## New York Journal of Mathematics

New York J. Math.19(2013) 395–421.

## Automorphisms of free groups. I

### Laurent Bartholdi

Abstract. We describe, up to degree n, the Lie algebra associated with the automorphism group of a free group of rankn. We compute in particular the ranks of its homogeneous components, and their structure as modules over the linear group.

Along the way, we infirm (but confirm a weaker form of) a conjec- ture by Andreadakis, and answer a question by Bryant–Gupta–Levin–

Mochizuki.

Contents

1. Introduction 396

1.1. Main results 397

1.2. Main points 398

1.3. Plan 399

1.4. Thanks 399

2. GLr(Q)-modules 399

2.1. Inflation 400

3. Free groups, their Lie algebras, and their automorphisms 401

3.1. Pronilpotent groups 402

3.2. Automorphisms 402

3.3. Structure ofJ 403

4. Free differential calculus 405

5. Structure ofM 406

5.1. Decomposition inGL_{r}(Q)-modules 408

6. Structure ofL 410

6.1. Generators ofA 411

7. Proofs of the main theorems 412

7.1. Theorem A 412

7.2. Theorem C 414

7.3. Theorems B and D 414

8. Examples and illustrations 415

Received March 29, 2013.

2010Mathematics Subject Classification. 20E36, 20F28, 20E05, 20F40.

Key words and phrases. Lie algebra; Automorphism groups; Lower central series.

ISSN 1076-9803/2013

395

8.1. r= 2 416

8.2. r= 3 and r= 4 417

References 420

1. Introduction

Let F denote a free group of rank r. The group-theoretical structure of
the automorphism groupAofF is probably exceedingly difficult to describe,
butA may be ‘graded’, following Andreadakis [1], into a more manageable
object. Let Fn denote the nth term of the lower central series of F, and
let A_{n} denote the kernel of the natural mapAut(F)→Aut(F/F_{n+1}). Then
A0/A1 =GLr(Z), andAn/An+1 are finite-rank freeZ-modules; furthermore,
[A_{n}, A_{m}]⊆A_{m+n}, and therefore

L =M

n≥1

An/An+1

has the structure of a Lie algebra.

Let, by comparison, Fb denote the limit of the quotients (F/Fn)n≥1; it
is a pronilpotent group, and F /b Fbn is naturally isomorphic to F/Fn. Let B
denote the automorphism group ofFb, and let similarlyB_{n}denote the kernel
of the natural map Aut(F)b → Aut(F /b Fbn+1). Then B0/B1 = GLr(Z) and
M =L

n≥1B_{n}/B_{n+1} is also a Lie algebra; furthermore, B_{n}/B_{n+1} are also
finite-rank freeZ-modules. In contrast toL, the structure ofM is well un-
derstood: it acts on the completion of the freeZ-Lie algebraL

n≥1Fbn/Fbn+1, and its elements may be described as “polynomial noncommutative first- order differential operators”, that is expressions

X

i

α_{i}X_{i}_{1}. . . X_{i}_{n} ∂

∂X_{i}_{0}

in the noncommuting variables X_{1}, . . . , X_{r}. The embedding F → Fb with
dense image induces a natural map L → M, which is injective but not
surjective.

The following problems appear naturally:

(1) Describe the closure of the image of L inM.

(2) Relate An to the lower central series (γn(A1))n≥1 ofA1. (3) Compute the ranks ofLn=An/An+1 and Mn=Bn/Bn+1.

Ad (1), Andreadakis observes that L1 =M1 andL2 =M2, while L3 6=

M3 forr ≤3.

Ad (2), Andreadakis conjectures [1, page 253] that A_{n} = γ_{n}(A_{1}), and
proves his assertion forr= 3,n≤3 and forr = 2. This is further developed
by Pettet [27], who proves that γ_{3}(A_{1}) has finite index in A_{3} for all r,
building her work on Johnson’s homomorphism [15]. Further results have
been obtained by Satoh [28,29] and, in particular, what amounts to our

Theorem C under a slightly stronger restriction on the parameter n. The arguments in [7] let one deduce TheoremsBand Dfrom Theorem A.

Ad (3), Andreadakis proves rankMn= r

n+ 1 X

d|n+1

µ(d)r^{(n+1)/d},

where µ denotes the M¨obius function, and computes for r = 3 the ranks rank(Ln) = 9,18,44 for n = 2,3,4 respectively. Pettet [27] generalizes these calculations to

rank(L2) = r^{2}(r−1)

2 , rank(L3) = r^{2}(r^{2}−4)

3 + r(r−1)

2 .

1.1. Main results. In this paper, we prove:

Theorem A. For allr,n, we have γ_{n}(A_{1})≤A_{n}, and A_{n}/γ_{n}(A_{1})is a finite
group. Moreover,

An=p

γn(A1),

that is, A_{n} is the set of all g∈A such that g^{k}∈γ_{n}(A_{1}) for some k6= 0.

On the other hand, forr = 3, n= 7, we have An/γn(A1) =Z/3.

Therefore, Andreadakis’s conjecture is false, but barely so.

Theorem B. If r ≥n >1, then we have the rank formula

(1) rankLn= r

n+ 1 X

d|n+1

µ(d)r^{(n+1)/d}− 1
n

X

d|n

φ(d)r^{n/d},

where φ denotes the Euler totient function.

As a byproduct, Andreadakis’s above calculations for r = 3 should be corrected to rank(L4) = 43.

In studying the structure of L, I found it useful to consider Ln =
A_{n}/A_{n+1} not merely as an abelian group, but rather as a GL_{r}(Z)-module
under the conjugation action of A_{0}/A_{1}, and then to appeal to the classi-
fication of GLr(Q)-representations by tensoring with Q. Theorem B is a
consequence of the following description of Ln⊗Q as aGL_{r}(Q)-module.

We start by a GLr(Q)-module decomposition of Mn⊗Q. It turns out thatMn⊗Qnaturally fits into an exact sequence

0−→Tn−→Mn⊗Q−→^{tr} An−→0,

whose terms we now describe. Let{x_{1}, . . . , x_{r}}denote a basis ofF. The first
subspaceTnconsists of theGLr(Q)-orbit in Mn⊗Qof the automorphisms

T_{w} :x_{i} 7→x_{i} for alli < r, x_{r}7→x_{r}w

for all choices ofw∈Fn+1∩hx_{1}, . . . , xr−1i, and are so called because of their
affinity to ‘transvections’. The second subspaceAn may be identified with
theGLr(Q)-orbit in Mn⊗Qof

Aa1...an :xi7→xi[xi, a1, . . . , an] for all i,

for all choices ofa_{1}, . . . , a_{n}∈F; here and below [u, v] denotes the commuta-
toru^{−1}v^{−1}uv, and [u1, . . . , un] denotes the left-normed iterated commutator
[[u_{1}, . . . , un−1], u_{n}].

