Braids, inner automorphisms and the Andreadakis problem
BBraids, inner automorphismsand the Andreadakis problem
Jacques
Darn´e
June 4, 2020
Abstract
In this paper, we generalize the tools that were introduced in [Dar19b] in orderto study the Andreadakis problem for subgroups of IA n . In particular, we studythe behaviour of the Andreadakis problem when we add inner automorphisms toa subgroup of IA n . We notably use this to show that the Andreadakis equalityholds for the pure braid group on n strands modulo its center acting on the freegroup F n − , that is, for the (pure, based) mapping class group of the n -puncturedsphere acting on its fundamental group. Introduction
In his 1962 PhD. thesis [And65], Andreadakis studied two filtrations on the groupof automorphisms of the free group F n . More precisely, these were filtrations on thesubgroup IA n of Aut( F n ) consisting of automorphisms acting trivially on the abelian-ization F abn ∼ = Z n . These filtrations were: • The lower central series IA n = Γ ( IA n ) ⊇ Γ ( IA n ) ⊇ · · · . • The filtration IA n = A ⊇ A ⊇ · · · now known as the Andreadakis filtration .He showed that there are inclusions A i ⊇ Γ i ( IA n ), and he conjectured that these wereequalities. This question became known as the Andreadakis conjecture , and it turnedout to be a very difficult one, which is still nowdays shrouded with mystery. Theinterest of this question (and its difficulty) lies notably in the fact that the definitionsof these filtrations are very different in nature, and thus what we understand aboutthem is too. For instance, it is very easy to test whether a given element lies in some A j , but there is no known efficient procedure for testing whether the same elementbelongs to Γ j or not. On the other hand, producing elements of Γ j is not difficult,but we do not know any good recipe for producing elements of A j (apart, of course,from the one producing elements of Γ j ). So far, these difficulties have been overcomeonly for very small values of n (cid:62) G of IA n . Namely, if G is such asubgroup, then we can ask whether the inclusions Γ i ( G ) ⊆ G ∩ A i are equalities. Theanswer is obviously negative for some subgroups which are embedded in IA n in a wrongway (take for instance a cyclic subgroup of Γ ( IA n )). But if G is nicely embedded in IA n , we can hope that these filtrations on G are equal, in which case we say that thesubgroup G of IA n satisfies the Andreadakis equality .1 a r X i v : . [ m a t h . A T ] J un nteresting examples of such subgroups include the pure braid group, embedded in IA n via Artin’s action on the free group, for which the Andreadakis equality was shownin [Dar19b]. Another example is the pure welded braid group, also acting on the freegroup via
Artin’s action. This subgroup is also the group of (pure) basis-conjugatingautomorphisms of F n , also known as the McCool group P Σ n . We still do not knowwhether the Andreadakis equality holds for this one. A version of the latter problemup to homotopy has been considered in [Dar19a]. In all these cases, the Andreadakisequality can be seen as a comparison statement between different kinds of invariantsof elements of the group (see [Dar19a]). Main results
Automorphisms of free groups
Consider the 2-sphere S with n marked points, and let us choose a basepoint differentfrom the marked points. The group of isotopy classes of orientation-preserving self-homeomorphisms of S fixing the base point and each marked point is isomorphicto the quotient P ∗ n of the pure braid group on n strands by its center. This groupsacts canonically on the fundamental group of the sphere with the n marked pointsremoved, which is free on n − § P ∗ n identifies with a subgroup of IA n − .Our main goal is the following theorem : Theorem 5.7.
The subgroup P ∗ n +1 of IA n satisfies the Andreadakis equality. It turns out that the subgroup P ∗ n +1 of IA n is generated by pure braids, togetherwith inner automorphisms of F n . Our strategy of proof relies on this fact. Indeed,our key result (Th. 4.4 and Cor. 5.1) is a theorem allowing us to decide, for a givensubgroup K of IA n , whether K · Inn( F n ) satisfies the Andreadakis equality when K does. The reader should note that such a result allowing us to pass from a subgroupto another one is quite exceptional : the Andreadakis problem (and, more generally,the lower central series) does not usually behave well when passing to smaller or biggergroups. In this regard, although P ∗ n +1 strictly contains P n and is contained in P Σ n ,we can hardly see our present result as a step further in the study of the Andreadakisproblem for P Σ n . It has to be considered as a new interesting example in itself, andalso as a good pretext to develop new tools in the study of the Andreadakis problem.We give two other applications of our key result (Th. 4.4 and Cor. 5.1). By apply-ing it to the subgroup K = IA + n of triangular automorphisms, we get the Andreadakisequality for a somewhat bigger subgroup (Th. 5.3). Moreover, by applying it to thesubgroup K = P Σ + n , we recover the result obtained (with some mistakes) in [Ibr19]showing that the Andreadakis equality holds for the group of partial inner automor-phisms defined and studied in [BN16]. Inner automorphisms
The question of whether the Andreadakis equality holds can be asked, more generally,for subgroups of IA G , where G is any group, and IA G is the group of automorphismsof G acting trivially on its abelianization G ab . In order to get our results about sub-groups of IA n , we need to show that the Andreadakis equality holds for the subgroup2nn( F n ) of inner automorphisms of F n . Although this result is fairly easy to get (see § L ( G ) denotedthe graded Lie ring obtained from the lower central series of G ): Corollary 2.2.
Let G be a group. The Andreadakis equality holds for Inn( G ) if andonly if every central element of L ( G ) is the class of some central element of G . We give several examples and counter-example, the main one being the case of thegroup G = P n of pure braids. In order to study it, we give a calculation of the centerof P n which can readily be adapted to a calculation of the center of its Lie ring, andwe prove: Theorem 3.10.
The subgroup
Inn( P n ) of IA ( P n ) ( ⊂ Aut( P n ) ) satisfies the Andrea-dakis equality. Outline of the paper:
The first section is devoted to recalling the needed definitionsand results from the theory of group filtrations, in particular filtrations on automor-phism groups and on braid groups. In §
2, we study the Andreadakis problem for innerautomorphisms, which turns out to be very much related to calculations of centers ofgroups and of their Lie rings. We then turn to the calculation of the center of thepure braid group, of which we give a version that generalizes easily to a calculation ofthe center of the associated Lie ring; this allows us to solve the Andreadakis problemfor inner automorphisms of the pure braid group ( § §
4, we prove our keyresult (Th. 4.4), giving a criterion for deducing the Andreadakis equality for a productof subgroups from the Andreadakis equality for these subgroups. Finally, the last sec-tion is devoted to applications to subgroups of automorphisms of free groups, namelytriangular automorphisms, triangular basis-conjugating automorphism, and the purebraid group on n strands modulo its center acting on F n − .In addition to our main results, we put in an appendix a comparison between theDrinfeld-Kohno Lie ring and the Lie ring of so-called braid-like derivations of the freeLie ring, boiling down to some rank calculations. In a second appendix, we write downa new proof of the faithfulness of the action of the braid group on n strands moduloits center on F n − , which involves less calculations than the ones in the literature, andwe gather some useful group-theoretic results. Contents
Introduction 11 Reminders 4
Inner automorphisms 10 IA n P ∗ n P ∗ n on F n − . . . . . . . . . . . . . . . . . . . . . . 277.B Residually nilpotent groups . . . . . . . . . . . . . . . . . . . . . . . . 287.C Splitting by the center . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 We recall here some of the basics of the general theory of (strongly central) groupfiltrations and the Andreadakis problem. Details may be found in [Dar19c, Dar19b].
Since the only filtrations we consider in the present paper are strongly central ones (inthe sense of [Dar19c]), we adopt Serre’s convention [Ser06] and we simply call them filtrations . The systematic study of such filtrations was initiated by Lazard [Laz54],who called them N -series . Notation 1.1.
Let G be a group. If x, y ∈ G , we denote by [ x, y ] their commutator xyx − y − , and we use the usual exponential notations x y = y − xy and y x = yxy − forconjugation in G . If A, B ⊆ G are subsets of G , we denote by [ A, B ] the subgroupgenerated by commutators [ a, b ] with a ∈ A and b ∈ B .4 efinition 1.2. A filtration G ∗ on a group G is a sequence of nested subgroups G = G ⊇ G ⊇ G ⊇ · · · satisfying: ∀ i, j (cid:62) , [ G i , G j ] ⊆ G i + j . If G ∗ and H ∗ are two filtrations on the same group G , we write G ∗ ⊆ H ∗ if G i ⊆ H i for all i . This relation, though not strictly speaking an inclusion relation, will be treatedlike one. The minimal filtration (for the inclusion relation) on a given group G is itslower central series Γ ∗ ( G ), defined as usual by Γ ( G ) = G and Γ i +1 ( G ) = [ G, Γ i ( G )]when i (cid:62)
1. Recall that G is called nilpotent (resp. residually nilpotent ) if Γ i ( G ) = { } for some i (resp. if (cid:84) Γ i ( G ) = { } ). Since the lower central series is the minimalfiltration on G , if G i = { } for some i (resp. if (cid:84) G i = { } ) for any filtration G ∗ on G = G , then G is nilpotent (resp. residually nilpotent). Convention 1.3.
Let G be a group endowed with a filtration G ∗ . Let g be an elementof G . If there is an integer d such that g ∈ G d − G d +1 , it is obviously unique. Wethen call d the degree of g with respect to G ∗ , and we denote by g the class of g in thequotient G d /G d +1 . If such a d does not exist (that is, if g ∈ (cid:84) G i ), we say that g hasdegree ∞ and we put g = 0.Recall that to a filtration G ∗ we can associate a graded Lie ring (that is, a gradedLie algebra over Z ): Proposition-definition 1.4. If G ∗ is a group filtration, the graded abelian group L ( G ∗ ) := (cid:76) G i /G i +1 become a Lie ring when endowed with the Lie bracket [ − , − ] induced by commutators in G . Precisely, with the above convention, this bracket isdefined by: ∀ x, y ∈ G, [ x, y ] := [ x, y ] . Convention 1.5.
When no filtration is specified on a group G , it is implied that G isendowed with its lower central series Γ ∗ ( G ). In particular, we denote L (Γ ∗ ( G )) simplyby L ( G ).Let us recall that the construction of the associated Lie ring from a filtration is afunctor from the obvious category of filtrations and filtration-preversing group mor-phisms to the category of graded Lie rings. This functor L : G ∗ (cid:55)→ L ( G ∗ ) will bereferred to as the Lie functor . We will use the fact that it is exact . Precisely, if G ∗ , H ∗ and K ∗ are group filtrations, a short exact sequence of filtrations is a short exactsequence H (cid:44) → G (cid:16) K of groups, such that the morphisms are filtration-preserving,and such that they induce a short exact sequence of groups H i (cid:44) → G i (cid:16) K i for allinteger i (cid:62)
1. As a consequence of the nine Lemma in the category of groups, the Liefunctor sends a short exact sequence of filtrations to a short exact sequence of gradedLie rings [Dar19c, Prop. 1.24].
