arXiv:1604.07566v2 [math.NT] 9 May 2017

THE COHOMOLOGY OF CANONICAL QUOTIENTS OF
FREE GROUPS AND LYNDON WORDS
IDO EFRAT

Abstract. For a prime number p and a free profinite group S, let
S (n,p) be the nth term of its lower p-central filtration, and S [n,p] the
corresponding quotient. Using tools from the combinatorics of words,
we construct a canonical basis of the cohomology group H 2 (S [n,p] , Z/p),
which we call the Lyndon basis, and use it to obtain structural results on
this group. We show a duality between the Lyndon basis and canonical
generators of S (n,p) /S (n+1,p) . We prove that the cohomology group
satisfies shuffle relations, which for small values of n fully describe it.

1. Introduction
Let p be a fixed prime number. For a profinite group G one defines the
lower p-central filtration G(n,p) , n = 1, 2, . . . , inductively by
G(1,p) = G,

G(n+1,p) = (G(n,p) )p [G, G(n,p) ].

Thus G(n+1,p) is the closed subgroup of G generated by the powers hp and
commutators [g, h] = g −1h−1 gh, where g ∈ G and h ∈ G(n,p) . We also set
G[n,p] = G/G(n,p) .
Now let S be a free profinite group on the basis X, and let n ≥ 2. Then
S
is a free object in the category of pro-p groups G with G(n,p) trivial. As
with any pro-p group, the cohomology groups H l (S [n,p]) = H l (S [n,p], Z/p),
l = 1, 2, capture the main information on generators and relations, respectively, in a minimal presentation of S [n,p]. The group H 1 (S [n,p]) is just the
L
dual (S [2,p] )∨ ∼
= x∈X Z/p, and it remains to understand H 2 (S [n,p]).
[n,p]

When n = 2 the quotient S [2,p] is an elementary abelian p-group, and
the structure of H 2 (S [2,p] ) is well-known. Namely, for p > 2 one has an
isomorphism
V
∼
H 1 (S [2,p]) ⊕ 2 H 1 (S [2,p] ) −
→ H 2 (S [2,p]),

2010 Mathematics Subject Classification. Primary 12G05, Secondary 20J06, 68R15.
Key words and phrases. Profinite cohomology, lower p-central filtration, Lyndon
words, Shuffle relations, Massey products.
This work was supported by the Israel Science Foundation (grant No. 152/13).
1

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IDO EFRAT

which is the Bockstein map on the first component, and the cup product
on the second component. Furthermore, taking a basis χx , x ∈ X, of
H 1 (S [2,p]) dual to X, there is a fundamental duality between pth powers
and commutators in the presentation of S and Bockstein elements and
cup products, respectively, of the χx (see [NSW08, Ch. III, §9] for details).
These facts have numerous applications in Galois theory, ranging from class
field theory ([Koc02], [NSW08]), the works by Serre and Labute on the prop Galois theory of p-adic fields ([Ser63], [Lab67]), the structure theory of
general absolute Galois groups ([MS96], [EM11]), the birational anabelian
phenomena ([Bog91], [Top16]), Galois groups with restricted ramification
([Vog05], [Sch10]), and mild groups ([Lab06], [For11], [LM11]), to mention
only a few of the pioneering works in these areas.
In this paper we generalize the above results from the case n = 2 to
arbitrary n ≥ 2. Namely, we give a complete description of H 2(S [n,p]) in
terms of a canonical linear basis of this cohomology group. This basis is
constructed using tools from the combinatorics of words – in particular, the
Lyndon words in the alphabet X, i.e., words which are lexicographically
smaller than all their proper suffixes (for a fixed total order on X). We
call it the Lyndon basis, and use it to prove several structural results on
H 2 (S [n,p]), and in particular to compute its size (see below).
The Lyndon basis constructed here can be most naturally described in
terms of central extensions, as follows: For 1 ≤ s ≤ n let U denote the
group of all unipotent upper-triangular (s + 1) × (s + 1)-matrices over the
ring Z/pn−s+1 . There is a central extension
0 → Z/p → U → U[n,p] → 1
(Proposition 6.3). It corresponds to a cohomology element γn,s ∈ H 2 (U[n,p]).
For a profinite group G and a continuous homomorphism ρ : G → U we
write ρ̄ : G[n,p] → U[n,p] for the induced homomorphism, and ρ̄∗ γn,s for the
pullback to H 2 (G[n,p]). Now for any word w = (x1 · · · xs ) in the alphabet X
we define a homomorphism ρw : S → U by setting the entry (ρw (σ))ij to be
the coefficient of the subword (xi · · · xj−1 ) in the power series Λ(σ), where
Λ : S → (Z/pn−s+1 )hhXii× is the Magnus homomorphism, defined on
the generators x ∈ X by Λ(x) = 1 + x (see §3 and §6 for more details).
The Lyndon basis is now given by:
Main Theorem. The pullbacks αw,n = (ρ̄w )∗ γn,s , where w ranges over all
Lyndon words of length 1 ≤ s ≤ n in the alphabet X, form a linear basis
of H 2 (S [n,p]) over Z/p.