Forr ≥n, the spaceAnisr^{n}-dimensional, and is isomorphic quaGL_{r}(Q)-
module with H1(F,Q)^{⊗n}, via Aa1...an ↔ a1 ⊗ · · · ⊗ an; hence the name
reminding the A_{a}_{1}_{...a}_{n} of their ‘associative’ origin.

Again for r ≥ n, we may define An as H1(F,Q)^{⊗n}, and then the ‘trace
map’ tr : Mn → An sends an automorphism to the trace of its Jacobian
matrix; compare [23].

Theorem C. Assume r ≥ n > 1, and identify An with H1(F,Q)^{⊗n}. Let
Z/n=hγi act onAn by cyclic permutation:

(a_{1}⊗ · · · ⊗a_{n})γ =a_{2}⊗ · · · ⊗a_{n}⊗a_{1}.

ThenLn⊗Q containsTn, and its image inAn is the subspace of “cycli- cally balanced” elements An(1−γ) spanned by all

a_{1}⊗ · · · ⊗a_{n}−a_{2}⊗ · · · ⊗a_{n}⊗a_{1}.

TheGL_{r}(Q)-decomposition ofV_{n}=H_{1}(F,Q)^{⊗n}mimicks that of the regu-
lar representation of the symmetric groupS_{n}, and is well described through
Young diagrams (see §2 for the definitions of Young diagram, tableaux and
major index). For example, the decomposition ofV_{n} in irreducibles is given
by all standard tableaux withnboxes. Lie elements inVn, which correspond
to inner automorphisms inAn, correspond to standard tableaux with major
index≡1 (mod n), as shown by Klyashko [17]. We show:

Theorem D. Ifr ≥n, the decomposition ofLn⊗Qin irreducibles is given as follows:

• All standard tableaux withn+1boxes, major index≡1 (mod n+1), and at mostr−1 rows, to which a column of length r−1 is added at the right.

• All standard tableaux withnboxes, at most r rows, and major index 6≡0 (modn).

The first class corresponds to Tn, and the second one to An.

In fact, numerical experiments show that TheoremBshould remain true under the weaker condition r≥n−1. Illustrations appear in §8.

1.2. Main points. The proofs of Theorems A,B,D follow from classical results in the representation theory ofGLr(Q). The proof of TheoremCuses results of Birman and Bryant–Gupta–Levin–Mochizuki to the respective ef- fects that a endomorphism is invertible if and only if its Jacobian matrix is invertible, and that in that case the trace of its Jacobian matrix is cyclically balanced.

In fact, these last authors ask whether that condition is sufficient for an endomorphism to be invertible; I give in §6 an example showing that it is not so.

1.3. Plan. §2briefly summarizes the representation theory of GL_{r}(Q).

§3 recalls some facts about the automorphism group of a free group in the language of representation theory and free differential calculus.

§4 recalls elementary properties of free differential calculus.

§5 and§6describe the Lie algebrasM and L respectively, both as alge-
bras and as GL_{r}(Q)-modules.

§7 proves the theorems stated above.

Finally, §8 provides some examples and illustrations of the main results.

Depending on the reader’s familiarity with the subject, she/he may skip to§5.

1.4. Thanks. I greatly benefited from discussions with Andr´e Henriques, Joel Kamnitzer and Chenchang Zhu, and wish to thank them for their pa- tience and generosity. I am also grateful to Steve Donkin, Donna Testerman, Takao Satoh and Naoya Enomoto for remarks and references that improved an earlier version of the text, and to the anonymous referee for his/her valuable remarks.

Some decompositions were checked using the computer software system GAP[11], and in particular its implementation of the “meataxe”. Extensive calculations led to the second statement of TheoremA.

There has been a big gap between the beginning and the end of my writing this text, and I am very grateful to Benson Farb for having: (1) encouraged me to finish the writeup; (2) given me the opportunity of doing it at the University of Chicago in a friendly and stimulating atmosphere.

2. GL_{r}(Q)-modules

Throughout this section we denote by V the natural GLr(Q)-module Q^{r}.
We consider only algebraic representations, i.e., those linear representa-
tions whose matrix entries are polynomial functions of the matrix entries of
GL_{r}(Q). The degree of such a representation is the degree of these polyno-
mial functions. IfW is a representation of degreen, then the scalar matrix
µ1 acts byµ^{n} on W.

A fundamental construction by Weyl (see [10,§ 15.3]) is as follows. The tensor algebra ofV decomposes as

T(V) =M

n≥0

V^{⊗n}= M

λpartition ofn

U_{λ}⊗W_{λ},

where U_{λ} and W_{λ} are respectively irreducible S_{n}- and GLr(Q)-modules.

Each irreducible S_{n}-representation appears exactly once in this construc-
tion, and those W_{λ} which are nonzero, i.e., for whichλhas at mostr lines,
describe all representations of GL_{r}(Q) exactly once, up to tensoring with a
power of the one-dimensional determinant representation.

Therefore, degree-nrepresentations ofGL_{r}(Q) are indexed by irreducible
representations ofS_{n}, i.e., by conjugacy classes of S_{n}, i.e., by partitions of
{1, . . . , n}, the parts corresponding to cycle lengths in the conjugacy class.

Partitions with more thanrparts yieldW_{λ}= 0, and therefore do not appear
in the decomposition of V^{⊗n}.

It is convenient to represent partitions as Young diagrams, i.e. diagrams of boxes. The lengths of the rows, assumed to be weakly decreasing, give the parts in a partition. Thus

is the partition 5 = 2 + 2 + 1. The natural representation V is described by a single box, and its symmetric and exterior powers are represented by a single row and a single column of boxes respectively. A standard tableau with shape λ, for λ a partition of n, is a filling-in of the Young diagram of λ with each one of the numbers {1, . . . , n} in such a way that rows and columns are strictly increasing rightwards and downwards respectively. For example,

1 3 2 5

4 ,

1 2 3 5

4 ,

1 2 3 4

5 ,

1 4 2 5 3 ,

1 3 2 4 5

are the standard tableaux with shape 2 + 2 + 1. For λa partition ofn, the
multiplicity of the GL_{r}(Q)-module W_{λ} inV^{⊗n} is the dimension of U_{λ}, and
is the number of standard tableaux with shape λ. The module Wλ may be
written as V^{⊗n}c_{λ} for some idempotent c_{λ} :QS_{n} → QS_{n}, called the Schur
symmetrizer. This amounts to writing

(2) W_{λ} =V^{⊗n}⊗_{Q}_{S}_{n}U_{λ},

and c_{λ} for the projection fromQS_{n} toU_{λ}.