Before introducing the Andreadakis problem, we introduce one of our main tools inits study, which is the
Johnson morphism associated to an action of a filtration onanother one. Recall that the categorical notion of an action of an object on anotherone in a protomodular category leads to the following definition:5 efinition 1.6.
Let K ∗ and H ∗ be filtrations on groups K = K and H = H . An action of K ∗ on H ∗ is a group action of K on H by automorphisms such that: ∀ i, j (cid:62) , [ K i , H j ] ⊆ H i + j , where commutators are computed in H (cid:111) K , that is : ∀ k ∈ K, ∀ h ∈ H, [ k, h ] = ( k · h ) h − . Given a group action of K on H , the above conditions are exactly the ones requiredfor the sequence of subgroups ( H i (cid:111) K i ) i (cid:62) to be a filtration on H (cid:111) K , denoted by H ∗ (cid:111) K ∗ . Then L ( H ∗ (cid:111) K ∗ ) is a semi-direct product of L ( H ∗ ) and L ( K ∗ ), encoding anaction of L ( K ∗ ) on L ( H ∗ ) by derivations, described explicitly by the formula: ∀ k ∈ K, ∀ h ∈ H, k · h = ( k · h ) h − . This action can also be seen as a morphism from L ( K ∗ ) to the Lie ring Der( L ( H ∗ )) ofderivations of L ( H ∗ ), called the Johnson morphism associated to the action of K ∗ on H ∗ : τ : L ( K ∗ ) −→ Der( L ( H ∗ )) k (cid:55)−→ (cid:16) h (cid:55)→ ( k · h ) h − (cid:17) . Actions of filtrations can be obtained from group actions via the following:
Proposition-definition 1.7.
Let K be a group acting on another group H by auto-morphisms, and let H ∗ be a filtration on H = H . Then, there is a greatest one amongfiltrations K ∗ on a subgroup K of K such that the action of K on H induces an actionof K ∗ on H ∗ . This filtration is denoted by A ∗ ( K, H ∗ ) and is defined by: A j ( K, H ∗ ) = { k ∈ K | ∀ i (cid:62) , [ k, H i ] ⊆ H i + j } . The filtration A ∗ (Aut( H ) , H ∗ ) , denoted simply by A ∗ ( H ∗ ) , is called the Andreadakisfiltration associated to H ∗ . When furthermore H ∗ = Γ ∗ H , we denote it by A ∗ ( H ) ,and we call it the Andreadakis filtration associated to H . The filtration A ∗ ( H ) is afiltration on A ( H ) =: IA H , which is the group of automorphisms of H acting triviallyon H ab . The following fact was the initial motivation for introducing such filtrations [Kal50]:
Fact 1.8.
In the above setting, if K acts faithfully on H , and if H i = { } for some i (resp. if (cid:84) H i = { } ), then A i − ( K, H ∗ ) = { } (resp. (cid:84) A i ( K, H ∗ ) = { } ). It particu-lar, A ( K, H ∗ ) must then be nilpotent (resp. residually nilpotent). Remark that if the morphism a : K → Aut( H ) represents the action of K on H ,then A ∗ ( K, H ∗ ) = a − ( A ∗ ( H ∗ )). We now briefly recall the description of lower central series of semi-direct product,which will be an important ingredient in the proof of our key theorem (Th. 4.4). Thisdescription is based on the following general construction:6 roposition-definition 1.9. [Dar19b, Def. 3.3].
Let G be a group, and let H be anormal subgroup of G . We define a filtration Γ G ∗ ( H ) on H by: (cid:40) Γ G ( H ) := H, Γ Gk +1 := [ G, Γ Gk ( H )] . If a group K acts on another group H by automorphisms, then we denote Γ H (cid:111) K ∗ ( H )only by Γ K ∗ ( H ). It is the minimal filtration on H which is acted upon by Γ ∗ ( K ) via the action of K on H . Proposition 1.10. [Dar19b, Prop. 3.4].
Let K be a group acting on another group H by automorphisms. Then: ∀ i (cid:62) , Γ i ( H (cid:111) K ) = Γ Ki ( H ) (cid:111) Γ i ( K ) . Moreover, under the right conditions, Γ K ∗ ( H ) is in fact equal to Γ ∗ ( H ): Proposition 1.11. [Dar19b, Prop. 3.5].
Let K be a group acting on another group H by automorphisms. The following conditions are equivalent: • The action of K on H ab is trivial. • [ K, H ] ⊆ [ H, H ] = Γ ( H ) . • The action of K on H induces an action of Γ ∗ K on Γ ∗ H . • Γ K ∗ ( H ) = Γ ∗ ( H ) . • ∀ i (cid:62) , Γ i ( H (cid:111) K ) = Γ i ( H ) (cid:111) Γ i ( K ) . • L ( H (cid:111) K ) ∼ = L ( H ) (cid:111) L ( K ) . • ( H (cid:111) K ) ab ∼ = H ab × K ab . When these conditions are satisfied, we say that the semi-direct product H (cid:111) K is an almost-direct one. Let G ∗ be a group filtration. The Andreadakis problem is concerned with the compar-ison of two filtrations on A ( G ∗ ), namely the Andreadakis filtration A ∗ ( G ∗ ) and thefiltration Γ ∗ ( A ( G ∗ )). The latter is the minimal filtration on the group A ( G ∗ ), hence A ∗ ( G ∗ ) contains Γ ∗ ( A ( G ∗ )). Now, if K is a subgroup of A ( G ∗ ) (that is, K acts on G by automorphisms preserving the G i , and the induced action on L ( G ∗ ) is trivial),then we can restrict these fitrations to K , and the filtrations so obtained must containΓ ∗ ( K ), which is the minimal filtration on K . Definition 1.12.
Let G ∗ be a group filtration and K be a subgroup of A ( G ∗ ). We saythat K satisfies the Andreadakis equality with respect to G ∗ if the following inclusionsare equalities: Γ ∗ ( K ) ⊆ K ∩ Γ ∗ ( A ( G ∗ )) ⊆ K ∩ A ∗ ( G ∗ ) . If K is a subgroup of IA G , we simply say that K satisfies the Andreadakis equality when it does with respect to Γ ∗ ( G ). Remark 1.13.
This definition can be generalized to groups acting on G (in a possiblynon-faithful way). Precisely, let K act on a group G via a morphism a : K → Aut( G ),and let G be endowed with a filtration G ∗ . Suppose that a ( K ) ⊆ A ( G ∗ ) (that is, K L ( G ∗ ) istrivial). Then we get inclusions of filtrations on K :Γ ∗ ( K ) ⊆ a − (Γ ∗ ( A ( G ∗ ))) ⊆ a − ( A ∗ ( G ∗ )) = A ∗ ( K, G ∗ ) . However, since Γ ∗ ( a ( K )) = a (Γ ∗ ( K )), these filtrations are equal if and only if a ( K )satisfies the Andreadakis equality with respect to G ∗ , so we can (and will) focus onsubgroups of Aut( G ) when studying the difference between such filtrations.The following result is deduced easily from the definitions, and will be our maintool in proving the Andreadakis equality for subgroups of IA G : Proposition 1.14. [Dar19c, Lem. 1.28].
Let K ∗ and H ∗ be group filtrations. TheJohnson morphism associated to a given action of K ∗ on H ∗ is injective if and only if K ∗ = A ∗ ( K , H ∗ ) . The Andreadakis problem for automorphisms of free groups:
The classicalsetting is the one when G = F n is the free group on n generators, and G ∗ is itslower central series. Then IA G , denoted by IA n , is the subgroup of Aut( F n ) made ofautomorphisms acting trivially on F abn ∼ = Z n . The Andreadakis filtration associatedto F n is a filtration on IA n simply denoted by A ∗ and referred to as the Andreadakisfiltration . Recall that F n is residually nilpotent, which implies that (cid:84) A i = { } , thus IA n is residually nilpotent (Fact 1.8). Since the Lie algebra of F n is the free Lie ring L n on n generators, the Johnson morphism associated to the action of A ∗ on Γ ∗ ( F n ),gives an injection (Prop. 1.14 above) τ : L ( A ∗ ) (cid:44) → Der( L n ) . We gather here the results we need about Artin’s braid groups and filtrations on them.Our main reference here is Birman’s book [Bir74]. The reader can also consult theoriginal papers of Artin [Art25, Art47].
We denote Artin’s braid group by B n and the subgroup of pure braids by P n . Recallthat B n is generated by σ , ..., σ n − and P n is generated by the A ij (for i < j ), whichare drawn as follows: σ i A ij = ( σ n − ··· σ i +1 ) σ i · · · · · · i − i + 1 i i + 2 · · · · · · · · · i j Convention 1.15.
It is often convenient to have A ij defined for all i, j (not onlyfor i < j ), using the formulas A ji = A ij and A ii = 1.8orgetting the ( n + 1)-th strand induces a projection P n +1 (cid:16) P n which is split(a section is given by adding a strand away from the other ones). The kernel of thisprojection identifies with the fundamental group of the plane with n punctures, whichis the free group F n . We thus get a decomposition into a semi-direct product: P n +1 ∼ = F n (cid:111) P n , which encodes an action of P n on F n via automorphisms. A basis of F n is given by theelements x i := A i,n +1 . A classical result of Artin [Bir74, Cor. 1.8.3 and Th. 1.9] says thatthis action is faithful, and its image is exactly the group Aut ∂C ( F n ) of automorphismspreserving the conjugacy class of each generator and fixing the following boundaryelement : · · · n + 1 ∂ n := x · · · x n = A ,n +1 · · · A n,n +1 . Since P n ∼ = Aut ∂C ( F n ), such automorphisms are also called braid automorphisms . Wewill often identify P n with Aut ∂C ( F n ) in the sequel.Remark that their are many possible choices of generators of P n , but the choicesthat we have made are coherent : they allow us to interpret the above split projec-tion P n +1 (cid:16) P n (defined by forgetting the ( n + 1)-th strand) as the projection fromAut ∂C ( F n +1 ) onto Aut ∂C ( F n ) induced by x n +1 (cid:55)→ β sends x n +1 to one of its conjugates, thus preserves the normal closure of x n +1 andinduces an automorphism of F n +1 /x n +1 ∼ = F n ). The section of this projection de-fined by adding a strand corresponds to extending canonically automorphisms of F n toautomorphisms of F n +1 fixing x n +1 . Remark 1.16.