COHOMOLOGY AND LYNDON WORDS

3

We further show a duality between the Lyndon basis and certain canonical elements σw ∈ S (n,p) , with w a Lyndon word of length ≤ n, generalizing
the above mentioned duality in the case n = 2 (see Corollary 8.3).
The cohomology elements ρ̄∗ γn,s include the Bockstein elements (for s =
1), the cup products (for n = s = 2), and more generally, the elements
of n-fold Massey products in H 2 (G[n,p]) (for n = s ≥ 2); see Examples
7.4. The full spectrum ρ̄∗ γn,s , 1 ≤ s ≤ n, appears to give new significant
“external” objects in profinite cohomology, which to our knowledge have
not been investigated so far in general.
Lyndon words are known to have tight connections with the shuffle algebra, and indeed, the αw,n for arbitrary words w of length ≤ n in X
satisfy natural shuffle relations (Theorem 9.4). In §10-§11 we show that
for n = 2, 3 these shuffle relations fully describe H 2 (S [n,p]), provided that
p > 2, p > 3, respectively (for n = 2 this was essentially known). Interestingly, related considerations arise also in the context of multiple zeta
values, see e.g. [MP00], although we are not aware of a direct connection.
The Lyndon words on X form a special instance of Hall sets, which are
well-known to have fundamental role in the structure theory of free groups
and free Lie algebras (see [Reu93], [Ser92]). In addition, the Lyndon words
have a triangularity property (see Proposition 4.4(b)). This property
allows us to construct certain upper-triangular unipotent matrices that
express a (semi-)duality between the σw and the cohomology elements αw,n .
We now outline the proof that the αw,n form a linear basis of H 2 (S [n,p]).
For simplicity we assume for the moment that X is finite. To each Lyndon
word w of length 1 ≤ s ≤ n one associates in a canonical way an element
τw of the s-th term of the lower central series of S (see §4). The cosets
n−s
of the powers σw = τwp
generate S (n,p) /S (n+1,p) (Theorem 5.3). Using
the special structure of the lower p-central filtration of U we define for
any two Lyndon words w, w ′ of length ≤ n a value hw, w ′in ∈ Z/p (see
§6). Thetriangularity property of Lyndon words implies that the matrix
hw, w ′in is unipotent upper-triangular, whence invertible. Turning now
to cohomology, we define a natural perfect pairing
(·, ·)n : S (n,p) /S (n+1,p) × H 2 (S [n,p]) → Z/p.

Cohomological computations show that, for Lyndon words w, w ′ of length

≤ n, one has hw, w ′in = (σw , αw′ ,n )n . Hence the matrix (σw , αw′ ,n )n is
also invertible. We then conclude that the αw′ ,n form a basis of H 2 (S [n,p])
(Theorem 8.5). This immediately determines the dimension of the latter

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cohomology group, in terms of Witt’s necklace function, which counts
the number of Lyndon words over X of a given length (Corollary 8.6).
In the special case n = 2, the theory developed here generalizes the
above description of H 2 (S [2,p]) in terms of the Bockstein map and cup
products (see §10 for details). Namely, the matrix (hw, w ′i2 ) is the identity
matrix, which gives the above duality between pth powers/commutators
and Bockstein elements/cup products. Likewise, the shuffle relations just
recover the basic fact that the cup product factors via the exterior product.
In §11 we describe our theory explicitly also for the (new) case n = 3.
While here we focus primarily on free profinite groups, it may be interesting to study the canonical elements ρ̄∗ γn,s for more general profinite
groups G, in particular, when G = GF is the absolute Galois group of a
field F . For instance, when n = 2, they were used in [EM11] (following
[3,p]
[MS96] and [AKM99]) and [CEM12] to describe the quotient GF . Triple
Massey products for GF (which correspond to the case n = s = 3) were
also extensively studied in recent years – see [EM15], [MT15b], [MT16],
[MT17], and [Wic12] and the references therein.
I thank Claudio Quadrelli and the anonymous referee for their careful
reading of this paper and for their very valuable comments and suggestions
on improving the exposition.
2. Words
Let X be a nonempty set, considered as an alphabet. Let X ∗ be the free
monoid on X. We view its elements as associative words on X. The length
of a word w is denoted by |w|. We write ∅ for the empty word, and ww ′
for the concatenation of words w and w ′ .
Recall that a magma is a set M with a binary operation (·, ·) : M ×
M → M. A morphism of magmas is a map which commutes with the
associated binary operations. There is a free magma MX on X, unique
up to an isomorphism; that is, X ⊆ MX , and for every magma (·, ·) : N ×
N → N and a map f0 : X → N there is a magma morphism f : MX → N
extending f0 . See [Ser92, Ch. IV, §1] for an explicit construction of MX .
The elements of MX may be viewed as non-associative words on X.
The monoid X ∗ is a magma with respect to concatenation, so the universal property of MX gives rise to a unique magma morphism f : MX → X ∗ ,
called the foliage (or brackets dropping) map, such that f (x) = x for
x ∈ X.

COHOMOLOGY AND LYNDON WORDS

5

Let H be a subset of MX and ≤ a total order on H. We say that (H, ≤)
is a Hall set in MX , if the following conditions hold [Reu93, §4.1]:
(i) X ⊆ H;
(ii) If h = (h′ , h′′ ) ∈ H \ X, then h < h′′ ;
(iii) For h = (h′ , h′′ ) ∈ MX \ X, one has h ∈ H if and only if
• h′ , h′′ ∈ H and h′ < h′′ ; and
• either h′ ∈ X, or h′ = (v, u) where u ≥ h′′ .
Given a Hall set (H, ≤) in MX we call H = f (H) a Hall set in X ∗ .
Every w ∈ H can be written as w = f (h) for a unique h ∈ H [Reu93, Cor.
4.5]. If w ∈ H \ X, then we can uniquely write h = (h′ , h′′ ) with h′ , h′′ ∈ H,
and call w = w ′w ′′ , where w ′ = f (h′ ) and w ′′ = f (h′′ ), the standard
factorization of w [Reu93, p. 89].
Next we fix a total order ≤ on X, and define a total order ≤alp (the
alphabetical order) on X ∗ as follows: Let w1 , w2 ∈ X ∗ . Then w1 ≤alp w2
if and only if w2 = w1 v for some v ∈ X ∗ , or w1 = vx1 u1 , w2 = vx2 u2 for
some words v, u1 , u2 and some letters x1 , x2 ∈ X with x1 < x2 . Note that
the restriction of ≤alp to X n is the lexicographic order.
In addition, we order Z≥0 × X ∗ lexicographically with respect to the
usual order on Z≥0 and the order ≤alp on X ∗ . We then define a second
total order  on X ∗ by setting
(2.1)