The major index of a tableau T is the sum of those entries j ∈ T such that j+ 1 lies on a lower row thatj in T. For example, the major indices in the example above are respectively 4, 5, 6, 7, 8.

The tensor algebra T(V) contains as a homogeneous subspace the Lie algebra generated byV; it is isomorphic to the free Lie algebraL(F). The homogeneous components Ln(F) are naturally GLr(Q)-modules, and their decomposition in irreducibles is described by Klyashko [17]:

Proposition 2.1. The decomposition in irreducibles of Ln(F) is given by those tableaux with nboxes whose major index is ≡1 (mod n).

2.1. Inflation. We use a construction of GL_{r}(Q)-modules fromGLr−1(Q)-
modules, calledinflation. Consider aGLr−1(Q)-module S. It is naturally a
Q^{r−1}oGLr−1(Q)-module, via the projectionQ^{r−1}oGLr−1(Q)→GLr−1(Q).

We may embed the affine group Aff :=Q^{r−1}oGLr−1(Q) in GLr(Q) as the
matrices with last row (0, . . . ,0,1). For an algebraic groupG, letP(G) de-
note the Hopf algebra of polynomial functions on G. We may then induce
S to aGL_{r}(Q)-moduleSe:=S⊗_{P(Aff)}P(GL_{r}(Q)). In fact, the inverse opera-
tion is easier to describe: restrict theGLr(Q)-moduleSetoAff, and consider

then the fixed points S of Q^{r−1}; this is an irreducible GLr−1(Q)-module.

See [14, II.2.11] for details.

This inflated moduleSehas the same degree asS, and moreover its decom-
position in irreducibles admits the same Young diagrams as S’s: indeed it
is immediate to check thatQ^{r−1}⊗_{P(Aff)}P(GLr(Q)) =Q^{r}. Every irreducible
submodule of S may be seen as a submodule of (Q^{r−1})^{⊗n} for some n, us-
ing (2). We may then describeS as (Q^{r−1})^{⊗n}⊗_{QS}_{n}U for some S_{n}-module
U. We get

Se= (Q^{r})^{⊗n}⊗_{QS}_{n}U.

3. Free groups, their Lie algebras, and their automorphisms LetGbe a group. We recall a standard construction due to Magnus [20].

Let (Gn)n≥1 be a chain of normal subgroups of G, with Gn+1 ⊆ Gn and
[G_{m}, G_{n}]⊆G_{m+n} for all m, n≥1.

Definition 3.1. The Lie ring associated with the series (G_{n}) is
L =L(G) =

∞

M

n=1

Ln,

withLn=G_{n}/G_{n+1}.

Addition within the homogeneous componentLnis inherited from group multiplication in Gn, and the Lie bracket on L is defined among homoge- neous elements by

Lm×Ln→Lm+n, (uG_{m+1}, vG_{n+1})7→[u, v]G_{m+n+1}.

A typical example is obtained by letting (G_{n}) be the lower central se-
ries (γn(G)) of G, defined by γ1(G) = G and γn+1(G) = [γn(G), G]. The
subgroupsA_{n}described in the introduction yield an interesting (sometimes
different) series.

We have an action ofGonG_{n}by conjugation, which factors to an action
of G/G_{1} on L sinceG_{1} acts trivially on G_{n}/G_{n+1}.

Conversely, if Gn is characteristic in G for all n, then we may set H =
GoAutGtheholomorphofG, and consider the sequence (G_{n})n≥1as sitting
insideH. The resulting Lie algebraL admits, by the above, a linear action
of Aut(G). The IA-automorphisms of G — those automorphisms that act
trivially onG/[G, G] — act trivially onL because [G, G]⊂G2, so the linear
groupGL(H_{1}G) =Aut(G)/IA(G) acts onL.

Lubotzky originally suggested to me that the structure of the groups A=Aut(F) and B =Aut(Fb) could be understood by considering their Lie algebras with GLr(Z)-action; see [2]. For fruitful developments of this idea see [6].

3.1. Pronilpotent groups. A pronilpotent group is a limit of nilpotent
groups. We recall some useful facts gleaned from [1]. LetF denote a (usual)
free group of rankr. GiveF a topology by choosing as basis of open neigh-
bourhoods of the identity the collection of subgroupsF_{n}inF’s lower central
series, and letFbbe the completion ofF in this topology. We naturally view
F as a dense subgroup ofFb.

In considering series (Fb_{n}) of subgroups ofFb, we further require that the
Fbnbe closed inFb. Let (Fbn) be the (closed) lower central series ofFb, defined
byFbn+1= [Fbn,Fb]. We have Fn=F∩Fbn, andFnis dense in Fbn. Therefore
Ln(Fb) = Ln(F), and by [22, Chapter 5] the module Ln(Fb) is Z-free, of
rank rn given by Witt’s formula

rn= 1 n

X

d|n

µ(d)r^{n/d},

whereµ denotes the Moebius function.

From now on, we reserve the symbolJ for this Lie algebraL(Fb). It is
naturally equipped with aGL_{r}(Z)-action.

3.2. Automorphisms. We next turn to the groupB of continuous auto-
morphisms ofF, in the usual compact-open topology. Sinceb F is dense inF,b
every automorphismφ∈B is determined by the imagesx^{φ}_{1}, . . . , x^{φ}_{r} ∈Fb of a
basis ofF. The images of these in F /b Fb2 =Z^{r} determine a homomorphism
B →GL_{r}(Z), which is onto just as for the homomorphismA→GL_{r}(Z).

The embeddingF 7→Fbinduces an embedding A→B which, as we shall see, does not have dense image.

LetB_{1} denote the kernel of the mapBGL_{r}(Z); more generally, denote
by πn the natural map πn : B → Aut(F/Fn+1) for all n ≥ 0, and set
B_{n} = kerπ_{n}. this defines a series of normal subgroups B =B_{0} > B_{1} >· · ·.
Furthermore, by a straightforward adaptation of [1, Theorem 1.1], we have
[B_{m}, B_{n}]⊂B_{m+n}, and therefore a Lie algebra

M =M

n≥1

Mn=M

n≥1

Bn/Bn+1.