The free subgroup of B n +1 generated by the A i,n +1 is in fact normalizednot only by P n but also by B n . The corresponding action of B n on F n (induced byconjugation in B n +1 ) is faithful, and its image is the group of automorphisms of F n permuting the conjugacy classes of the x i and fixing the boundary element. A braid acts on F abn ∼ = Z n via the associated permutation of the basis. As a conse-quence, the pure braid group P n acts trivially on F abn , thus is a subgroup of IA n . Since IA n is residually nilpotent, so is P n . The Lie ring associated to its lower central serieswas first determined rationnally in [Koh85], and it was shown not to have torsion in[FR87], where its ranks where computed. Details about the complete description over Z that we now recall may be found in the appendix of [Dar19b]. Definition 1.17.
The
Drinfeld-Kohno Lie ring is DK n := L ( P n ).9ince P n acts trivially on F abn , the semi-direct product P n +1 ∼ = F n (cid:111) P n is an almost-direct one, thus it induces a decomposition of the associated Lie rings: L ( P n +1 ) ∼ = L n (cid:111) L ( P n ) , where L n = L ( F n ) is the free Lie ring on n generators X i := t i,n +1 = A i,n +1 (for i (cid:62) n ).Thus, L ( P n ) decomposes as an iterated semi-direct product of free Lie rings. Fromthat, it is not difficult to get a presentation of this Lie ring: Proposition 1.18. [Dar19b, Prop A.3].
The Drinfeld-Kohno Lie ring DK n = L ( P n ) is generated by the t ij = A ij ( (cid:54) i, j (cid:54) n ), under the relations: t ij = t ji , t ii = 0 ∀ i, j, [ t ij , t ik + t kj ] = 0 ∀ i, j, k, [ t ij , t kl ] = 0 if { i, j } ∩ { k, l } = ∅ . Recall that the decomposition DK n +1 ∼ = L n (cid:111) DK n is encoded in the correspondingJohnson morphism τ : DK n → Der( L n ). The following result allows us to identify DK n with a Lie subring of the Lie ring of derivations of the free Lie ring: Theorem 1.19. [Dar19b, Th. 6.2].
The subgroup P n of IA n satistfies the Andreadakisequality. This means exactly that the corresponding Johnson morphism τ : DK n → Der( L n ) is injective. In the sequel, we identify DK n with τ ( DK n ). It is not difficult to see that DK n is in fact a Lie subring of the Lie ring Der ∂t ( L n ) of tangential derivations (derivationssending each X i to [ X i , w i ] for some w i ∈ L n ) vanishing on the boundary element ,defined by: ∂ n = X + · · · + X n = t ,n +1 + · · · + t n,n +1 ∈ L n ⊂ DK n +1 . Definition 1.20.
A derivation of the free Lie ring L n is called braid-like if it is anelement of Der ∂t ( L n ). It is called a braid derivation if it stands inside DK n .Not all braid-like derivations are braid derivations. The difference between the twoLie subrings of Der( L n ) is investigated in the appendix ( § DK n onto DF n − giving the above decomposition DK n ∼ = L n − (cid:111) DK n − can be seen as the restriction of the projection Der ∂t ( L n ) (cid:16) Der ∂t ( L n − ) induced by X n (cid:55)→ This first section is devoted to the study of the Andreadakis problem for inner auto-morphisms of a group G , with respect to any filtration G ∗ on the group. We prove ageneral criterion (Th. 2.1), involving a comparison between the center of G and thecenter of the Lie ring associated to G ∗ . Since most of our applications will be to thecase when G ∗ is the lower central series of G , we spell out the application of our generalcriterion to this case in Cor. 2.2. We then turn to examples, including the easy case ofthe free group. Our most prominent application will be to the pure braid group, whichwill be the goal of the next section. 10 .1 A general criterion Recall that for any element g of a group G , the inner automorphism c g associated to g is defined by c g ( x ) = g x (= gxg − ) for all x ∈ G . The map c : g (cid:55)→ c g is a groupmorphism whose image is the normal subgroup Inn( G ) of Aut( G ). The kernel of thismorphism is the set of elements g ∈ G such that for all x ∈ G , gxg − = x , which is thecenter Z ( G ) of G . As a consequence, Inn( G ) ∼ = G/ Z ( G ).The same story can be told for Lie algebras: a Lie algebra g acts on itself via theadjoint action, the image of this action ad : g → Der( g ) is the Lie algebra ad( g ) ofinner derivations, and the kernel of ad is the center z ( g ) of g , so that ad( g ) ∼ = g / z ( g ).We now explain how these two stories are related to the Andreadakis problem.Let G ∗ be a filtration on G = G , and let A ∗ ( G ∗ ) by the associated Andreadakisfiltration. The filtration G ∗ acts on itself via the adjoint action, which is induced bythe action of G on itself by inner automorphisms. The latter is represented by themorphism c : G → Aut( G ) described above, and the fact that it induces an action of G ∗ on itself is reflected in the fact that c sends G ∗ to A ∗ ( G ∗ ). Thus, if we considerthe image of G ∗ under the corestriction π : G (cid:16) Inn( G ) of c , we get an inclusion offiltration π ( G ∗ ) ⊆ A ∗ ( G ∗ ). The following Proposition gives a criterion for the inclusion π ( G ∗ ) ⊆ Inn( G ) ∩ A ∗ ( G ∗ ) to be an equality: Theorem 2.1.
Let G ∗ be a filtration on G = G . Its image in Inn( G ) coincides with Inn( G ) ∩ A ∗ ( G ∗ ) if and only if the inclusion z ( L ( G ∗ )) ⊇ L ( G ∗ ∩ Z ( G )) is an equality,that is, exactly when every central element of L ( G ∗ ) is the class of some central elementof G . If G ∗ is the lower central series of G , whose image in Inn( G ) is the lower centralseries of Inn( G ), then A ∗ (Γ ∗ G ) is the usual Andreadakis filtration on IA ( G ). Corollary 2.2.
For a group G , the following conditions are equivalent: • The Andreadakis equality holds for
Inn( G ) . • Every central element of L ( G ) is the class of some central element of G . • The canonical projection L ( G/ Z ( G )) (cid:16) L ( G ) / z ( L ( G )) = ad( L ( G )) is an iso-morphism. This introduces a motivation for solving the Andreadakis problem for inner au-tomorphisms of a group : it allows one to compute the Lie algebra of the quotient G/ Z ( G ). We will apply this later to the pure braid group P n ( § Proof of Cor. 2.2.
As a direct application of Theorem 2.1, the first assertion is equiv-alent to the inclusion z ( L ( G )) ⊇ L (Γ ∗ ( G ) ∩ Z ( G )) being an equality, which is clearlyequivalent to the second assertion. In order to see that it is also equivalent to the thirdcondition, let us consider the short exact sequence of filtrations :Γ ∗ ( G ) ∩ Z ( G ) Γ ∗ ( G ) Γ ∗ ( G/ Z ( G )) . By applying the Lie functor, we get an isomorphism: L ( G/ Z ( G )) ∼ = L ( G ) / L (Γ ∗ ( G ) ∩ Z ( G )) . As a consequence, the inclusion z ( L ( G )) ⊇ L (Γ ∗ ( G ) ∩ Z ( G )) induces a canonical pro-jection L ( G/ Z ( G )) (cid:16) L ( G ) / z ( L ( G )). The latter is an isomorphism if an only if theformer is an equality. 11 roof of Theorem 2.1. Let us denote by π ( G ∗ ) the image of G ∗ under π : G (cid:16) Inn( G ),and the inclusion π ( G ∗ ) ⊆ Inn( G ) ∩ A ∗ ( G ∗ ) by i . The latter induces a morphism i ∗ : L ( π ( G ∗ )) → L (Inn( G ) ∩ A ∗ ( G ∗ )), whose injectivity is equivalent to the two filtrationson Inn( G ) being equal. By definition of π and i , we have a commutative square: π ( G ∗ ) G ∗ Inn( G ) ∩ A ∗ ( G ∗ ) A ∗ ( G ∗ ) . i cπ By taking the associated graded, we get the left square in: L ( π ( G ∗ )) L ( G ∗ ) L (Inn( G ) ∩ A ∗ ( G ∗ )) L ( A ∗ ( G ∗ )) Der( L ( G ∗ )) . i c ad π τ The fact that the triangle on the right commutes can be seen via an abstract argument( c represents the adjoint action of G ∗ , hence c represents the adjoint action of L ( G ∗ )),or can be obtained via a direct calculation using the usual explicit description of theJohnson morphism : τ ( c g ) : x (cid:55)→ c g ( x ) x − = [ g, x ] = [ g, x ] = ad g ( x ) . (2.2.1)From this fact and the injectivity of the Johnson morphism, we deduce that the kernelof c is z ( L ( G ∗ )). The Lie subring L ( G ∗ ∩ Z ( G )), on the other hand, appears asthe kernel of π by applying the Lie functor to the following short exact sequence offiltrations: G ∗ ∩ Z ( G ) G ∗ π ( G ∗ ) . π Now, π induces a surjection from ker( c ) = ker( i π ) onto ker( i ) whose kernelis exactly ker( π ) (this can be seen as an application of the usual exact sequence0 → ker( v ) → ker( uv ) → ker( u ) → coker( v ) to u = i and v = π ). Thus we have:ker( i ) ∼ = z ( L ( G ∗ )) L ( G ∗ ∩ Z ( G )) , (2.2.2)whence the conclusion. Remark 2.3.
The isomorphism (2.2.2) gives more information than the statement ofTheorem 2.1, which is the case when this kernel is trivial. However, we will mainly usethe latter case in the sequel.
We now apply Corollary 2.2 in order to give examples of groups whose group of innerautomorphism satisfies (or not) the Andreadakis equality.