w1  w2

⇐⇒

(|w1 |, w1 ) ≤ (|w2|, w2 )

with respect to the latter order on Z≥0 × X ∗ .
A nonempty word w ∈ X ∗ is called a Lyndon word if it is smaller
in ≤alp than all its non-trivial proper right factors. Equivalently, no nontrivial rotation leaves w invariant, and w is lexicographically strictly smaller
than all its rotations 6= w ([CFL58, Th. 1.4], [Reu93, Cor. 7.7]). We denote
the set of all Lyndon words on X by Lyn(X), and the set of all such words
of length n (resp., ≤ n) by Lynn (X) (resp., Lyn≤n (X)). The set Lyn(X),
totally ordered with respect to ≤alp , is a Hall set [Reu93, Th. 5.1].
Example 2.1. The Lyndon words of length ≤ 4 on X are
(x) for x ∈ X,
(xy), (xxy), (xyy), (xxxy), (xxyy), (xyyy) for x, y ∈ X, x < y,
(xyz), (xzy), (xxyz), (xxzy), (xyxz), (xyyz), (xyzy), (xyzz),
(xzyy), (xzyz), (xzzy) for x, y, z ∈ X, x < y < z,
(xyzt), (xytz), (xzyt), (xzty), (xtyz), (xtzy) for x, y, z, t ∈ X, x < y < z < t

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The necklace map (also called the Witt map) is defined for integers
n, m ≥ 1 by
1X
µ(d)mn/d .
ϕn (m) =
n
d|n

Here µ is the Möbius function, defined by µ(d) = (−1)k , if d is a product of
k distinct prime numbers, and µ(d) = 0 otherwise; Alternatively, 1/ζ(s) =
P∞
s
n=1 µ(n)/n , where ζ(s) is the Riemann zeta function and s is a complex
number with real part > 1. When m = q is a prime power, ϕn (q) also
counts the number of irreducible monic polynomials of degree n over a
field of q elements [Reu93, §7.6.2]. We also define ϕn (∞) = ∞. One has
[Reu93, Cor. 4.14]
(2.2)

| Lynn (X)| = ϕn (|X|).
3. Power series

We fix a commutative unital ring R. Recall that a bilinear map M ×
N → R of R-modules is non-degenerate if its left and right kernels are
trivial, i.e., the induced maps M → Hom(N, R) and N → Hom(M, R) are
injective. The bilinear map is perfect if these two maps are isomorphisms.
Let RhXi be the free associative R-algebra on X. We may view its
elements as polynomials over R in the non-commuting variables x ∈ X.
The additive group of RhXi is the free R-module on the basis X ∗ . We grade
RhXi by total degree. Let RhhXii be the ring of all formal power series in
the non-commuting variables x ∈ X and coefficients in R. For f ∈ RhhXii
P
we write fw for the coefficient of f at w ∈ X ∗ . Thus f = w∈X ∗ fw w.
There are natural embedings X ∗ ⊆ RhXi ⊆ RhhXii, where we identify
w ∈ X ∗ with the series f such that fw = 1 and fw′ = 0 for w ′ 6= w. There
is a well-defined non-degenerate bilinear map of R-modules
X
fw gw ;
(3.1)
RhhXii × RhXi → R, (f, g) 7→
w∈X ∗

See [Reu93, p. 17]. For every integer n ≥ 0, it restricts to a non-degenerate
bilinear map on the homogenous components of degree n.
∗
We may identify the additive group of RhhXii with RX via the map
f 7→ (fw )w . When R is equipped with a profinite ring topology (e.g. when R
∗
is finite), the product topology on RX induces a profinite group topology
on RhhXii. Moreover, the multiplication map of RhhXii is continuous,
making it a profinite ring. The group RhhXii× of all invertible elements in
RhhXii is then a profinite group.

COHOMOLOGY AND LYNDON WORDS

7

Next we recall from [FJ08, §17.4] the following terminology and facts
on free profinite groups. Let G be a profinite group and X a set. A map
ψ : X → G converges to 1, if for every open normal subgroup N of G,
the set X \ ψ −1 (N) is finite. We say that a profinite group S is a free
profinite group on basis X with respect to a map ι : X → S if
(i) ι : X → S converges to 1 and ι(X) generates S as a profinite group;
(ii) For every profinite group G and a map ψ : X → G converging to 1,
there is a unique continuous homomorphism ψ̂ : S → G such that
ψ = ψ̂ ◦ ι on X.
A free profinite group on X exists, and is unique up to a continuous isomorphism. We denote it by SX . The map ι is then injective, and we identify
X with its image in SX .
We define the (profinite) Magnus homomorphism ΛX,R : SX → RhhXii×
as follows (compare [Efr14, §5]):
P
i i
Assume first that X is finite. For x ∈ X one has 1 = (1+x) ∞
i=0 (−1) x
in RhhXii, so 1 + x ∈ RhhXii× . Hence, by (ii), the map ψ : X → RhhXii× ,
x 7→ 1 + x, uniquely extends to a continuous homomorphism ΛX,R : SX →
RhhXii×.
Now suppose that X is arbitrary. Let Y range over all finite subsets of
X. The map ψ : X → SY , which is the identity on Y and 1 on X \ Y , converges to 1. Hence it extends to a unique continuous group homomorphism
SX → SY . Also, there is a unique continuous R-algebra homomorphism
RhhXii → RhhY ii, which is the identity on Y and 0 on X \ Y . Then
SX = lim SY , RhhXii = lim RhhY ii, RhhXii× = lim RhhY ii× .
←−
←−
←−
We define ΛX,R = lim ΛY,R . It is functorial in both X and R in the natural
←−
way. Note that ΛX,R (x) = 1 + x for x ∈ X.
In the sequel, X will be fixed, so we abbreviate S = SX and ΛR = ΛX,R .
For σ ∈ S and a word w ∈ X ∗ we denote the coefficient of w in ΛR (σ)
by ǫw,R (σ). Thus
X
ΛR (σ) =
ǫw,R (σ)w.
w∈X ∗