Every elementφ∈B_{1} is determined by the elementsx^{−1}_{1} x^{φ}_{1}, . . . , x^{−1}_{r} x^{φ}r ∈
Fb_{2}; and, conversely, every choice of f_{1}, . . . , f_{r} ∈Fb_{2} determines a homomor-
phism φ_{0} :F →Fb defined on the basis by x^{φ}_{i}^{0} =x_{i}f_{i}, and extended multi-
plicatively; the so-defined mapφ0 extends to a continuous map φ:Fb →F,b
since φ^{−1}(Fb_{n}) contains Fb_{n} and the (Fb_{n}) are a basis for the topology on F.b
Finally, φ is onto, because φπn is onto for all n ≥ 0; indeed, a family of
elements{x^{φπ}_{1} ^{n}, . . . , x^{φπ}r ^{n}}generates the nilpotent groupF/Fnif and only if
it generates its abelianizationF/F2.

By [1,§4] or [19, Theorem 5.8], theZ-moduleMn is free of rank r·r_{n+1}.
Furthermore,B0/B1∼=GLr(Z), so M is naturally equipped with aGLr(Z)-
action.

3.3. Structure of J. In this § we set V = H_{1}(F ,b Z) ∼= Z^{r}, and study
the structure ofJ as aGLr(Z)-module. It is well-known [20] that, as a Lie
algebra, J is the freeZ-algebra generated byV.

Fb_{n}is topologically spanned byn-fold commutators of elements ofFb, which
can be written as functionsf =f(v_{1}, . . . , v_{n}), wheref is now also seen as a
commutator expression, evaluated at elementsvi ∈Fb. Of the other hand, if
we considerf as an element of Jn=Fbn/Fbn+1, thesevi should actually be
seen as elements ofV =F /b Fb2, sincef(v1, . . . , vn)∈Fbn+1 as soon as one of
thev_{i} belongs toFb_{2}.

The action of GLr(Z) on commutator expressions f = f(v1, . . . , vn) is
diagonal: f^{ρ}=f(v^{ρ}_{1}, . . . , v^{ρ}n). This yields immediately

Lemma 3.2. The representation Jn of GLr(Z) has degree n.

The following result dates back to the origins of the study of free Lie rings [4], and is even implicit in Witt’s work; it appears in the language of operads in [12, Proposition 5.3].

Theorem 3.3. The decomposition of Jn as a GL_{r}(Z)-module is given by
inclusion-exclusion as follows:

Jn= 1 n

M

d|n

µ(d)(ψdV)^{⊗n/d},

where ψ_{d} is the d-th Adams operation (keeping the underlying vector space,
raising eigenvalues to the d-th power).

Although this formula is explicit and allows fast computation of character
values, it is not quite sufficient to write downJnconveniently — it would be
better to expressJnasV^{⊗n}⊗_{Z}_{S}_{n}Snfor an appropriateS_{n}-representation
Sn.

Let us assume for a moment thatK is a ring containing a primitiven-th root of unity ε, and that V is a freeK-module of rank r. Then by [17] we have

(3) S_{n}= Ind^{S}^{n}

Z/nK_{ε},

whereZ/n acts on Kε∼=K by multiplication by ε. Furthermore, if K con-
tains _{n}^{1}, Klyashko gives an isomorphism between the functorsV 7→Jn(V)
and

V 7→C_{n}(V) = Hom_{K[Z/n]}(K_{ε}, V^{⊗n})'

v ∈V^{⊗n}|vγ=εv},
whereZ/n=hγi acts onV^{⊗n} by permutation of the factors.

We may not assume that Z contains n-th roots of unity — it does not;

however, Sn is defined over Z and may be constructed without reference

to any ε. Numerous authors [16,18] have studied the decomposition in irreducibles of the induction from a cyclic subgroup of a one-dimensional representation. We reproduce it here in our notation.

Proposition 3.4 ([18]). The multiplicity of the irreducible representation
U_{λ} in S_{n} is the number of standard Young tableaux of shape λ and major
index congruent to 1 modulon.

The following result seems new, and constructs efficiently the representa-
tion S_{n} without appealing ton-th roots of unity:

Proposition 3.5. Inside S_{n}, consider the following subgroups: a cyclic
subgroup Z/n generated by a cycle γ of length n; its automorphism group
(Z/n)^{∗}; and its subgroup(n/d)Z/ngenerated byγ^{n/d}, isomorphic to a cyclic
group of order d. Then

(4) Sn=M

d|n

µ(d)Ind^{S}_{((n/d)}^{n}

Z/n)o(Z/n)^{∗}1,

where 1 denotes the trivial representation.

For concreteness, we may identify S_{n} with the symmetric group of Z/n.

Then γ is the permutation i7→ i+ 1 (mod n), and (Z/n)^{∗} is the group of
permutations of the form i7→ki (modn) for allk coprime to n.

Proof. The proof proceeds by direct computation of the characters of the left- and right-hand side of (4), using the expression (4).

To simplify notation, we will write C = Z/n. We write elements of
CoC^{∗} as (m, u). For m ∈ C we write m^{∗} = n/gcd(m, n) its order in C.

We enumerate C^{∗}={u_{1}, . . . , u_{φ(n)}}.

It suffices actually to prove that the inductions of Kε and 1 to CoC^{∗}
are isomorphic. Let α denote the character of Ind^{CoC}_{C} ^{∗}K_{ε}; then

α(m, u) =

(µ(m^{∗})_{φ(m}^{φ(n)}∗) ifu= 1,

0 otherwise,

whereφis Euler’s totient function. Indeed Ind^{C}_{C}^{o}^{C}^{∗}α(m, u) is aφ(n)×φ(n)-
monomial matrix; it is the product of a diagonal matrix with entries ε^{µ}^{1},
. . . , ε^{µ}^{φ(n)} and the permutational matrix given byu’s natural action onC^{∗}.
This matrix has trace 0 unless u= 1, in which case its trace is φ(n)/φ(m^{∗})
the sum of all primitivem^{∗}-th roots of unity.

Letβ_{d}denote the character of Ind^{CoC}_{C}_{n/d}^{∗}

oC^{∗}1. Then by similar reasoning
β(m, u) =

(gcd(^{n}_{d}, u−1) if gcd(^{n}_{d}, u−1)|m,

0 otherwise.