The first counter-examples we can give are groups withno center and an abelian Lie ring (that is, a Lie ring reduced to the abelianization),such as the symmetric group Σ n ( n (cid:62) n /A n ∼ = Z / he braid group : Another slightly more interesting counter-example in the braidgroup B n , whose classical generators σ , ..., σ n − are all conjugate. Its Lie algebra isreduced to its abelianization B abn ∼ = Z , generated by the common class σ of all σ i . Thecenter of B n is cyclic, generated by ξ n = ( σ · · · σ n − ) n . Thus its image in B abn ∼ = Z isnot equal to all of B abn , but to n ( n − Z ⊂ Z .Such examples are not very interesting to us, since filtrations on them do not containa lot of information (they contain only information about the abelianization). We nowdescribe a way of obtaining nilpotent (or residually nilpotent) counter-examples. Constructing counter-examples as semi-direct products :
Let G be a group,and α be an automorphism of G . Consider the semi-direct product G (cid:111) Z encoding theaction of Z on G through powers of α . This semi-direct product is an almost-directone if and only if α ∈ IA ( G ). Under this condition, Γ ∗ ( G (cid:111) Z ) = Γ ∗ ( G ) (cid:111) Γ ∗ ( Z ), sothat G (cid:111) Z is nilpotent (resp. residually nilpotent) whenever G is.Let us denote by t the generator of Z acting via α on G , and consider its class t in L ( G (cid:111) Z ). This element is central in L ( G (cid:111) Z ) = L ( G ) (cid:111) Z if and only if itsaction on L ( G ) is trivial. This happens exactly when α ∈ A ( G ). Indeed, for any class g ∈ L i ( G ), we have [ t, g ] = [ t, g ] = [ α, g ], and this is trivial in L i +1 ( G ) if and only if[ α, g ] ∈ Γ i +2 G . Thus, [ t, g ] = 0 for all g if and only if for all i (cid:62)
1, [ α, Γ i G ] ⊆ Γ i +2 G .When is t the class of a central element in G (cid:111) Z ? Lifts of t to G (cid:111) Z are elementsof the form gt with g ∈ Γ G . Using Proposition 3.1 below, we see that such an elementis central in G (cid:111) Z if and only if t acts on G via c g − . Thus, t is the class of a centralelement of G (cid:111) Z if and only if α is an inner automorphism c x ( x ∈ Γ G ), in whichcase ( x − , t ) is the only central element whose class in ( G (cid:111) Z ) is t .We conclude that this construction gives counter-examples whenever there existautomorphisms α in A which are not inner. Such automorphisms exist for the freegroup, or for the free nilpotent group of any class at least 3, giving counter-examplesof any nilpotency class greater than 3. Let G = F n be the free group on n generators, with n (cid:62)
2. It is centerless, so thatInn( F n ) ∼ = F n . For short, we will often denote Inn( F n ) only by F n . We can apply theabove machinery to the lower central series Γ ∗ ( F n ). Since the associated graded ring L ( F n ) is free, whence centerless. As a consequence, we deduce directly from Corollary2.2 the following : Corollary 2.4.
The subgroup
Inn( F n ) of IA n satisfies the Andreadakis equality. We do not need the full strength of the above theory of filtrations on inner auto-morphisms in order to get this result. In fact, it may be enlightening to write down adirect proof of this corollary.
Direct proof of Cor. 2.4. If w ∈ F n , suppose that c w ∈ A j , and let us show that w ∈ Γ j ( F n ). The hypothesis means that: ∀ x ∈ F n , c w ( x ) ≡ x (mod Γ j +1 ( F n )) . But c w ( x ) x − = [ w, x ], and [ w, x i ] ∈ Γ j +1 ( F n ) implies that the class w of w in L ( F n )either is of degree at least j or commutes with the generator x i . The latter is possibleonly if w ∈ Z x i (see the classical Cor. 3.6 recalled below), which can be true only forone value of i . Thus w must be of degree at least j , which means that w ∈ Γ j ( F n ).13e can give yet another proof, which looks more like a simple version of the abovetheory in this particular case, and enhances the fact (needed later) that the Lie ring ofInn( F n ) identifies with the ring of inner derivations of L n . Yet another proof of Cor. 2.4.
The Andreadakis equality is equivalent to the associ-ated Johnson morphism τ : L ( F n ) → Der( L n ) being injective (Prop. 1.14). Because ofthe direct calculation (2.2.1), this morphism can be identified with ad : L n → Der( L n ),whose kernel is the center of L n , which is trivial. Thus the result. We now turn to showing the Andreadakis equality for inner automorphisms of thepure braid group, a goal achived in Theorem 3.10. In order to do this, we recoverthe classical computation of the center of P n in a way that can be adapted easily to acomputation of the center of its Lie ring DK n . The following easy result, which we have already used once in 2.2.1, will be our maintool in computing the center of P n , and that of DK n : Proposition 3.1.
Let K be a group acting on another group H by automorphisms.Then the center of H (cid:111) K consists of elements hk (with h ∈ H and k ∈ K ) such that: • k is central in K , • h is a fixed point of the action of K on H (that is, K · h = { h } ), • k acts on H via c − h .Similarly, let k be a Lie algebra acting on another Lie algebra h by derivations. Thenthe center of h (cid:111) k consists of elements x + y (with x ∈ h and y ∈ k ) satisfying: • y is central in k , • x is a fixed point of the action of k on h (that is, k · x = { } ), • y acts on h via − ad( x ) .Proof. Let h ∈ H and k ∈ K . The condition hk ∈ Z ( H (cid:111) K ) means exactly thatfor all h (cid:48) ∈ H and all k (cid:48) ∈ K , we have hkh (cid:48) k (cid:48) = h (cid:48) k (cid:48) hk . For h (cid:48) = 1, this gives hkk (cid:48) = k (cid:48) hk = k (cid:48) hk (cid:48)− k (cid:48) k , from which we deduce h = k (cid:48) hk (cid:48)− and kk (cid:48) = k (cid:48) k . This mustbe true for all k (cid:48) ∈ K , whence the first two conditions. Suppose that these are satisfied.Then we write the condition for hk to belong to Z ( H (cid:111) K ) as hkh (cid:48) k − kk (cid:48) = h (cid:48) k (cid:48) hk (cid:48)− k (cid:48) k ,which becomes hkh (cid:48) k − = h (cid:48) h , that is k · h (cid:48) = h − h (cid:48) h (which has to hold for all h (cid:48) ∈ H ).The latter is exactly the third condition in our statement.The case of Lie algebras is quite similar. Let x ∈ h and y ∈ k . The element x + y belongs to z ( h (cid:111) k ) if and only if for all x (cid:48) ∈ h and y (cid:48) ∈ k , the bracket [ x + y, x (cid:48) + y (cid:48) ] =[ x, x (cid:48) ] + [ y, y (cid:48) ] + y · x (cid:48) − y (cid:48) · x is trivial. For x (cid:48) = 0, this gives [ y, y (cid:48) ] = 0 and y (cid:48) · x , forall y (cid:48) ∈ k , whence the first two conditions. Suppose that these are satisfied. Then thecondition for x + y to belong to z ( h (cid:111) k ) can be written as [ x, x (cid:48) ] + y · x (cid:48) = 0, that is, y · x (cid:48) = − [ x, x (cid:48) ] (which has to hold for all x (cid:48) ∈ h ).14 .2 The center of the pure braid group Consider the pure braid group P n on n generators. By applying Proposition 3.1 to thesemi-direct product decomposition P n ∼ = F n − (cid:111) P n − recalled in § Proposition 3.2.
The center of P n (whence of B n if n (cid:62) ) is cyclic, generated by theelement ξ n defined by ξ = 1 and ξ n +1 = ∂ n · ξ n . As a braid automorphism, ξ n = c − ∂ n . Remark 3.3.
The relation ξ n +1 = ∂ n · ξ n gives the usual formula for this centralelement [Bir74, cor. 1.8.4]: ξ n = ( A n A n · · · A n − ,n )( A ,n − · · · A n − ,n − ) · · · ( A A ) A . · · ·· · · n − n Proof of Prop. 3.2.
The result is clear for n (cid:54)
2. We thus suppose n (cid:62) Z ( P n ) is not trivial: it contains the braid automorphism c ∂ n .We use the decomposition P n ∼ = F n − (cid:111) P n − to show that this element generates Z ( P n ).Let wβ ∈ Z ( F n − (cid:111) P n − ). Proposition 3.1 implies that β should act on F n − via c − w .From Lemma 3.5 , we deduce that w ∈ (cid:104) ∂ n − (cid:105) . As a consequence: Z ( P n ) ⊆ (cid:68) ∂ n − c − ∂ n − (cid:69) . (3.3.1)The central element c ∂ n must be equal to ∂ kn − c − k∂ n − for some k . But since theprojection from P n onto P n − is induced by x n (cid:55)→
1, we see that it sends c ∂ n to c ∂ n − ,whence k = 1, and equality in (3.3.1). Moreover, we have obtained the inductionrelation c ∂ n = ∂ − n − c ∂ n − , which implies the relation we wanted for ξ n , if we define ξ n tobe c − ∂ n .Finally, we remark that for n (cid:62)
3, the center of B n has to be a subgroup of P n (whence of Z ( P n )), since its image in Σ n must be in the center of Σ n , which is trivial.The other inclusion can be obtained by showing directly that ξ n commutes with theclassical generators of B n (which is obvious from the geometric picture).An element of the free group is called primitive when it is part of a basis of the freegroup. Recall the following easy result: Lemma 3.4.
In the free group F n , the centralizer of any primitive element w is thecyclic group generated by w .Proof. Suppose that ( x = w, x , ..., x n ) is a basis of F n . If g ∈ F n , the relation x gx − g − = 1 cannot hold in the free group if any letter different from x appears inthe reduced expression of g in the letters x i , whence our claim.15 emma 3.5. In Aut( F n ) , the intersection between P n and Inn( F n ) is cyclic, generatedby c ∂ n , which is a central element of P n .Proof. Let β ∈ P n be a braid automorphism which is also an inner automorphism c w for some w ∈ F n . Then c w must fix ∂ n , that is, w must commute with ∂ n . However, ∂ n is a primitive element of F n . Indeed, ( ∂ n , x , ..., x n ) is a basis of F n , the changeof basis being given by ∂ n = x · · · x n and x = ∂ n x − n · · · x − . As a consequence, w ∈ (cid:104) ∂ n (cid:105) . Thus P n ∩ Inn( F n ) ⊆ (cid:104) c ∂ n (cid:105) . This inclusion is in fact an equality, since c ∂ n is a braid automorphism. Moreover, for any braid automorphism β ∈ P n , we have βc ∂ n β − = c β ( ∂ n ) = c ∂ n , hence c ∂ n is central in P n . The above computation of Z ( P n ) can readily be adapted to compute the center of theLie ring of P n , which is the Drinfeld-Kohno Lie ring DK n . We use the decomposition DK n ∼ = L n (cid:111) DK n − , induced by the decomposition P n ∼ = F n − (cid:111) P n − (see § Lemma 3.6.