By the construction of ΛR , we have ǫ∅,R (σ) = 1. Since ΛR is a homomorphism, for every σ, τ ∈ S and w ∈ X ∗ one has
X
ǫu1 ,R (σ)ǫu2 ,R (τ ).
(3.2)
ǫw,R (στ ) =
w=u1 u2

We will also need the classical discrete version of the Magnus homomorphism. To define it, assume that X is finite, and let FX be the free

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group on basis X. There is a natural homomorphism FX → SX . The dis×
crete Magnus homomorphism Λdisc
Z : FX → ZhhXii is defined again
by x 7→ 1 + x. There is a commutative square
(3.3)

FX
Λdisc
Z

/

SX
ΛZp




ZhhXii× 


/

Zp hhXii× .

4. Lie algebra constructions
Recall that the lower central filtration G(n,0) , n = 1, 2, . . . , of a profinite group G is defined inductively by
G(1,0) = G,

G(n+1,0) = [G(n,0) , G].

Thus G(n+1,0) is generated as a profinite group by all elements of the form
[σ, τ ] with σ ∈ G(n,0) and τ ∈ G. One has [G(n,0) , G(m,0) ] ≤ G(n+m,0) for
every n, m ≥ 1 (compare [Ser92, Part I, Ch. II, §3]).
Proposition 4.1. Let S = SX and let σ ∈ S. Then:
(a) σ ∈ S (n,0) if and only if ǫw,Zp (σ) = 0 for every w ∈ X ∗ with 1 ≤
|w| < n;
(b) σ ∈ S (n,p) if and only if ǫw,Zp (σ) ∈ pn−|w|Zp for every w ∈ X ∗ with
1 ≤ |w| < n.
Proof. In the discrete case (a) and (b) are due to Grün and Magnus (see
[Ser92, Part I, Ch. IV, th. 6.3]) and Koch [Koc60], respectively. The results
in the profinite case follow by continuity.
For other approaches see [Mor12, Prop. 8.15], [CE16, Example 4.5], and
[MT15a, Lemma 2.2(d)].

We will need the following profinite analog of [Fen83, Lemma 4.4.1(iii)].
Lemma 4.2. Let σ ∈ S (n,0) and τ ∈ S (m,0) , and let w ∈ X ∗ have length
n + m. Write w = u1 u2 = u′2 u′1 with |u1| = |u′1 | = n and |u2 | = |u′2| = m.
Then
ǫw,Zp ([σ, τ ]) = ǫu1 ,Zp (σ)ǫu2 ,Zp (τ ) − ǫu′2 ,Zp (τ )ǫu′1 ,Zp (σ).
Proof. By Proposition 4.1(a), we may write ΛZp (σ) = 1 + P + O(n + 1) and
ΛZp (τ ) = 1 + Q + O(m + 1), where P, Q ∈ Zp hhXii are homogenous of degrees n, m, respectively, and where O(r) denotes a power series containing
only terms of degree ≥ r. Then
ΛZp ([σ, τ ]) = 1 + P Q − QP + O(n + m + 1).

COHOMOLOGY AND LYNDON WORDS

9

(compare e.g., [Mor12, Proof of Prop. 8.5]). By (3.2), it follows that
ǫw,Zp ([σ, τ ]) = (P Q − QP )w = Pu1 Qu2 − Qu′2 Pu′1
= ǫu1 ,Zp (σ)ǫu2 ,Zp (τ ) − ǫu′2 ,Zp (τ )ǫu′1 ,Zp (σ).

L∞

The commutator map induces on the graded ring n=1 S (n,0) /S (n+1,0) a
graded Lie algebra structure [Ser92, Part I, Ch. II, Prop. 2.3]. Let d be the
L
n
n+1
is a
ideal in the Zp -algebra Zp hhXii generated by X. Then ∞
n=1 d /d
Lie algebra with the Lie brackets defined on homogenous components by
[f¯, ḡ] = f g − gf for f ∈ dn , g ∈ dm [Ser92, p. 25]. By Proposition 4.1(a),
ΛZp induces a graded Zp -algebra homomorphism
gr ΛZp :

∞
M
n=1

S (n,0) /S (n+1,0) →

∞
M

dn /dn+1 ,

σS (n+1,0) 7→

n=1

X

ǫw,Zp (σ)+dn+1.

|w|=n

Then Lemma 4.2 means that gr ΛZp is a Lie algebra homomorphism.
For w ∈ Lyn(X) we inductively define an element τw of S and a noncommutative polynomial Pw ∈ ZhXi ⊆ Zp hXi as follows:
• If w = (x) has length 1, then τw = x and Pw = x;
• If |w| > 1, then we take the standard factorization w = w ′w ′′ of w
with respect to the Hall set Lyn(X) (see §2), and set
τw = [τw′ , τw′′ ],

Pw = Pw′ Pw′′ − Pw′′ Pw′ .