The result now follows from the next, elementary lemma, whose proof is immediate, by noting that the left- and right-hand sides are multiplicative,

and agree whenn is a prime power.

Lemma 3.6. For any `|n, we have X

`|d|n

µ(d)n

d =µ(`)φ(n) φ(`).

If 1< u < n, we also have X

`|d|n

µ(d) gcd ^{n}_{d}, u−1

= 0.

4. Free differential calculus

We recall the basic notions from [9]. Let again F denote a free group of
rank r, with basis{x_{1}, . . . , x_{r}}. Define derivations

∂

∂xi

:ZF →ZF
by the rules _{∂x}^{∂}

ixi = 1, _{∂x}^{∂}

i(x^{−1}_{i} ) =x^{−1}_{i} , and _{∂x}^{∂}

i(x^{±1}_{j} ) = 0 ifi6=j, extended
to ZF linearity and by the Leibniz rule _{∂x}^{∂}

i(uv) = _{∂x}^{∂u}

iv^{o} +u_{∂x}^{∂v}

i, where o:ZF →Zdenotes the augmentation map.

A simple calculation proves the formula

(5) ∂

∂xi

[u, v] =u^{−1}v^{−1}

(u−1)∂v

∂xi

−(v−1)∂u

∂xi

.

In particular, if u∈γ_{n}(F), then moduloγ_{n+2}(F) we have

∂

∂xi

[u, x_{i}]≡(u−1)−(x_{i}−1)∂u

∂xi

, ∂

∂xi

[u, x_{j}]≡ −(x_{j}−1)∂u

∂xi

ifj6=i.

Denote by $ ≤ ZF the kernel of o; then for all u ∈ ZF we have the

“fundamental relation” [9, (2.3)]

u−u^{o} =

r

X

i=1

∂u

∂xi

(xi−1).

WriteXi =xi−1 for i∈ {1, . . . , r}, and consider the ring
R=ZhhX_{1}, . . . , X_{r}ii

of noncommutative formal power series. The map τ : x_{i} 7→ X_{i} + 1 defines
an embedding of F in R, which extends to an embedding τ : ZF → R.

Let $denote also the fundamental ideal hX_{1}, . . . , Xri of R; this should be
no cause of confusion, since $^{τ} =$∩(ZF)^{τ}. The ring R is graded, with
homogeneous component R_{n} of rank r^{n}, spanned by words of length n in
X_{1}, . . . , X_{r}.

The dense subalgebra of R generated by the Xi is free of rank r; it is therefore a Hopf algebra, isomorphic to the enveloping algebra of the free Lie algebra J. From now on, we consider J as a Lie subalgebra of Rin this manner.

5. Structure of M

We are ready to understand the Lie algebraM associated with the auto- morphism group B ofFb.

The module V =H1(Fb;Z) naturally identifies with F /b Fb2. Its dual, V^{∗},
identifies with homomorphismsFb→Z.

Theorem 5.1. TheGL_{r}(Z)-moduleMn is isomorphic to V^{∗}⊗Jn+1. The
isomorphism ρ:V^{∗}⊗Jn+1 →Mn is defined on elementary tensorsα⊗f
by

α⊗f 7→ {x_{i}7→xif^{α(x}^{i}^{)}},
and extended by linearity.

The proof is inspired from [24]; see also [19, Lemma 5.7].

Proof. Consider an elementary tensor α⊗f. There is a unique endomor-
phism φ:Fb → Fb satisfying x^{φ}_{i} =x_{i}f^{α(x}^{i}^{)}, so ρ’s image in contained inB.

Next, {x_{i}f^{α(x}^{i}^{)}}is a basis of Fb, since it spansF /b Fb_{2}, soφis invertible.

The mapρ is well-defined: iff ∈Fn+2, then the automorphism
xi 7→xif^{α(x}^{i}^{)}

of Fb belongs toBn+1, so the automorphism α⊗f is may be defined indif-
ferently for an element f ∈Jn+1 or its representativef ∈F_{n+1}.

Let us denote temporarily byx^{∗}_{1}, . . . , x^{∗}_{r} the dual basis of V^{∗}, defined by
x^{∗}_{i}(x_{j}) =1_{ij}.

We construct a map σ :B_{n} → V^{∗} ⊗Fb_{n+1}. Let φ∈ B_{n} be given. Then
x^{φ}_{i} ≡xi mod Fbn+1, sox^{φ}_{i} =xifi for somefi ∈Fbn+1. We set

φ^{σ} =

r

X

i=1

x^{∗}_{i} ⊗f_{i}.

If φ∈B_{n+1}, then f_{i} ∈Fb_{n+2}, so φ^{σ} ∈V^{∗}⊗Fb_{n+2}. It follows thatσ induces
a well-defined mapMn→ V^{∗}⊗Jn+1. Furthermore the maps ρ and σ are
inverses of each other.

Next, we check thatρ is linear. Consider (α⊗f)^{ρ}=φand (β⊗g)^{ρ}=χ.

Then

x^{φχ} = (xf^{α(x)})^{χ}= (xf^{α(x)})g^{β(xf}^{α(x)}^{)}=xf^{α(x)}g^{β(x)} =x^{ρ(α⊗f+β⊗g)}.
Finally, we show that the GL_{r}(Z)-actions are compatible. Choose an
element ofGLr(Z), and lift it to someµ∈B. Consider (α⊗f)^{ρ}=φ. Then
(α⊗f)^{µ}=α^{0}⊗f^{µ}, whereα^{0} ∈V^{∗} is defined byα^{0}(x) =α(x^{µ}^{−1}), so

x^{φ}^{µ} =x^{µ}^{−1}^{φµ}=

x^{µ}^{−1}f^{α(x}^{µ}

−1)

µ

=x(f^{µ})^{α(x}^{µ}

−1) =x^{(α}^{0}^{⊗f}^{µ}^{)}^{ρ}.
The Lie bracket onM may be expressed via the identification ρ of The-
orem 5.1.

Theorem 5.2. Consider α, β ∈ V^{∗} and f = f(v_{0}, . . . , v_{m}) ∈ Jm+1 and
g =g(w0, . . . wn) ∈Jn+1. Then the bracket Mm×Mn → Mm+n is given
by

[α⊗f, β⊗g] =α⊗

m

X

i=0

β(v_{i})f(v_{0}, . . . , vi−1, g, v_{i+1}, . . . , v_{m})

−β⊗

n

X

i=0

α(w_{i})g(w_{0}, . . . , wi−1, f, w_{i+1}, . . . , w_{n}).