In the free Lie ring L ( Z n ) , the centralizer of an element x ∈ Z n is Z · xd ,where d is the gcd of the coefficients of x .Proof. Consider a basis ( x = x/d, x , ..., x n ) of Z n . In the tensor algebra T ( Z n ), whichis the enveloping ring of L ( Z n ), the centralizer of dx consists of all polynomials in x only. Among these, the only ones belonging to L ( Z n ) (the only primitive ones, in thesense of Hopf algebras) are the linear ones.We also have an analogue of Lemma 3.5 in this context (see Def. 1.20 for thedefinition of braid-like derivation): Lemma 3.7. In Der( L n ) , the intersection between Der ∂t ( L n ) and Inn( L n ) is cyclic,generated by ad ∂ n , which is a central element of Der ∂t ( L n ) .Proof. Let X ∈ L n such that the inner derivation ad X is braid-like. Then ad X ( ∂ n ) = 0,that is, X must be in the centralizer of ∂ n . We deduce from Lemma 3.6 that X ∈ Z · ∂ n .Thus Der ∂t ( L n ) ∩ ad( L n ) ⊆ Z · ∂ n . This inclusion is in fact an equality, since ad ∂ n is abraid-like derivation. Moreover, for any braid-like derivation d , we have: (cid:2) d, ad ∂ n (cid:3) = d (cid:0)(cid:2) ∂ n , − (cid:3)(cid:1) − (cid:2) ∂ n , d ( − ) (cid:3) = (cid:2) d (cid:0) ∂ n (cid:1) , − (cid:3) = 0 , hence ad ∂ n is central in Der ∂t ( L n ).Finally, we can use Lemmas 3.6 and 3.7 to compute the center of DK n : Proposition 3.8.
The center of DK n is cyclic, generated by the element ξ n definedby ξ = 0 and ξ n +1 = ∂ n + ξ n . As a braid derivation, ξ n = − ad( ∂ n ) . Remark 3.9.
Out of the relation ξ n +1 = ∂ n + ξ n , we get a formula for this centralelement: ξ n = (cid:88) i 2. We thus suppose n (cid:62) ∂ n of Lemma 3.7 is in fact a braidderivation. Indeed, DK n contains all braid-like derivations of degree 1 (see Prop. 6.1).Alternatively, we can use formula (2.2.1) to see that ad ∂ n identifies with the class ofthe braid automorphism c ∂ n via the Johnson morphism τ : DK n (cid:44) → Der ∂t ( L n ).Since DK n identifies with a Lie subring of Der ∂t ( L n ), Lemma 3.7 implies that z ( DK n ) is not trivial: it contains the braid derivation ad ∂ n . We use the decom-position DK n ∼ = L n − (cid:111) DK n − to show that this element generates z ( DK n ). Let X + d ∈ L n − (cid:111) DK n − . Proposition 3.1 implies that d should act on L n − via − ad X .From Lemma 3.7 , we deduce that X ∈ Z · ∂ n − . As a consequence: z ( DK n ) ⊆ Z · ( ∂ n − − ad ∂ n − ) ⊂ L n − (cid:111) DK n − . (3.9.1)The central element ad ∂ n must then be equal to k · ( ∂ n − − ad ∂ n − ) for some integer k . But since the projection from DK n onto DK n − is induced by X n (cid:55)→ 0, we seethat it sends ad ∂ n to ad ∂ n − , whence k = 1, and equality in (3.9.1). Moreover, we haveobtained the induction relation ad ∂ n = − ∂ n − + ad ∂ n − ), which implies the relation wewanted for ξ n , if we define ξ n to be − ad ∂ n . Let us denote, as usual, the quotient P n / Z ( P n ) by P ∗ n . In a similar fashion, we denoteby DK ∗ n the quotient of the Drinfeld-Kohno Lie ring DK n by its center. Theorem 3.10. The subgroup Inn( P n ) of IA ( P n ) ( ⊂ Aut( P n ) ) satisfies the Andrea-dakis equality. Equivalently: L ( P ∗ n ) ∼ = DK ∗ n . Proof. The only central elements of DK n are multiples of ξ n , which is the class of thecentral element ξ n ∈ P n (Prop. 3.8). Thus, our claim is obtained as a direct applicationof Cor. 2.2 to G = P n . Corollary 3.11. The Lie algebra DK ∗ n , and hence the group P ∗ n , are centerless. As aconsequence, the subgroup Inn( P ∗ n ) of IA ( P ∗ n ) also satisfies the Andreadakis equality.Proof. We use the fact that the only non-trivial central elements of DK n are in degreeone. Let x ∈ z ( DK ∗ n ) be the class of x ∈ DK n . Let y be any element of DK n Thenthe bracket [ x, y ] must be in z ( DK n ) (since its class in DK ∗ n is trivial). But since itsdegree is at least 2, it must be trivial. Hence x ∈ z ( DK n ), whence x = 0.In order to deduce that P ∗ n is centerless, we can use the fact that it is residuallynilpotent, since P n is (Lemma 7.7), and thus any non-trivial element in its center wouldgive a non-trivial class in its Lie algebra, which must be central too.The last statement is then a direct application of Cor. 2.2 to G = P ∗ n . Remark 3.12. Part of the results of Th. 3.10 and its corollary can be deduced fromthe (classical) calculation of the center of P n and from the classical (non-canonical)splitting P n ∼ = P ∗ n × Z ( P n ) recalled in the appendix (Cor. 7.9). In fact, if we apply theLie functor to this direct product, we get: DK n = L ( P n ) ∼ = L ( P ∗ n ) × L ( Z ( P n )) ∼ = L ( P ∗ n ) × Z · ξ n , ξ n = (cid:80) t ij of thegenerator ξ n of Z ( P n ). Thus we get the computation L ( P ∗ n ) ∼ = DK n /ξ n . Moreover,we see easily that P ∗ n is centerless (Prop. 7.8). However, neither does this imply that ξ n generates the center of DK n , nor do we get the statements about Andreadakisequalities without computing this center. I A n Here we turn to the proof of our key result (Th. 4.4), which generalises [Dar19b,Prop. 4.1]. In order to do this, we study filtrations on products HK , where H and K are subgroups of a given group G , such that K normalizes H . Namely, we investigatethe behaviour of the lower central series of HK , and of a filtration G ∗ ∩ ( HK ) inducedby a filtration G ∗ of G , with respect to the product decomposition of HK . Let G be a group, and let H and K be subgroups of G , such that K normalizes H .Then K acts on H by conjugation in G , we can form the corresponding semi-directproduct H (cid:111) K and we get a well-defined morphism H (cid:111) K → G given by ( h, k ) (cid:55)→ hk .The image of this morphism is the subgroup HK of G . Its kernel, given by the elements( h, k ) ∈ H (cid:111) K such that hk = 1, is isomorphic to H ∩ K , via k (cid:55)→ ( k − , k ). Thus weget a short exact sequence of groups: H ∩ K (cid:44) → H (cid:111) K (cid:16) HK. The surjection on the right induces a surjection Γ i ( H (cid:111) K ) (cid:16) Γ i ( HK ), for all i (cid:62) i ( H (cid:111) K ) = Γ Ki ( H ) (cid:111) Γ i ( K ) (see Prop. 1.10), this implies: Proposition 4.1. Let K and H be subgroups of a group G , such that K normalizes H . Then: ∀ i (cid:62) , Γ i ( HK ) = Γ Ki ( H )Γ i ( K ) . Moreover, the kernel of Γ i ( H (cid:111) K ) (cid:16) Γ i ( HK ) consists of elements k ∈ H ∩ K suchthat ( k − , k ) ∈ Γ i ( H (cid:111) K ): it is Γ Ki ( H ) ∩ Γ i ( K ) ⊆ H ∩ K . Thus we get a short exactsequence of filtrations:Γ K ∗ ( H ) ∩ Γ ∗ ( K ) (cid:44) → Γ K ∗ ( H ) (cid:111) Γ ∗ ( K ) (cid:16) Γ ∗ ( HK ) . Let G ∗ be a filtration, and let H and K be subgroups of G = G such that K normalizes H . Then we can consider two filtrations on HK : the induced filtration G ∗ ∩ ( HK ) andthe product of induced filtrations ( G ∗ ∩ H )( G ∗ ∩ K ). The former obviously containsthe latter. We now describe a criterion for this inclusion to be an equality. Proposition 4.2. In the above setting, the following assertions are equivalent:(i) G ∗ ∩ ( HK ) = ( G ∗ ∩ H )( G ∗ ∩ K ) , (ii) Inside L ( G ∗ ) , L ( G ∗ ∩ H ) ∩ L ( G ∗ ∩ K ) = L ( G ∗ ∩ ( H ∩ K )) . emark 4.3. The case considered in [Dar19b, Prop. 4.1] was exactly the case when H ∩ K = 1. In this context, H and K were called G ∗ -disjoint when they satisfied theequivalent conditions of the proposition. Proof of Proposition 4.2. Suppose that (ii) does not hold. Then there exists an element x ∈ L ( G ∗ ∩ H ) ∩L ( G ∗ ∩ K ) − L ( G ∗ ∩ ( H ∩ K )) for some i (cid:62) 1. Then x = h = k for some h ∈ G i ∩ H and some k ∈ G i ∩ K . Since h = k in L i ( G ∗ ), the element g = h − k is in G i +1 .It is also obviously in HK . However, we claim that g / ∈ ( G i +1 ∩ H )( G i +1 ∩ K ). Indeed, ifwe could write g as a product h (cid:48) k (cid:48) with h (cid:48) ∈ G i +1 ∩ H and k (cid:48) ∈ G i +1 ∩ K , then we wouldhave h − k = h (cid:48) k (cid:48) , whence kk (cid:48)− = hh (cid:48) ∈ H ∩ K . And by construction, hh (cid:48) = h = x in G i /G i +1 . But this would imply that x ∈ L ( G ∗ ∩ ( H ∩ K )), a contradiction. Thus g must be a counter-example to (i).Conversely, suppose (i) false. Then there exists g ∈ G j ∩ ( HK ), such that for all( h, k ) ∈ H (cid:111) K satisfying g = hk , neither h nor k belongs to G j (if h or k belongs to G j , so does the other one, since their product g does). Then h ≡ k − (cid:54)≡ G j ).For all such ( h, k ), there exists i < j such that h, k ∈ G i − G i +1 . Let us take ( h, k )such that this index i is maximal. We show that the element h = − k ∈ L i ( G ∗ ) givesa counter-example to the equality (ii). Indeed, it is clear that h = − k belongs to L i ( G ∗ ∩ H ) ∩ L i ( G ∗ ∩ K ). Suppose now that this element should belong to L i ( G ∗ ∩ ( H ∩ K )). Then there would exist x ∈ G i ∩ H ∩ K such that h = − k = x . Thismeans that h − x = − ( x + k ) = 0 or, equivalently: hx − = − xk = 0. But then g = hk = ( hx − )( xk ), with hx − and xk in G i +1 , contradicting the maximality of i .Thus h cannot belong to L i ( G ∗ ∩ ( H ∩ K )), whence our claim. We are now able to state our key theorem, which is an improvement on [Dar19b,Th. 4.2]. We will apply it to the case when G = IA n and G ∗ = A ∗ is the Andreadakisfiltration, but we still give a general statement. Theorem 4.4. Let G ∗ be a filtration, and let H and K be subgroups of G = G suchthat [ K, H ] ⊆ [ H, H ] . Suppose that in the Lie ring L ( G ∗ ) , the intersection of the Liesubrings L ( G ∗ ∩ H ) and L ( G ∗ ∩ K ) is L ( G ∗ ∩ ( H ∩ K )) . Then: (cid:40) G ∗ ∩ K = Γ ∗ ( K ) G ∗ ∩ H = Γ ∗ ( H ) ⇒ G ∗ ∩ ( HK ) = Γ ∗ ( HK ) . Recall that in the case when G = IA n and G ∗ = A ∗ is the Andreadakis filtration,the Lie ring L ( A ∗ ) embeds into the Lie ring Der( L n ) of derivations of the free Lie ring, via the Johnson morphism τ . As a consequence, the hypothesis about Lie subrings of L ( A ∗ ) can be checked there. Proof of Theorem 4.4. Remark that [ K, H ] ⊆ [ H, H ] implies in particular that K nor-malizes H . Thus, we can apply the results of § K, H ] ⊆ [ H, H ] is exactlythe condition needed for Γ K ∗ ( H ) to be equal to Γ ∗ ( H ), we get:Γ ∗ ( HK ) = Γ ∗ ( H )Γ ∗ ( K ) . Then we use Proposition 4.2 to get: G ∗ ∩ ( HK ) = ( G ∗ ∩ H )( G ∗ ∩ K ) , whence the result. 