For w ∈ Lynn (X) one has τw ∈ S (n,0) . Moreover:
Proposition 4.3. Let n ≥ 1. The cosets of τw , with w ∈ Lynn (X), generate S (n,0) /S (n+1,0) .
Proof. See [Reu93, Cor. 6.16] for the discrete version. The profinite version
follows by taking closure.

Let ≤alp and  be the total orders on X ∗ defined in §2. The importance
of the Lyndon words in our context, beside forming a Hall set, is part (b)
of the following Proposition, called the triangularity property.
Proposition 4.4. Let w ∈ Lyn(X). Then
(a) ΛZp (τw ) − 1 − Pw is a combination of words of length > |w|.
(b) Pw − w is a combination of words of length |w| which are strictly
larger than w with respect to ≤alp .
(c) ΛZp (τw ) − 1 − w is a combination of words which are strictly larger
than w in .

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IDO EFRAT

Proof. (a) Since gr ΛZp is a Lie algebra homomorphism, for w ∈ Lynn (X)
we have by induction
(gr ΛZp )(τw S (n+1,0) ) = Pw + dn+1 ,
and the assertion follows. See also [Reu93, Lemma 6.10(ii)].
(b)

See [Reu93, Th. 5.1] and its proof.

(c)

This follows from (a) and (b).



5. Generators for S (n,p)/S (n+1,p)
Let π be an indeterminate over the ring Z/p and let Z/p[π] be the polynoL
mial ring. Let A• = ∞
n=1 An be a graded Lie Z/p-algebra with Lie bracket
[·, ·]. Suppose that there is a map Z/p[π] × A• → A• , (α, ξ) 7→ αξ, which
is Z/p-linear in Z/p[π], such that πξ ∈ As+1 for ξ ∈ As , and such that for
every ξ1 , ξ2 ∈ As ,
(
πξ1 + πξ2 ,
if p > 2 or s > 1,
(5.1)
π(ξ1 + ξ2 ) =
πξ1 + πξ2 + [ξ1 , ξ2 ], if p = 2, s = 1.
By induction, this extends to:
Lemma 5.1. Let r, k, s ≥ 1 and let ξ1 , . . . , ξk ∈ As . Then
(P
k
k
X
π r ξi ,
if p > 2 or s > 1,
πr (
ξi ) = Pi=1
P
k
r
r−1
[ξi , ξj ], if p = 2, s = 1.
i=1 π ξi +
i<j π
i=1
We write hT i for the submodule of A• generated by a subset T .

Lemma 5.2. Let n ≥ 2 and for each 1 ≤ s ≤ n let Ts be a subset of
As . When p = 2 assume also that [τ1 , τ2 ] ∈ T2 ∪ {0} for every τ1 , τ2 ∈ T1 .
If the sets π n−s hTs i, s = 1, 2, . . . , n, generate An , then the sets π n−s Ts ,
s = 1, 2, . . . , n, also generate An .
Proof. When p > 2 or s > 1 Lemma 5.1 shows that π n−s hTs i = hπ n−s Ts i.
When p = 2 and s = 1, it shows that
π n−1 hT1 i ⊆ hπ n−1 T1 i + hπ n−2T2 i ⊆ An .
Therefore the subgroup of An generated by the sets π n−s Ts , s = 1, 2, . . . , n,
contains the sets π n−s hTs i, s = 1, 2, . . . , n, and hence equals An .

Motivated by e.g., [Laz54], [Ser63], [Lab67], we now specialize to a graded
Lie algebra defined using the lower p-central filtration. We refer to [NSW08,
Ch. III, §8] for the following facts. For the free profinite group S on the
basis X and for n ≥ 1 we set grn (S) = S (n,p) /S (n+1,p) . It is an elementary

COHOMOLOGY AND LYNDON WORDS

11

abelian p-group, which we write additively. The commutator map induces
a map [·, ·] : grn (S) × grm (S) → grn+m (S), which endows a graded Lie
L
p
(r,p)
algebra structure on gr• (S) = ∞
n=1 grn (S). The map τ 7→ τ maps S
(r+1,p)
r
into S
, and induces a map πr : grr (S) → grr+1 (S). The map (π , ξ) 7→
πr (ξ) for ξ ∈ grr (S) extends to a map Z/p[π] × gr• (S) → gr• (S) which is
Z/p-linear in the first component and which satisfies (5.1).
n−s

Theorem 5.3. Let n ≥ 1. The cosets of τwp
w ∈ Lyns (X), generate S (n,p) /S (n+1,p).

, with 1 ≤ s ≤ n and

Proof. The case n = 1 is immediate, so we assume that n ≥ 2. For every
1 ≤ s ≤ n let Ts be the set of cosets of τw , w ∈ Lyns (X), in grs (S).
By Proposition 4.3, hTs i is the image of S (s,0) in grs (S). Hence π n−s hTs i
consists of the cosets in grn (S) of the pn−s -powers of S (s,0) . One has
n
Y
n−s
(S (s,0) )p
S (n,p) =
s=1

[NSW08, Prop. 3.8.6]. Thus the sets π n−s hTs i, s = 1, 2, . . . , n, generate
grn (S).
Further, T1 consists of the cosets of x, with x ∈ X, and T2 consists
of the cosets of commutators [x, y] with x < y in X. Moreover, when
p = 2 the cosets of [x, y] and [y, x] = [x, y]−1 in gr2 (S) coincide. Lemma
5.2 therefore implies that even the sets π n−s Ts , s = 1, 2, . . . , n, generate
grn (S), as required.