Proof. Write (α⊗f)^{ρ} = φ and (β⊗g)^{ρ} = χ. Then φ^{−1} = (α⊗f^{−φ}^{−1})^{ρ}
and χ^{−1} = (β⊗f^{−χ}^{−1})^{ρ}; indeed

(x^{φ})^{φ}^{−1} = (xf^{α(x)})^{φ}^{−1} =xf^{−φ}^{−1}^{α(x)}f^{α(x)φ}^{−1} =x.

We then compute

x^{[φ,χ]}=x^{φ}^{−1}^{χ}^{−1}^{φχ}= (xf^{−φ}^{−1}^{α(x)})^{χ}^{−1}^{φχ}

= (xg^{−χ}^{−1}^{β(x)})^{φχ}f^{−α(x)φ}^{−1}^{χ}^{−1}^{φχ}

= (xf^{α(x)})^{χ}g^{−β(x)χ}^{−1}^{φχ}f^{−α(x)φ}^{−1}^{χ}^{−1}^{φχ}

=xg^{β(x)}f^{α(x)χ}g^{−β(x)χ}^{−1}^{φχ}f^{−α(x)φ}^{−1}^{χ}^{−1}^{φχ}

≡x

f^{α(x)χ}f^{−α(x)φ}^{−1}^{χ}^{−1}^{φχ} g^{β(x)}g^{−β(x)χ}^{−1}^{φχ}

mod Fbn+m+2

≡xf^{α(x)(χ−1)}g^{β(x)(1−φ)}=x^{(α⊗f}^{χ−1}^{−β⊗g}^{φ−1}^{)}^{ρ} mod Fb_{n+m+2}.
Now, again computing moduloFbn+m+2, we have

f^{χ}=f(v_{0}g^{β(v}^{0}^{)}, . . . , v_{m}g^{β(v}^{m}^{)})≡f

m

Y

i=0

f(v_{0}, . . . , vi−1, g, v_{i+1}, . . . , v_{m})^{β(v}^{i}^{)},

and similarly for g, so the proof is finished.

In fact, the dual basis {x^{∗}_{i}} of V^{∗} is naturally written _{∂}

∂xi ; in that language, Theorem 5.1can be rephrased in an isomorphism

ρ:

r

X

i=1

fi

∂

∂xi

7→

φ:xj 7→xj r

Y

i=1

f

∂xj

∂xi

i =xjfj

betweenM and order-1 differential operators onR. Moreover, if P fi ∂

∂xi ∈
Mn thenfi ∈Fbn+1, sofi−1∈ R_{n+1}. Theorem 5.2 then expresses the Lie
bracket onM as a kind of “Poisson bracket”: forY ∈ R_{n+1}andZ ∈ R_{m+1},
we have

Y ∂

∂x_{i}, Z ∂

∂x_{j}

= ∂Y

∂x_{j}Z ∂

∂x_{i} − ∂Z

∂x_{i}Y ∂

∂x_{j} ∈V^{∗}⊗ R_{n+m+1}.

The representationMn can also be written in terms of representations of the symmetric group, as follows. The representation Mn⊗det has degree

n+r, and therefore may be written as V^{⊗(n+r)}⊗_{Z}_{S}_{n+r} T_{n}, for some rep-
resentation Tn of S_{n+r}. Recall that Sn denotes the representation of S_{n}
corresponding to the Lie submoduleJn⊂ R_{n}.

Proposition 5.3. Let (−1)denote the sign representation of S_{r−1}. Then
T_{n}= Ind^{S}_{S}^{n+r}

n+1×Sr−1S_{n+1}⊗(−1).

Proof. Let W, W^{0} be two GL_{r}(Z)-representations, of degreesm, m^{0} respec-
tively. Then they may be writtenW =V^{⊗m}⊗_{Z}_{S}_{m}T andW^{0} =V^{⊗m}^{0}⊗_{Z}_{S}

m0

T^{0} for representations T, T^{0} of S_{m},S_{m}^{0} respectively. Their tensor product
W ⊗W^{0} then satisfies

W ⊗W^{0} ∼=V^{⊗(m+m}^{0}^{)}⊗_{Z}_{S}

m+m0 Ind^{S}_{S}^{m+m}^{0}

m×S_{m}0T ⊗T^{0}.

The proposition then follows from Theorem 5.1, with W = V^{∗} and W^{0} =
Jn+1, sinceV^{∗}⊗det =V^{⊗r−1}⊗_{Z}_{S}_{r−1} (−1).

5.1. Decomposition in GLr(Q)-modules. We now turn to the funda-
mental decomposition of the module Mn. Its GL_{r}(Z)-module structure
seems quite complicated; so we content ourselves with a study of theGL_{r}(Q)-
module Mn⊗Q. We define the following two submodules of Mn⊗Q. The
first,Tn, is spanned by theGL_{r}(Q)-orbit of the automorphisms

Tw :xi 7→xi for alli < r, xr7→xrw

for all choices of w ∈ F_{n+1}∩ hx_{1}, . . . , xr−1i. The second subspace, An, is
spanned by the automorphisms

Aa1...an :xi7→xi[xi, a1, . . . , an] for all i,
for all choices of a_{1}, . . . , a_{n}∈F.

Lemma 5.4. We have Mn⊗Q=Tn⊕An quaGL_{r}(Q)-modules.

Proof. We may view Tn and An as submodules of V^{∗} ⊗Jn+1 via The-
orem 5.1. Here, Tn is spanned by all those α ⊗f(v0, . . . , vn) such that
α(v_{i}) = 0 for all i ∈ {0, . . . , n}, when f ranges over n-fold commutators.

On the other hand, An is spanned by theP

ix^{∗}_{i} ⊗f(xi, v1, . . . , vn) when f
ranges over n-fold commutators. We conclude that Tn∩An = 0, and it
remains to check, by dimension counting, that Tn+An=Mn⊗Q.

By the Littlewood–Richardson rule, the moduleT_{n} from Proposition 5.3
is a sum of irreducible representations of S_{n+r} of all possible skew shapes
λobtained by playing the “jeu du taquin” on a column of heightr−1 (the
Young diagram of the sign representation) and shapesµappearing in S_{n+1}.
In the diagram λ, the column of height r−1 occupies either the places
(1,1), . . . ,(r−1,1), or the places (2,1), . . . ,(r,1). We shall see that the first
case corresponds to summands of Tn, and the second case corresponds to
summands of An.