19 Applications The subgroup H = Inn( F n ) ∼ = F n of IA n is an ideal candidate for applying our keyresult (Th. 4.4). We begin by spelling out the consequences of Th. 4.4 in this particularcase (Cor. 5.1). It turns out that for each of the subgroups K which were shown tosatisfy the Andreadakis equality in [Dar19b], we can apply Cor. 5.1 to show that HK does too. Moreover, two of the subgroups so obtained had already been consideredin the literature. These are the group of partial inner automorphisms of [BN16], andthe subgroup P ∗ n +1 of IA n presented in the introduction, which was first studied in[Mag34]. We are going to apply Theorem 4.4 with H = Inn( F n ) and G = IA n endowed with theAndreadakis filtration G ∗ = A ∗ , for three different subgroups K of IA n . We recordin Corollary 5.1 below the consequences of Theorem 4.4 in this context. For short, weoften denote by F n the subgroup H = Inn( F n ) ∼ = F n of IA n , and by F n K its productwith a given subgroup K of IA n . Corollary 5.1 (to Th. 4.4) . Let K be a subgroup of IA n , and τ : L ( K ) → Der( L n ) be the corresponding Johnson morphism. Suppose that K satisfies the Andreadakisequality, which means that τ is injective. Suppose, moreover, that every element of τ ( L ( K )) ∩ ad( L n ) comes from an element of K ∩ Inn( F n ) (precisely, it equals τ ( x ) , forsome x ∈ K ∩ Inn( F n ) ). Then F n K satisfies the Andreadakis equality.Proof. We apply Theorem 4.4 to the subgroups H = Inn( F n ) and K of G = IA n endowed with the Andreadakis filtration G ∗ = A ∗ . • The hypothesis [ K, H ] ⊆ [ H, H ] comes from the fact that IA n (whence K ) nor-malizes Inn( F n ) and acts trivially on F abn . • Cor. 2.4 says that H = Inn( F n ) satisfies the Andreadakis equality. • By hypothesis, K does too.Thus, we are left with proving that the following inclusion (which is true in general) isin fact an equality in L ( A ∗ ): L ( A ∗ ∩ F n ) ∩ L ( A ∗ ∩ K ) ⊇ L ( A ∗ ∩ ( F n ∩ K )) . If we embed L ( A ∗ ) into Der( L n ) via the Johnson morphism, the Lie ring L ( A ∗ ∩ F n )identifies with the Lie subring of inner derivations ad( L n ) (see § K satisfies the Andreadakis equality, the Lie ring L ( A ∗ ∩ K ) identifies with τ ( L ( K )).As for L ( A ∗ ∩ ( F n ∩ K )), it is exactly the set of τ ( x ), for x ∈ K ∩ Inn( F n ), whence theresult. We recall the definition of the subgroup IA + n of triangular automorphisms [Dar19b,Def. 5.1]: Definition 5.2. Fix ( x , ..., x n ) an ordered basis of F n . The subgroup IA + n of IA n consists of triangular automorphisms , i.e. automorphisms ϕ acting as: ϕ : x i (cid:55)−→ ( x w i i ) γ i , where w i ∈ (cid:104) x j (cid:105) j
The subgroup F n IA + n of IA n satisfies the Andreadakis equality.Proof. Remark that a triangular automorphism ϕ has to fix x . As a consequence, thederivation τ ( ϕ ) sends x to ϕ ( x ) x − = 0. The only inner derivations vanishing on x are the multiples of ad( x ) (Lemma 3.6). However, ad( x ) is the image by τ of c x , and c x is a triangular automorphism. Thus, the hypotheses of Corollary 5.1 are satisfied,whence the desired conclusion. The triangular McCool P Σ + n group was first considered in [CPVW08], where its Liealgebra was computed. It was shown to satisfy the Andreadakis equality in [Dar19b].The subgroup F n P Σ + n of IA n (or, more precisely, F n P Σ − n , which is obtained fromthe definition of F n P Σ + n with the opposite order on the generators) is exactly the partialinner automorphism group I n defined and studied in [BN16]. Indeed, I n is defined asthe subgroup generated by the automorphism c ki for k (cid:62) i , defined by: c ki : x j (cid:55)−→ (cid:40) x x i j if j (cid:54) k,x j else.Equivalently, it is generated by the c ni = c x − i (which generate Inn( F n )), together withthe c ki c − k − ,i = χ ki for k > i , which generate P Σ − n . Theorem 5.4. The subgroup I n = F n P Σ + n of IA n satisfies the Andreadakis equality.Proof. The proof is exactly the same as the proof of Theorem 5.3, using the fact that P Σ + n satisfies the Andreadakis equality [Dar19b][Cor. 5.5], and remarking that c x isnot only in IA + n , but in P Σ + n . The Artin action of P n on F n is by group automorphisms fixing ∂ n = x · · · x n , thereis an induced action of P n on the quotient F n /∂ n ∼ = F n − (which is free on x , ..., x n − since ( x , ..., x n − , ∂ n ) is a basis of F n ). This action is not faithful anymore, since thegenerator c ∂ n of Z ( P n ) (see Prop. 3.2) acts trivially modulo ∂ n . However, by a classicaltheorem of [Mag34], of which we give a simple proof in our appendix ( § Z ( P n ) : the above induces a faithful action of P ∗ n / Z ( P n ) on F n − .The latter is exactly the action on the free group of the (pure, based) mapping classgroup of the punctured sphere presented in the introduction, described in a purelyalgebraic fashion. Indeed, P n identifies with the boundary-fixing mapping class groupof the n -punctured disc, and the Artin action corresponds to the canonical action onthe fundamental group of this space. Then, collapsing the boundary to a point (whichwe choose as basepoint) to get an action on the n -punctured sphere corresponds exactlyto taking the quotient by ∂ n = x · · · x n , at the level of fundamental groups.From the description of Artin’s action, we see that automorphisms of F n − ob-tained from the above action must send each generator x i to one of its conjugatesand, since x − n ≡ x · · · x n − (mod ∂ n ), they must also send the boundary element ∂ n − = x · · · x n − to one of its conjugates. This means exactly that this action is via elements of Inn( F n − ) P n − : 21 emma 5.5. The subgroup F n P n of IA n is the subgroup of all automorphisms of F n sending each generator x i to one of its conjugates, and sending the boundary element ∂ n = x · · · x n to one of its conjugates.Proof. The conditions of the lemma obviously describe a subgroup G of Aut( F n ) (it isan intersection of stabilizers for the action of Aut( F n ) on the set of conjugacy classesof F n ), and it contains P n and Inn( F n ). Now let σ ∈ G . Then σ ( ∂ n ) = ∂ wn for some w ,hence c w ◦ σ fixes ∂ n (where c w : x (cid:55)→ w x is the inner automorphism associated to w ).As a consequence, c w σ ∈ P n , whence σ ∈ F n P n , and our claim.In fact, the proof of Magnus’ theorem (Th 7.1) that we give consists in showing thatthe above action of P n on F n − induces an isomorphism between P ∗ n and F n − P n − . Wewill make use of the fact that their is another isomorphism between these two groups,given by: Lemma 5.6. The kernel of the map ( w, β ) (cid:55)→ c w β from F n (cid:111) P n ∼ = P n +1 to F n P n ⊂ Aut( F n ) is exactly the center of P n +1 . As a consequence, F n P n ∼ = P ∗ n +1 .Proof. The center of P n +1 is generated by ( ∂ n +1 , c − ∂ n +1 ) (Prop. 3.2), which is obviouslysent to the identity. Conversely, if ( w, β ) is such that c w = β − , then c w ∈ P n ∩ Inn( F n ).Lemma 3.5 implies that w must be a power of ∂ n +1 , whence the result.We now give the main application of our key result (Th. 4.4): Theorem 5.7. The subgroup P ∗ n +1 ∼ = F n P n of IA n satisfies the Andreadakis equality.Proof. We apply Corollary 5.1 to the subgroup K = P n of IA n . The Andreadakisequality holds for P n : this was proved in [Dar19b, Th. 6.2], recalled as Th. 1.19 above.We need to check the other hypothesis. Recall that L ( P n ), which is the Drinfeld-Kohno Lie ring DK n , is identified to a Lie subring of Der ∂t ( L n ) as in § ∂t ( L n ) with ad( L n ) is (linearly) generated by ad ∂ n (Lemma 3.7).Now, ad ∂ n is the image by τ of c ∂ n , which is a braid automorphism, whence ourconclusion. Remark 5.8. This does not depend of an identification of P ∗ n +1 with F n P n : one canchoose the more algebraic isomorphism of Lemma 5.6, or the more geometric (andarguably more interesting) isomorphism given by Magnus’ theorem (Th 7.1). Thelatter allows us to reformulate our result in more geometric terms : the Johnson