6. The pairing hw, w ′in

For a commutative unitary ring R and a positive integer m, let Um (R)
be the group of all m × m upper-triangular unipotent matrices over R. We
write Im for the identity matrix in Um (R), and Eij for the matrix with 1
at entry (i, j) and 0 elsewhere. As above, X will be a totally ordered set.
For the following fact we refer, e.g., to [Efr14, Lemma 7.5]. We recall
from §3 that ǫu,R (σ) is the coefficient of the word u ∈ X ∗ in the formal
power series ΛR (σ) ∈ RhhXii× .
Proposition 6.1. Given a profinite ring R and a word w = (x1 · · · xs ) in
X ∗ there is a well defined homomorphism of profinite groups
ρw
R : S → Us+1 (R),

σ 7→ (ǫ(xi ···xj−1 ),R (σ))1≤i<j≤s+1.

Remark 6.2. In particular, for each x ∈ X the map χx,R = ǫ(x),R : S → R
is a group homomorphism, where R is considered as an additive group.

12

IDO EFRAT

The homomorphisms χx,R , x ∈ X, are dual to the basis X, in the sense
that χx,R (x) = 1, and χx,R (y) = 0 for x 6= y in X.
Proposition 6.3. Let 1 ≤ s ≤ n. For R = Z/pn−s+1 one has:
(a) Us+1 (R)(n,p) = Is+1 + Zpn−s E1,s+1 .
(b) Us+1 (R)(n,p) is central in Us+1(R).
Proof. (a) We follow the argument of [MT15a, Lemma 2.4]. Take X =
{x1 , . . . , xs } be a set of s elements, let S = SX , and let w = (x1 · · · xs ). The
matrices ρw
R (xi ) = Is+1 + Ei,i+1 , i = 1, 2, . . . , s, generate Us+1 (R) [Wei55, p.
w
55], so ρR is surjective. Therefore it maps S (n,p) onto Us+1 (R)(n,p) .
By Proposition 4.1(b), for σ ∈ S (n,p) and u ∈ X ∗ of length 1 ≤ |u| ≤ s
one has ǫu,Zp (σ) ∈ pn−|u| Zp .
If |u| < s, then ǫu,Zp (σ) ∈ pn−|u| Zp ⊆ pn−s+1 Zp . Hence ǫu,R (σ) = 0 in
this case.
If |u| = s, then ǫu,Zp (σ) ∈ pn−s Zp , so ǫu,R (σ) ∈ pn−s R.
n−s
n−s
Moreover, τwp
∈ (S (s,0) )p
≤ S (n,p) . By Proposition 4.4(c), ΛZp (τw ) =
1 + w + f , where f is a combination of words strictly larger than w in .
n−s
Hence ΛZp (τwp ) = 1 + pn−s w + g, where g is also a combination of words
n−s
strictly larger than w in , which implies that ǫw,R (τwp ) = pn−s · 1R .
(n,p)
Consequently, Us+1 (R)(n,p) = ρw
) = Is+1 + Zpn−s E1,s+1 .
R (S
(b) It is straightforward to see that Is+1 +ZE1,s+1 is central in Us+1 (R),
so the assertion follows from (a).

See [Bor04, §2] for a related analysis of the lower p-central filtration of
Us+1(Z/pn−s+1 ).
Consider the obvious isomorphism
∼

ιn,s : pn−s Z/pn−s+1 Z −
→ Z/p,

apn−s (mod pn−s+1 ) 7→ a (mod p).

In view of Proposition 6.3(a), we may define a group isomorphism
∼

ιUn,s : Us+1 (Z/pn−s+1)(n,p) −
→ Z/p,

(aij ) 7→ ιn,s (a1,s+1 ).

Next let w, w ′ ∈ X ∗ be words of lengths 1 ≤ s, s′ ≤ n, respectively,
n−s
n−s
where w is Lyndon. We have τwp
∈ (S (s,0) )p
≤ S (n,p) . By Propon−s
n−s
′
sition 4.1(b), ǫw′ ,Zp (τwp ) ∈ pn−s Zp , and therefore ǫw′ ,Z/pn−s′ +1 (τwp ) ∈
′
′
pn−s Z/pn−s +1 Z. We set
(6.1)

n−s

hw, w ′in = ιn,s′ (ǫw′ ,Z/pn−s′ +1 (τwp

)) ∈ Z/p.

Alternatively,
(6.2)

′

n−s

(τ p
hw, w ′in = ιUn,s′ (ρw
Z/pn−s′ +1 w

)).

COHOMOLOGY AND LYNDON WORDS

13

Let  be as in (2.1).
Proposition 6.4. Let w, w ′ be words in X ∗ of lengths 1 ≤ s, s′ ≤ n,
respectively, with w Lyndon.
(a) If w ′ ≺ w, then hw, w ′in = 0;
(b) hw, win = 1;
(c) If w ′ contains letters which do not appear in w, then hw, w ′in = 0;
(d) If s < s′ < 2s, then hw, w ′in = 0.
n−s

Proof. (a), (b) Proposition 4.4(c) implies that ΛZp (τwp ) − 1 − pn−s w
is a combination of words strictly larger than w with respect to , and
′
the same therefore holds over the coefficient ring Z/pn−s +1 . Hence, if
n−s
w ′ ≺ w, then ǫw′ ,Z/pn−s′ +1 (τwp ) = 0, so hw, w ′in = 0. If w = w ′, then
n−s
ǫw,Z/pn−s+1 (τwp ) = pn−s · 1Z/pn−s+1 , whence hw, win = 1.
(c)

n−s

Here we clearly have ǫw′ ,Z/pn−s′ +1 (τwp

) = 0.

(d) Since τw ∈ S (s,0) , one may write ΛZp (τw ) = 1 + P + O(s′ + 1), where
P is a combination of words w ′′ of length s ≤ |w ′′ | ≤ s′ , and O(s′ + 1)
denotes a combination of words of length ≥ s′ + 1 (Proposition 4.1(a)).
n−s
Since s′ < 2s, this implies that ΛZp (τwp ) = 1 + pn−s P + O(s′ + 1). In
n−s
particular, ǫw′ ,Zp (τwp ) ∈ pn−s Zp , and therefore
n−s

ǫw′ ,Z/pn−s′ +1 (τwp

′

) ∈ pn−s (Z/pn−s +1 ) = {0},

since s < s′ . Hence hw, w ′in = 0.