In the first case, the original representation µ of S_{n+1} subsists, on the
condition that it contains at most r−1 lines. These summands therefore

precisely describe those representations of S_{n+1} on Ln+1 that come from
Fn+1∩ hx_{1}, . . . , xr−1i.

In the second case, the “jeu du taquin” procedure asks us to remove box
(1,2) from µ to fill position (1,1), and to propagate this hole in µ. This
amounts to restricting Sn+1 to the natural subgroup S_{n} of S_{n+1}. Recall
that Sn+1 = Ind^{S}_{C}^{n+1}χ, for a primitive character χ of the cyclic group C
generated by a cycle of lengthn+ 1. By Mackey’s theorem,

Res^{S}_{S}^{n+1}

n Sn+1= Ind^{S}_{C∩S}^{n}

nRes^{C}_{C∩S}_{n}χ= Ind^{S}_{1}^{n}1,

sinceCSn=S_{n+1} and C∩S_{n}= 1. Now the GLr(Q)-representation asso-
ciated with the S_{n}-representation QS_{n} is the full spaceV^{⊗n}, which spans

An naturally.

The correspondence X_{i}_{1}. . . X_{i}_{n} 7→ A_{x}_{i}

1...x_{in} defines a linear map θ_{n}^{0} :
R_{n}→An.

Lemma 5.5. If r ≥ n then θ^{0}_{n} is bijective, and makes An isomorphic to
V^{⊗n} qua GL_{r}(Q)-module.

Proof. It is clear thatθ^{0}_{n}is onto, and is compatible with theGLr(Q)-action.

Continuing with the argument of the previous lemma, the Young dia-
grams λand µ automatically have at mostr rows because r ≥n; so, since
Res^{S}_{S}^{n+1}

n Sn+1 is the regular representation, An∼=V^{⊗n}has the same dimen-

sion as R_{n}, soθ^{0}_{n} is injective.

Note, however, that θ_{n}^{0} is not injective for n > r, and that the θ^{0}_{n} do
not assemble into an algebra homomorphism R → M. There does exist,
however, an algebra homomorphismθ:R →M, defined asθ^{0}_{1} onV and ex-
tended multiplicatively toR. It givesM the “matrix-like” algebra structure
(compare with (6))

Y ∂

∂xi

· Z ∂

∂xj

= ∂Y

∂xj

Z ∂

∂xi

.

There does not seem to be any simple formula for the componentsθn of θ,
which are “deformations” ofθ^{0}_{n}.

Lemma 5.6. θ is an algebra homomorphismR →M, that is injective up to degree r. Its image is L

n≥0An. Its restriction θ J is injective, and has as image the inner automorphisms of M.

Proof. Consider the following filtration of R_{n}:

R^{i}_{n}=hproducts of elements ofR_{1} involving≥iLie bracketsi.

Then R^{0}_{n}=R_{n}, andR^{n−1}_{n} =Jn. Let R_{n}=Ln−2

i=0 R^{i}_{n}/R^{i+1}_{n} be the associ-
ated graded. A direct calculation gives

A_{a}_{1}_{...a}_{m}·A_{b}_{1}_{...b}_{n} =A_{a}_{1}_{...a}_{m}_{b}_{1}_{...b}_{n}+

m

X

j=1

A_{a}_{1}_{,...,[a}_{j}_{,b}_{1}_{,...,b}_{n}_{],...,a}_{m}.

Therefore θ_{n} = θ^{0}_{n} on Jn, and the associated graded maps θ^{0}_{n} and θ_{n}
coincide; therefore, by Lemma 5.5, the mapθn is injective ifn≤r.

This also shows that A =L

n≥0An is closed under multiplication. The
Lie subalgebra J of R then naturally corresponds under θ to the span of
the A_{[a}_{1}_{,...,a}_{n}_{]}, namely to inner automorphisms, acting by conjugation by

[a_{1}, . . . , a_{n}].

The following description is clear from the Young diagram decomposition of Tn given in Lemma5.4:

Lemma 5.7. The module Tn is isomorphic to the inflation of the module
Jn+1(hx_{1}, . . . , xr−1i) from GLr−1(Q) to GLr(Q).

6. Structure of L

We now turn to the Lie subalgebra L of M, associated with the au- tomorphism group of F. The main tool in identifying, within Mn, those automorphisms ofFbwhich “restrict” to automorphisms ofF is provided by Birman’s theorem. For an endomorphismφ:F →F, we define itsJacobian matrix and reduced Jacobian matrix

(6) Dφ= ∂(x^{φ}_{i})

∂xj

!r

i,j=1

∈M_{r}(ZF), Dφ=Dφ−1.

Theorem 6.1 ([3]). The map φ : F → F is invertible if and only if its Jacobian matrix Dφis invertible over ZF.

Ifφ∈Mn, thenφmay be written in the formP

f_{i}⊗_{∂x}^{∂}

i withf_{i} ∈F_{n+1}.
Then _{∂x}^{∂f}^{i}

j ∈$^{n}, so Dφ∈Mr($^{n}). By the chain rule, the Jacobian matrix
of a product of automorphisms is the product of their Jacobian matrices.

Consider automorphisms φ ∈ Mm, ψ ∈ Mn, so that Dφ ∈ M_{r}($^{m}) and
Dψ∈M_{r}($^{n}). Then D[φ, ψ]∈M_{r}($^{m+n}), and

D[φ, ψ]≡[Dφ, Dψ] (mod$^{m+n+1}).

The following result by Bryant, Gupta, Levin and Mochizuki gives a nec-
essary condition for invertibility, which we will show is sufficient in many
cases. Recall thatZF has an augmentation ideal$, and that$^{n}/$^{n+1}can
be naturally mapped into R_{n} via τ : (x_{i}_{1} −1)· · ·(x_{i}_{n} −1) 7→ X_{i}_{1}· · ·X_{i}_{n}.
The cyclic group Z/n=hγi naturally acts on R_{n} by cyclic permutation of
the variables:

(X_{i}_{1}· · ·X_{i}_{n})^{γ}=X_{i}_{2}· · ·X_{i}_{n}X_{i}_{1}.

LetR^{+}_{n} denote the subspace of “cyclically balanced” elements
R^{+}_{n} =R_{n}·(1−γ) ={u∈ R_{n}: u·(1 +γ+· · ·+γ^{n−1}) = 0}.

