial flexibility in this example: the degrees of the y ij may beadjusted, and the last relations may be changed to 3 a ij x ij = 3 c ij y ij for well-chosen a ij , c ij . The variables x ij themselves may be replaced by 3 b ij x ij for well-chosen b ij . Finally, some extra linear conditions may be imposed on the variables, such as x = x = x = x = x = 0. After some experimentation, we arrived at thefollowing reasonably small example: A = h e , e , e , e , y (2) i for i ∈ { , . . . , } , y (3) ij for 0 ≤ i < j ≤ | i e i = y i , − i e j + 3 − j e i = 3 − i − j y ij for ( i, j ) ∈ { (0 , , (0 , , (1 , , (2 , }i (7)with ω = 3 [ e , e , e ]. Proposition 4.1. For the Lie ring A defined in (7) we have ω ∈ δ ( A ) \ γ ( A ) and ω ∈ γ ( A ) .Proof. Expanding ω associatively, we get ω = 3 ( e e e − e e e − e e e + e e e ) . We may rewrite it as ω = − e (3 e + 3 e ) e − e (3 e + 3 e ) e + e (3 e + 3 e ) e + e (3 e + 3 e ) e + (3 e + 3 e ) e e + e e (3 e + 3 e ) − (3 e + 3 e ) e e − e e (3 e + 3 e ) . Each of the summands belongs to ̟ ( A ): they are all products of e k , e ℓ and3 − i e j + 3 − j e i for some { i, j, k, ℓ } = { , , , } . The binomial term equals3 − i − j y ij = 3 k +2 ℓ y ij , so the summand is the product of 3 k e k , 3 ℓ e ℓ and y ij ,namely the product of y k , y ℓ , y ij , of respective degrees 2 , , ω does not belong to γ ( A ) but that 3 ω does, we compute nilpotentquotients of A . We did the calculation using two different programs: lienq byCsaba Schneider and LieRing [16] for GAP [8] by Willem de Graaf and SerenaCical`o. (cid:3) In the next subsection, we give a direct proof that the associated group has3-torsion in δ /γ .4.3. A small, finite 3-group G with δ ( G ) = γ ( G ). We consider the group G given by the presentation (7), namely G = h e , e , e , e , y (2) i for i ∈ { , . . . , } , y (3) ij for 0 ≤ i < j ≤ | e i i = y i , e − i j e − j i = y − i − j ij for ( i, j ) ∈ { (0 , , (0 , , (1 , , (2 , }i (8)with ω = [ e , e , e ] . Proposition 4.2. In the group defined by (8) we have ω ∈ δ ( G ) .Proof. We will use the following well-known identity, which holds for any element x ∈ G and d ≥ x d − d X k =1 (cid:18) dk (cid:19) ( x − k . We compute modulo ̟ ( G ), and from now on write ≡ to mean equivalence modulo ̟ ( G ). We get ω − ≡ ([ e , e , e ] − 1) since [ e , e , e ] ∈ γ ( G )= 3 [ e , e ] e − (cid:0) ([ e , e ] − e − − ( e − e , e ] − (cid:1) = 3 [ e , e ] e − (cid:0) e − e − (( e − e − − ( e − e − e − − ( e − e − e − (( e − e − − ( e − e − (cid:1) ; OMOTOPY GROUPS OF SPHERES AND DIMENSION QUOTIENTS 17 and since 3 is divisible by the product of exponent of e , e , e modulo γ ( G ), ≡ (cid:0) ( e − e − e − − ( e − e − e − − ( e − e − e − 1) + ( e − e − e − (cid:1) . Let us write f i := e i − i ∈ { , , , } . Then, as in the proof of Proposition 4.1,we have ω − ≡ ( f f f − f f f − f f f + f f f )= − f (3 f + 3 f ) f − f (3 f + 3 f ) f + f (3 f + 3 f ) f + f (3 f + 3 f ) f + (3 f + 3 f ) f f + f f (3 f + 3 f ) − (3 f + 3 f ) f f − f f (3 f + 3 f ) . Next, using (9) we have e − j i e − i j − e − j i − 1) + ( e − i j − 1) + ( e − j i − e − i j − − j f i + (cid:18) − j (cid:19) f i + (cid:18) − j (cid:19) f i + · · · + 3 − i f j + (cid:18) − i (cid:19) f j + (cid:18) − i (cid:19) f j + · · · + 3 − i − j f i f j + · · · = y − i − j ij − − i − j ( y ij − 1) + (cid:18) − i − j (cid:19) ( y ij − + · · · Again using (9), the relations e i i = y i imply 3 i f i ∈ ̟ . Now 3 − i − j divides (cid:0) − j (cid:1) / i , so (cid:0) − j (cid:1) f i ∈ − i − j ̟ . Similarly, 3 − i − j divides (cid:0) − j (cid:1) and (cid:0) − j (cid:1) so all terms with a binomial co¨efficient belong to 3 − i − j ̟ + ̟ . Thesame holds for all terms in the last two rows. We therefore have3 − j f i + 3 − i f j ∈ − i − j ̟ + ̟ . We note 3 − i − j = 3 k +2 ℓ whenever { i, j, k, ℓ } = { , , , } . Returning to com-putations modulo ̟ , we consider a typical summand f k (3 − j f i + 3 − i f j ) f ℓ inour decomposition of ω − 1. We write 3 − j f i + 3 − i f j = 3 − i − j u + v with u ∈ ̟ , v ∈ ̟ to get f k (3 − j f i + 3 − i f j ) f ℓ = f k (3 k +2 ℓ u + v ) f ℓ = (3 k f k ) u (3 ℓ f ℓ ) + f k vf ℓ where each summand belongs to ̟ . (cid:3) Proposition 4.3. In the group defined by (8) , the element ω defined above doesnot belong to γ ( G ) , but its cube does.Proof. The proof is computer-assisted. It suffices to exhibit a quotient G of G inwhich the image of ω does not belong to γ ( G ) but its cube does, and we shallexhibit a finite 3-group as quotient. To make the computations more manageable, we replace the generators y i and y ij by generators z , . . . , z , and impose the choices y = [ z , z ] , y = [ z , z ] , y = [ z , z ] , y = [ z , z ] ,y = 1 , y = [ z , z , z ] , y = [ z , z , z ] , y = [ z , z , z ] . In this manner, we obtain an 8-generated group h e , . . . , e , z , . . . , z i . We nextimpose extra commutation relations: [ e , z ] , [ e , z ] , [ e , z ] , [ e , z ] , [ e , z ].We compute a basis of left-normed commutators of length at most 6 in thatgroup; notice that ω may be expressed as [ z , z , z , z , z , e ] , and impose extrarelations making γ cyclic and central.The resulting finite presentation may be fed to the program pq by EamonnO’Brien [23], to compute the maximal quotient of 3-class 17. This is a group oforder 3 , and can (barely) be loaded in the computer algebra system GAP [8] soas to check (for safety) that the relations of G hold, and that the element ω has anon-trivial image in it.Finally, the order of the group may be reduced by iteratively quotienting bymaximal subgroups of the centre that do not contain ω . 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(2001), no. 3, 489–513, DOI 10.1017/S030500410100487X.MR1816806 D´epartement de Math´ematiques et Applications, ´Ecole Normale Sup´erieure, Paris E-mail address : [email protected] Laboratory of Modern Algebra and Applications, St. Petersburg State University,14th Line, 29b, Saint Petersburg, 199178 Russia and St. Petersburg Department ofSteklov Mathematical Institute E-mail address ::