7. Transgressions
Given a profinite group G and a discrete G-module A, we write as usual
C (G, A), Z i (G, A), and H i (G, A) for the corresponding group of continuous i-cochains, group of continuous i-cocycles, and the ith profinite cohomology group, respectively. For x ∈ Z i (G, A) let [x] be its cohomology
class in H i (G, A).
For a normal closed subgroup N of G, let trg : H 1 (N, A)G → H 2 (G/N, AN )
be the transgression homomorphism. It is the map d0,1
of the Lyndon–
2
Hochschild–Serre spectral sequence associated with G and N [NSW08, Th.
2.4.3]. We recall the explicit description of trg, assuming for simplicity that
the G-action on A is trivial [NSW08, Prop. 1.6.6]: If x ∈ Z 1 (N, A), then
there exists y ∈ C 1 (G, A) such that y|N = x and (∂y)(σ1 , σ2 ) depends only
on the cosets of σ1 , σ2 modulo N, so that ∂y may be viewed as an element
of Z 2 (G/N, A). For any such y one has trg([x]) = [∂y].
i

14

IDO EFRAT

We fix for the rest of this section a finite group U and a normal subgroup
N of U satisfying:
(i) N ∼
= Z/p; and
(ii) N lies in the center of U.
We set Ū = U/N, and let it act trivially on U. We denote the image of
u ∈ U in Ū by ū. We may choose a section λ of the projection U → Ū such
that λ(1̄) = 1. We define a map δ ∈ C 2 (Ū, N) by
δ(ū, ū′ ) = λ(ū) · λ(ū′ ) · λ(ūū′ )−1 .
It is normalized, i.e., δ(ū, 1) = δ(1, ū) = 1 for every ū ∈ Ū.
We also define y ∈ C 1 (U, N) by y(u) = uλ(ū)−1 . Note that y|N = idN .
Lemma 7.1. For every u, u′ ∈ U one has
δ(ū, ū′ ) · y(u) · y(u′) = y(uu′).
Proof. Since y(u) and y(u′) are in N, they are central in U, so
δ(ū, ū′ ) · y(u) · y(u′) = λ(ū) · λ(ū′) · λ(ūū′)−1 · y(u) · y(u′)
= y(u) · λ(ū) · y(u′) · λ(ū′ ) · λ(ūū′ )−1
= uu′ λ(ūū′ )−1 = y(uu′).

For the correspondence between elements of H 2 and central extensions
see e.g., [NSW08, Th. 1.2.4].
Proposition 7.2. Using the notation above, the following holds.
(a) δ ∈ Z 2 (Ū, N);
(b) One has trg(idN ) = −[δ] for the transgression map trg : H 1 (N, N)U →
H 2 (Ū, N).
(c) The cohomology class [δ] ∈ H 2 (Ū, N) corresponds to the equivalence
class of the central extension
1 → N → U → Ū → 1.

(7.1)
Proof. (a), (b):

For u, u′ ∈ U Lemma 7.1 gives

(∂y)(u, u′) = y(u) · y(u′) · y(uu′)−1 = δ(ū, ū′ )−1 .
This shows that δ is a 2-cocycle, and that (∂y)(u, u′) depends only on the
cosets ū, ū′ . Further, idN ∈ Z 1 (N, N). By the explicit description of the
transgression above, trg(idN ) = −[δ].

COHOMOLOGY AND LYNDON WORDS

(c)

15

Consider the set B = N × Ū with the binary operation
(u, v̄) ∗ (u′, v̄ ′ ) = (δ(v̄, v̄ ′ )uu′, v̄v̄ ′ ).

The proof of [NSW08, Th. 1.2.4] shows that this makes B a group, and [δ]
corresponds to the equivalence class of the central extension
1 → N → B → Ū → 1.

(7.2)

Moreover, the map h : U → B, u 7→ (y(u), ū) is clearly bijective. We claim
that it is a homomorphism, whence an isomorphism. Indeed, for u, u′ ∈ U
Lemma 7.1 gives:
h(u) ∗ h(u′ ) = (y(u), ū) ∗ (y(u′), ū′ ) = (δ(ū, ū′ )y(u)y(u′), ūū′ )
= (y(uu′), ūū′ ) = h(uu′ ).
We obtain that the central extension (7.2) is equivalent to the central
extension (7.1).

Next let Ḡ be a profinite group, and let ρ̄ : Ḡ → Ū be a continuous
∼
homomorphism. Let ι : N −
→ Z/p be a fixed isomorphism (see (i)). Set
(7.3)

α = (ρ̄∗ ◦ ι∗ )([δ]) = −(ρ̄∗ ◦ ι∗ ◦ trg)(idN ) ∈ H 2 (Ḡ, Z/p),

where the second equality is by Proposition 7.2(b). Then α corresponds to
the equivalence class of the central extension
ι−1 ×1

0 → Z/p −−−→ U ×Ū Ḡ → Ḡ → 1,

(7.4)

where U ×Ū Ḡ is the fiber product with respect to the natural projection
U → Ū and to ρ̄; See [Hoe68, Proof of 1.1].
Suppose further that there is a profinite group G, a closed normal subgroup M of G, and a continuous homomorphism ρ : G → U such that
Ḡ = G/M, ρ(M) ≤ N, and ρ̄ : Ḡ → Ū is induced from ρ. The functoriality
of transgression yields a commutative diagram
H 1 (N, N)U

ι∗
∼

H 1 (N, Z/p)U
/

trg



H 2 (Ū, N)

ρ∗

H 1 (M, Z/p)G
/

trg

trg
ι∗
∼

/



H 2 (Ū, Z/p)

ρ̄∗

/



H 2 (Ḡ, Z/p).