Theorem 6.2([5]). Let J ∈Mr(ZF) be such that J−1∈Mr($^{n}). IfJ is
invertible and n≥2, then

(1) the trace of J−1 belongs to($^{n}∩[$, $]) +$^{n+1};

(2) tr(J−1)^{τ} ∈ R^{+}_{n}.

Returning to our description of automorphisms φ ∈ B as P fi ∂

∂xi, we get the

Corollary 6.3. If n ≥2 and φ= P
f_{i}_{∂x}^{∂}

i ∈ Mn is in the closure of Ln, thenP ∂fi

∂xi ∈ R^{+}_{n}.

Proof. We have x^{φ}_{i} = x_{i}f_{i}, so (Dφ)_{i,j} = 1_{i,j}+x_{i}_{∂x}^{∂f}^{i}

j, and x_{i}X_{i}_{1}. . . X_{i}_{n} ≡
Xi1. . . Xin (mod$^{n+1}) for alli, i1, . . . , in∈ {1, . . . , r}. We get tr(Dφ−1) =
P

i

∂fi

∂xi, and we apply Theorem6.2.

The authors of [5] ask whether the condition in Theorem 6.2 could be
sufficient forJ to be invertible and therefore for an endomorphismφ:F →F
to be an automorphism.^{1} This is not so; for example, consider r = 2 and
n= 4, in which case all automorphisms in A4 are interior. The map

φ:x7→x[[x, y],[[x, y], y]], y7→y
is an element ofB_{4}\A_{4}. However,

∂

∂x[[x, y],[[x, y], y]]

τ

= ∂

∂X[[X, Y],[[X, Y], Y]] =Y XY^{2}−Y^{2}XY
is in R^{+}_{4}.

On the other hand, we shall see below that the condition r ≤n implies the sufficiency of Theorem 6.2’s condition.

6.1. Generators of A. Generators of A1, and therefore ofL, have been identified by Magnus. He showed in [21] that the following automorphisms generate A1:

K_{i,j,k}:

(x_{i} 7→x_{i}[x_{j}, x_{k}]

x_{`}7→x_{`} for all `6=i.

In particularKi,j :=Ki,i,j conjugates xi by xj, leaving all other generators
fixed. If we lete_{i,j} denote the elementary matrix with a ‘1’ in position (i, j)
and zeros elsewhere, then the Jacobian matrix ofKi,j is readily computed:

Lemma 6.4.

DK_{i,j,k} ≡X_{j}e_{i,k}−X_{k}e_{i,j} (mod$^{2}).

Proof. This follows directly from (5).

Lemma 6.5. For every φ∈M with Dφ= (u_{i,j})_{i,j} we have
tr[Dφ, DK_{i,j,k}] = [u_{k,i}, Xj]−[uj,i, X_{k}].

1Added in proof: examples of the insufficiency of the condition in Theorem6.2for the invertibility ofJhave also been given in [25] and [13, Example 4.4].

We now write Ω = {1, . . . , r}^{∗} as index set of a basis of R, and for
ω = ω1. . . ωn ∈ Ω we write Xω = Xω1· · ·Xωn. We also write ‘i ∈ ω’ to
mean there is an index j such thatω_{j} =i. We also write ‘∗’ for an element
of Rthat we don’t want to specify, because its value will not affect further
calculations.

For i, j ∈ {1, . . . , r} and ω = ω_{1}. . . ω_{n} ∈ Ω with n ≥ 2 such that i6= j
and i6∈ω, choose k6=i, ωn−1 and define inductively

Ki,ω,j = [K_{i,ω}_{1}_{...ω}_{n−1}_{,k}, K_{k,ω}_{n}_{,j}].

Lemma 6.6. For j6=i6∈ω we have

DK_{i,ω,j} =X_{ω}e_{i,j}−X_{ω}_{1}_{...ω}_{n−1}_{j}e_{i,ω}_{n}.

Proof. The induction starts with n = 1, and follows from Lemma 6.4.

Then, forn≥2, choosek as above and compute:

DK_{i,ω,j} = [DK_{i,ω}_{1}_{...ω}_{n−1}_{,k}, DK_{k,ω}_{n}_{,j}]

= [X_{ω}_{1}_{...ω}_{n−1}e_{i,k}− ∗e_{i,ω}_{n−1}, X_{ω}_{n}e_{k,j}−X_{j}e_{k,ω}_{n}]

=Xωei,j−Xω1...ωn−1jei,ωn,

since fors∈ {j, ω_{n}}andt∈ {k, ω_{n−1}}the two termsei,ωn−1e_{k,s} and the four

terms e_{k,s}e_{i,t} vanish.

Define next, fori6=j6=k6=iand i6∈ω,

L_{i,ω,j,k} = [K_{i,ω}_{2}_{...ω}_{n}_{k,j}, Kj,ω1,i].

Lemma 6.7. For i6=j 6=k6=i6∈ω we have

DL_{i,ω,j,k}−1 =X_{ω}_{2}_{...ω}_{n}_{kω}_{1}e_{i,i}−X_{ωk}e_{j,j}+X_{ωj}e_{j,k}− ∗e_{i,ω}_{1}.
Proof. Again this is a direct calculation, using Lemma 6.6:

DLi,ω,j,k= [DKi,ω2...ωnk,j, DKj,ω1,i]

= [X_{ω}_{2}_{...ω}_{n}_{k}ei,j−Xω2...ωnje_{i,k}, Xω1ej,i−Xiej,ω1]

=X_{ω}_{2}_{...ω}_{n}_{kω}_{1}e_{i,i}−X_{ωk}e_{j,j}+X_{ωj}e_{j,k}−X_{ω}_{2}_{...ω}_{n}_{ki}e_{i,ω}_{1},
since fors∈ {j, k}the two terms e_{j,ω}_{1}e_{i,s}, and fort∈ {i, ω_{1}}the two terms

ei,kej,t, vanish.

7. Proofs of the main theorems

7.1. Theorem A. We start by Theorem A from the Introduction. Recall that for a subgroupH ≤Gwe write

√

H={g∈G:g^{k}∈H for somek6= 0}.

It is clear that γ_{n}(A_{1}) ≤ A_{n}, since (A_{n}) is a central series. Moreover,
consider φ ∈ A, written x^{φ}_{i} = x_{i}f_{i} for all i. Assume φ^{k} ∈ A_{n} for some n.

Then x^{φ}_{i}^{k} ∈x_{i}F_{n+1}, so f_{i}^{k} ∈F_{n+1}. Now √

F_{n+1} =F_{n+1}, so f_{i} ∈F_{n+1}, and
therefore

pAn=An.