kernelsassociated with the action of P ∗ n on the fundamental group of the n -punctured sphereare exactly the terms of the lower central series of P ∗ n .22 Appendix 1: Braid and braid-like derivations Recall (from § P n on F n induces an action of Γ ∗ ( P n ) onΓ ∗ ( F n ) and thus an action of the Drinfeld-Kohno Lie ring L ( P n ) = DK n on the freeLie ring L ( F n ) = L n . Moreover, the Johnson morphism encoding the latter is in factan injection: τ : DK n (cid:44) → Der( L n ) . We identify DK n with its image in Der( L n ), and its elements are called braid deriva-tions . It is easy to see that this image is contained in the Lie subring Der ∂t ( L n ) of braid-like derivations . The goal of the present section is to compare these two Liesubrings of Der( L n ). As graded abelian groups, both DK n and Der ∂t ( L n ) are fairly well understood. Sincethe free Lie ring L n does not have torsion, neither do they. Moreover, we can computeexplicitly their ranks in each degree, from the ranks of the free Lie algebra, denoted by d ( n, k ) := rk k ( L n ), which are in turn given by Witt’s formula [Ser06, I.4]: d ( n, k ) = (cid:88) st = k µ ( s ) n t , (6.0.1)where µ is the usual M¨obius function.In order to compute the ranks of DK n , consider the decomposition: DK n ∼ = L n − (cid:111) ( L n − (cid:111) ( · · · (cid:111) L ) · · · ) . As an immediate consequence, we get: rk k ( DK n ) = n − (cid:88) l =1 d ( l, k ) . (6.0.2)We now compute the ranks of Der ∂t ( L n ). We first recall that given any choice of n elements δ i of L n , there exists a unique derivation δ of L n sending each generator X i to δ i . Moreover, for each X i , the only elements Y of L n such that [ Y, X i ] = 0 are integralmultiples of X i (see Lemma 3.6 below). As a consequence, the map sending ( t i ) ∈ L nn tothe derivation δ : X i (cid:55)→ [ X i , t i ] is well-defined, and its kernel is Z · X × · · · × Z · X n ∼ = Z n (concentrated in degree 1). Moreover, by definition of tangential derivations, its imageis Der t ( L n ), whence: rk k (Der t ( L n )) = (cid:40) n ( n − 1) if k = 1 ,n · d ( n, k ) if k (cid:62) . Now, Der ∂t ( L n ) is the kernel of the linear map (raising the degree by 1) ev ∂ : δ (cid:55)→ δ ( ∂ n ) = δ ( X ) + · · · + δ ( X n ) from Der t ( L n ) to L n . This map is surjective. Indeed, L n is generated by so-called linear monomials (this is easy to prove – see for instance theappendix of [Dar19a]), which are of the form [ X i , t ] ( t ∈ L n ), and [ X i , t ] is the imageby ev ∂ of the derivation δ sending X i to [ X i , t ] and all other X j to 0. From this, wededuce: rk k (cid:0) Der ∂t ( L n ) (cid:1) = rk k (Der t ( L n )) − rk k +1 ( L n ) (6.0.3)= (cid:40) n ( n − − n ( n − = n ( n − if k = 1 ,n · d ( n, k ) − d ( n, k + 1) if k (cid:62) . (6.0.4)23 .B Comparison in degree In order to understand the image of DK n by the Johnson morphism DK n (cid:44) → Der ∂t ( L n ),we compare formulas 6.0.2 and 6.0.3.In degree one, we get: rk ( DK n ) = n − (cid:88) k =1 k = n ( n − (cid:0) Der ∂t ( L n ) (cid:1) . In fact, the Johnson morphism in degree one is given by the explicit morphism: τ : t ij (cid:55)−→ τ ( t ij ) : X l (cid:55)→ [ X i , X j ] if l = i, [ X j , X i ] if l = j, DK n on L n is encodedin L n (cid:111) DK n ∼ = DK n +1 and from the following relations in DK n +1 (see Prop. 1.18):[ t ij , t l,n +1 ] = [ t i,n +1 , t j,n +1 ] if l = i, [ t j,n +1 , t i,n +1 ] if l = j, τ , which is known to be injective, is easily seen to be surjective. Indeed,if δ ∈ Der ∂t ( L n ) is homogeneous of degree 1, then the equality δ ( X + · · · + X n ) = 0implies that the coefficient λ ij of [ X i , X j ] in δ ( X i ) equals the coefficient of [ X j , X i ] in δ ( X j ), from which we deduce that δ = (cid:80) λ ij τ ( t ij ), the sum being taken on ( i, j ) with i < j .Since DK n is generated in degree one, we deduce: Proposition 6.1. The Drinfeld-Kohno Lie ring DK n identifies, via the Johnson mor-phism, with the Lie subring of Der ∂t ( L n ) generated in degree one. In degree 2, using Witt’s fomula (6.0.1) for k = 2 and k = 3 (which give d ( n, 2) = n ( n − / d ( n, 3) = ( n − n ) / ( DK n ) = n ( n − n − (cid:0) Der ∂t ( L n ) (cid:1) . We can in fact write an explicit formula for τ , and show directly that it is anisomorphism: τ : [ t ik , t jk ] (cid:55)−→ δ ijk := [ τ ( t ik ) , τ ( t jk )] : X l (cid:55)→ [ X i , [ X j , X k ]] if l = i, [ X j , [ X k , X i ]] if l = j, [ X k , [ X i , X j ]] if l = k, δ ijk ( X + · · · + X n ) = 0 is exactly the Jacobi identity for X i , X j and X k . 24 roposition 6.2. The Johnson morphism τ : DK n → Der ∂t ( L n ) is an isomorphism indegree .Proof. We already know that τ is injective : we only need to show that it is surjective.Let δ ∈ Der ∂t ( L n ) be homogeneous of degree 2. Recall that L n is N n -graded, sincethe antisymmetry and Jacobi relation only relate parenthesized monomials with thesame letters. We denote by deg : L n → N n the corresponding degree, sending X i to e i . Since δ is tangential, for all i , δ ( X i ) can be written as a sum of λ ij [ X i , [ X i , X j ]]and of λ ijk [ X i , [ X j , X k ]] (with j < k ), for some integral coefficients λ ij , λ ijk . Then,in the decomposition of δ ( X ) + · · · + δ ( X n ) into N n -homogeneous components, thecomponents are: (cid:40) λ ij [ X i , [ X i , X j ]] for i (cid:54) = j,λ ijk [ X i , [ X j , X k ]] + λ jik [ X j , [ X i , X k ]] + λ kij [ X k , [ X i , X j ]] for i < j < k, of respective degrees 2 e i + e j and e i + e j + e k . The first ones are trivial if and onlyif λ ij = 0 for all i, j . The second ones are trivial if and only if they are multiplesof Jacobi identities, that is, if λ ijk = − λ jik = λ kij . Indeed, the component of degree e i + e j + e k of L n is the quotient of the free abelian group generated by [ X i , [ X j , X k ]],[ X j , [ X i , X k ]] and [ X k , [ X i , X j ]] by the Jacobi identity. When these conditions hold, wesee immediately that δ = (cid:80) λ ijk δ ijk ∈ Im( τ ). In order to compute ranks in degree 3, we use Witt’s fomula in degree 4, which gives d ( n, 4) = ( n − n ) / 4. We find:rk ( DK n ) = ( n − n − · n ( n − , rk (cid:0) Der ∂t ( L n ) (cid:1) = n ( n + 1) · n ( n − . Hence the following: Proposition 6.3. The inclusion DK n ⊂ Der ∂t ( L n ) is a strict one: not all braid-likederivations come from braids. We can in fact say more: Proposition 6.4. The cokernel of the inclusion of graded abelian groups DK n ⊂ Der ∂t ( L n ) is a graded abelian group whose rank in degree k (cid:62) is given by a poly-nomial function of n , whose leading term is k n k .Proof. Recall that in degree k , the rank of Der ∂t ( L n ) is n · d ( n, k ) − d ( n, k + 1) (for-mula (6.0.3)). Witt’s formula (6.0.1) implies that d ( n, k ) is a polynomial function of n , whose leading term is n k /k (since µ (1) = 1). Moreover, the second non-trivial termis in degree k/p , where p is the least prime factor of k , thus its degree is at most k/ n · d ( n, k ) − d ( n, k + 1) is a polynomial function of n , with leadingterm n · n k k − n k +1 k + 1 = n k +1 k ( k + 1) , and no term of degree k . 25ow, recall that in degree k , the rank of DK n is given by formula (6.0.2):rk k ( DK n ) = n − (cid:88) l =1 d ( l, k ) = n − (cid:88) l =1 k ( l k − l k/p + · · · ) = 1 k (cid:32) n − (cid:88) l =1 l k − n − (cid:88) l =1 l k/p + · · · (cid:33) . From Faulhaber’s formula for sums of powers (recalled below), we know that S α ( n ) isa polynomial function of n , whose leading terms are given by: S α ( n ) = n α +1 α + 1 − n α · · · Thus, the first two terms of rk k ( DK n ) are those of k S k ( n − k S k ( n − 1) = 1 k (cid:18) ( n − k +1 k + 1 − ( n − k · · · (cid:19) = n k +1 k ( k + 1) − k · n k + · · · We find, as announced, that the difference rk k (Der ∂t ( L n )) − rk k ( DK n ) is a polynomialfunction of n , whose leading term is k n k .We have used in the proof the usual formula (known as Faulhaber’s formula) forsums of powers. For any integer α (cid:62) S α ( n ) := n (cid:88) l =0 l α = 1 α + 1 α (cid:88) j =0 (cid:18) α + 1 j (cid:19) B j · n α +1 − j , where the B j are the Bernoulli numbers, defined by: ze z e z − ∞ (cid:88) j =0 B j z j j ! . Recall that the above formula can be obtained by a quite straightforward calculationof the exponential generating series: ∞ (cid:88) α =0 S α ( n ) z α α ! = n (cid:88) l =1 e lz = e z · e nz − e z − . P ∗ n We gather here some classical results about P ∗ n = P n / Z ( P n ), with some new proofs.Our first goal is a classical theorem of Magnus [Mag34] about the faithfulness of itsaction on the fundamental group of the punctured sphere. We give a complete proofwhich avoids any difficult calculation with group presentations. This proof is partiallybuilt on ideas described in the sketch of proof of [Bir74, Lem. 3.17.2]. The proof usesthe Hopf property in a crucial way, which leads us to recall some basic facts aboutHopfian groups. Finally, we describe the splitting P n ∼ = P ∗ n × Z ( P n ), which can beused to give an alternative proof of part of the results of § .A A faithful action of P ∗ n on F n − Recall that the pure braid group P n acts faithfully on F n via the Artin action. Sincethis action is by group automorphisms fixing ∂ n = x · · · x n , there is an induced actionof P n on the quotient F n /∂ n ∼ = F n − (see § Theorem 7.1. [Mag34, Formula (23)]. The Artin action induces a faithful action of P ∗ n = P n / Z ( P n ) on F n /∂ n ∼ = F n − . Moreover, this induces an isomorphism between P ∗ n and the subgroup F n − P n − of Aut( F n − ) . Remark 7.2. The restriction to pure braids is not an important one: we can deducereadily that the result holds for the action of B ∗ n on F n − , by showing that the kernelof this action is contained in P n . Indeed, let β ∈ B n and σ β its image in Σ n . If β actstrivially on F n /∂ n ∼ = F n − , then it acts trivially on F abn /∂ n . But the action of B n on F abn ∼ = Z n is via the canonical action of Σ n on Z n , and Z n / ( X + · · · X n ) is a faithfulrepresentation of Σ n , hence σ β = 1. Proof of Theorem 7.1. Let us denote by π : P n (cid:16) P ∗ n the canonical projection and by χ : P ∗ n (cid:55)→ Aut( F n − ) the action under scrutiny. We first show that Im( χ ) = F n − P n − . From the definition of the action, thecharacterization of braid automorphisms of F n and the description of F n − P n − givenin Lemma 5.5, it is obvious that χ takes values in F n − P n − . The map s : P n − (cid:44) → P n extending braid automorphisms by x n (cid:55)→ x n is easily seen to satisfy χπs ( β ) = β for all β ∈ P n − , hence P n − ⊂ Im( χ ). Moreover, for all j (cid:54) n − 1, the inner automorphism c x ··· x j is the image by χ of the braid automorphism: C j : x t (cid:55)−→ (cid:40) ( x ··· x j ) x t if t (cid:54) j, ( x j +1 ··· x n ) − x t if t > j. Since the x · · · x j form a basis of F n , we conclude that Inn( F n ) ⊂ Im( χ ). We now show that the action χ is faithful. Lemma 5.6 implies that F n − P n − is isomorphic to P ∗ n . Thus χ restricts to a morphism from P ∗ n to P ∗ n . We have justshowed that this endomorphism is surjective. We claim that its surjectivity impliesthat it is an automorphism of P ∗ n . Indeed, since P n embeds in IA n ( via the Artinaction), it is residually nilpotent. Hence, by Lemma 7.7, P ∗ n is too. Thus, Proposition7.5 implies that it is Hopfian , which means exactly that surjective endomorphisms of P ∗ n are automorphisms. Remark 7.3. The braid automorphism C j is easy to understand as a geometric braid.Here is a drawing of this braid: · · ·· · · · · ·· · · j j + 1 j + 2 n C j identifies with the (commutative) product of theDehn twists along γ and γ : γ γ × × · · · × j × j + 1 × j + 2 · · · × n Remark 7.4. Instead of using the Hopf property, one could define explicitly the inverseof χ : P ∗ n → P ∗ n using the explicit lifts of the generators given in the proof. However,showing that this inverse is well-defined directly from the presentation of P ∗ n doesinvolve quite a bit of calculation, which one would need to do in a clever way in ordernot to get lost. This method would be closer to the original proof of Magnus [Mag34,Formula (23)]. Our method is closer to the sketch of proof of [Bir74, Lem. 3.17.2], butthe latter seems to miss the fact that the endomorphism of P ∗ n at the end of our proofis a non-trivial one, whose injectivity is not obvious. Recall that a group G is called Hopfian if every surjective endomorphism of G is anautomorphism. The relevance of this property in our context relies on the following: Proposition 7.5. Residually nilpotent groups of finite type are Hopfian.Proof. Let G be a residually nilpotent groups of finite type and π : G → G be surjectiveendomorphism. The Lie ring L ( G ) is generated in degree one. This implies on the onehand, that every L k ( G ) is finitely generated, and on the other hand, that L ( π ) issurjective ( L ( π ) is, being the induced endomorphism of G ab ). Because of Lemma 7.6,this implies that all the L k ( π ) are isomorphisms.Suppose now that there exists x ∈ ker( π ) such that x (cid:54) = 1. Since G is residuallynilpotent, there exists k (cid:62) x ∈ Γ k G − Γ k +1 G . Then x ∈ ker( L k ( π )), whichis impossible. Lemma 7.6. Abelian groups of finite type are Hopfian.Proof. Let A be an abelian group of finite type and π : A → A be a surjective endomor-phism. If A has no torsion, then the rank of ker( π ) must be trivial, hence ker( π ) = 0,whence the result in this case.In general, let Tors( A ) denotes the torsion subgroup of A . Then π induces a com-mutative diagram whose rows are short exact sequences:Tors( A ) A A/ Tors( A )Tors( A ) A A/ Tors( A ) . π π π π is surjective, the induced endomorphism π of A/ Tors( A ) has to be too. But A/ Tors( A ) is a free abelian group of finite type. As a consequence of the first partof the proof, π has to be an isomorphism. Then, the snake lemma implies that theinduced endomorphism π of Tors( A ) is surjective. But Tors( A ) is finite, so π mustbe an isomorphism too. We can then apply the snake lemma again to conclude that π is injective. Lemma 7.7. If G is a residually nilpotent group, then so is G/ Z ( G ) .Proof. Let x ∈ G such that ¯ x ∈ Γ n ( G/ Z ( G )). Then x ∈ Z ( G )Γ n ( G ), whence [ G, x ] ⊆ Γ n +1 ( G ). Thus, if ¯ x ∈ (cid:84) Γ n ( G/ Z ( G )), then [ G, x ] ⊆ (cid:84) Γ n +1 ( G ) = { } , which meansthat x ∈ Z ( G ), and ¯ x = 1. We now show that there is a (non-canonical) splitting P n ∼ = P ∗ n × Z ( P n ) (Cor. 7.9),which we replace in the following general context: Proposition 7.8. For a group G , the following conditions are equivalent: • G/ Z G is centerless, and there exists an isomorphism G ∼ = Z G × ( G/ Z G ) . • The canonical projection G (cid:16) G/ Z G splits. • The canonical map Z G (cid:44) → G (cid:16) G ab is injective and its image is a direct factorof G ab .When G is of finite type, or Hopfian, these are equivalent to: • There exists an isomorphism G ∼ = Z G × ( G/ Z G ) .Proof. Suppose that the third condition is satisfied. We denote by π : G (cid:16) G ab the canonical projection. Let us identify Z G with its image in G ab , and let W be adirect complement of Z G in G ab . Then we claim that G decomposes as the directproduct of its subgroups Z G and π − ( W ). Indeed, if g ∈ G , then its class π ( g )decomposes as z + w with z ∈ Z G and w ∈ W , and gz − ∈ π − ( W ), which impliesthat g ∈ Z G · π − ( W ). Moreover, the intersection of the two subgroups is trivial bydefinition of W , and elements of Z G commute with elements of π − ( W ), whence ourclaim. Then the canonical projection p : G (cid:16) G/ Z G has to induce an isomorphism π − ( W ) ∼ = G/ Z G , whose inverse is a section of p .If p has a section s , then G decomposes as a semi-direct product ZG (cid:111) ( G/ Z G ),which has to be a direct product, since the conjugation action of G on Z G is trivial.Moreover, if z ∈ Z ( G/ Z G ), then s ( z ) is central in Z G × ( G/ Z G ) ∼ = G , thus z = ps ( z )is trivial, by definition of p .Now suppose that G/ Z G is centerless. Then Z ( Z G × ( G/ Z G )) = Z G × 1. Recallthat the canonical map Z G (cid:44) → G (cid:16) G ab is functorial in G . As a consequence, anisomorphism between G and Z G × ( G/ Z G ) has to induce a commutative square: Z G G ab Z G × Z G × ( G/ Z G ) ab , ∼ = ∼ = whence the third condition. 29 f G is Hopfian, then any direct factor of G is Hopfian: if G ∼ = H × K and u is asurjective endomorphism of H , then u × H × K ∼ = G ,thus is an automorphism, and u must be injective too. Thus, if G ∼ = Z G × ( G/ Z G ), then Z G is Hopfian. But we also have an induced isomorphism Z G ∼ = Z G ×Z ( G/ Z G ). If wecompose this isomorphism with the first projection, we get a surjective endomorphismof Z G , which has to be injective, forcing Z ( G/ Z G ) to be trivial, whence the result. If G is of finite type and G ∼ = Z G × ( G/ Z G ), then Z G is a quotient of G , thusit is abelian of finite type, hence Hopfian, and the same reasoning as in the case when G is Hopfian leads to the desired conclusion. Corollary 7.9. There exists an isomorphism P n ∼ = Z ( P n ) × P ∗ n .Proof. The center Z ( P n ) injects in P abn ∼ = Z { t ij } i Their is no canonical choice of splitting of P n (cid:16) P ∗ n : the splittingdepends on a choice of direct complement W of Z ( P n ) inside P abn . For instance, wecan choose W = W kl , generated by all the t ij for ( i, j ) (cid:54) = ( k, l ), so that the cor-responding section s kl sends the class of A ij to A ij if ( i, j ) (cid:54) = ( k, l ). In the litera-ture, authors often choose s , . Another natural choice is the section corresponding to W = { (cid:80) λ ij t ij | (cid:80) λ ij = 0 } , which sends the class of A ij A − kl to A ij A − kl , for all i, j, k, l . References [And65] Stylianos Andreadakis. On the automorphisms of free groups and free nilpotentgroups. Proc. London Math. Soc. (3) , 15:239–268, 1965. 1[Art25] Emil Artin. Theorie der Z¨opfe. Abh. Math. Sem. Univ. Hamburg , 4(1):47–72,1925. 8[Art47] E. Artin. Theory of braids. Ann. of Math. (2) , 48:101–126, 1947. 8[Bar16] Laurent Bartholdi. 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