The image of idN ∈ H 1 (N, N) in H 1 (M, Z/p)G is
(7.5)

θ = ι ◦ (ρ|M ) ∈ H 1 (M, Z/p).

By (7.3) and the commutativity of the diagram,
(7.6)

α = − trg(θ) ∈ H 2 (Ḡ, Z/p)

16

IDO EFRAT

Remark 7.3. Suppose that Ū is abelian and that Ḡ acts trivially on Ū.
A 2-cocycle representing α is
(σ̄, σ̄ ′ ) 7→ ι(λ(ρ̄(σ̄)) · λ(ρ̄(σ̄ ′ )) · λ(ρ̄(σ̄σ̄ ′ ))−1 ).
But λ(ρ̄(σ̄)) · λ(ρ̄(σ̄ ′ )) · λ(ρ̄(σ̄σ̄ ′ ))−1 is a 2-cocycle representing the image
of ρ̄ under the connecting homomorphism H 1 (Ḡ, Ū) → H 2 (Ḡ, N) arising
from (7.1). Thus α is the image of ρ̄ under the composition
ι

∗
H 1 (Ḡ, Ū) → H 2 (Ḡ, N) −
→
H 2 (Ḡ, Z/p).

Example 7.4. We give several examples of the above construction with
the group U = Us+1 (Z/pn−s+1 ) (where 1 ≤ s ≤ n), a continuous homomorphism ρ : G → U, where G is a profinite group, and the induced homomorphism ρ̄ : G[n,p] → U[n,p] . Note that assumptions (i) and (ii) then hold for
N = U(n,p) , by Proposition 6.3. We will be especially interested in the case
where G = S = SX is a free profinite group, M = S (n,p) , ρ = ρw
Z/pn−s+1 for
∗
w
[n,p]
a word w ∈ X of length 1 ≤ s ≤ n, and ρ̄ = ρ̄Z/pn−s+1 : S
→ U[n,p] is
the induced homomorphism. In this setup we write αw,n for α.
(1) Bocksteins. For a positive integer m and a profinite group Ḡ, the
connecting homomorphism arising from the short exact sequence of trivial
Ḡ-modules
0 → Z/p → Z/pm → Z/m → 0
is the Bockstein homomorphism
Bockm,Ḡ : H 1 (Ḡ, Z/m) → H 2 (Ḡ, Z/p).
Let U = U2 (Z/pn ) (i.e., s = 1). There is a commutative diagram of central
extensions
1
/

U(n,p)
≀

/

/

/

≀



0

U

pn−1 Z/pn Z

/

1

≀




/

U[n,p]

Z/pn
/

Z/pn−1
/

0.

For a profinite group Ḡ and a homomorphism ρ̄ : Ḡ → Z/pn−1 , Remark 7.3
therefore implies that α = Bockpn−1 ,Ḡ (ρ̄).
In particular, for x ∈ X take
(x)

ρ̄ = ρ̄Z/pn : Ḡ = S [n,p] → U2 (Z/pn )[n,p].
Identifying U2 (Z/pn )[n,p] = Z/pn−1 , we obtain
α(x),n = Bockpn−1 ,S [n,p] (ǫ(x),Z/pn−1 ).

COHOMOLOGY AND LYNDON WORDS

17

(2) Massey products. Let n = s ≥ 2, so U = Un+1 (Z/p). Let Ḡ be a
profinite group, let ρ̄ : Ḡ → U[n,p] be a continuous homomorphism, and let
ρi,i+1 : Ḡ → Z/p denote its projection on the (i, i + 1)-entry, i = 1, 2, . . . , n.
By a result of Dwyer [Dwy75, Th. 2.6], the extension (7.4) then corresponds
to a defining system for the n-fold Massey product
hρ12 , ρ23 , . . . , ρn,n+1i ⊆ H 2 (Ḡ, Z/p)
(see [Efr14, Prop. 8.3] for the profinite analog of this fact). Thus, for a
fixed Ḡ and for homomorphisms ρ̄1 , . . . , ρ̄n : Ḡ → Z/p, with ρ̄ varying over
all homomorphisms such that ρ̄i,i+1 = ρ̄i , i = 1, 2, . . . , n, the cohomology
element α ranges over the elements of the Massey product hρ̄1 , . . . , ρ̄n i.
In particular, for Ḡ = S [n,p] and for a word w = (x1 · · · xn ) ∈ X ∗ of
length n, the cohomology elements αw,n range over the Massey product
hǫ(x1 ),Z/p , . . . , ǫ(xn ),Z/p i ⊆ H 2 (S [n,p], Z/p), where the ǫ(xi ),Z/p are viewed as
elements of H 1 (S [n,p], Z/p).
(3) Cup products. In the special case n = s = 2, the Massey product
contains only the cup product. Hence for every profinite group Ḡ and a
homomorphism ρ̄ : Ḡ → U3 (Z/p) the cohomology element α ∈ H 2 (Ḡ, Z/p)
is the cup product ρ̄12 ∪ ρ̄23 . In particular, for w = (xy) we have
α(xy),Z/p = ǫ(x),Z/p ∪ ǫ(y),Z/p ∈ H 2 (S [2,p], Z/p).
8. Cohomological duality
Let S = SX be again a free profinite group on the totally ordered set X,
and let it act trivially on Z/p. Let n ≥ 2, so S (n,p) ≤ S p [S, S]. Then the
inflation map H 1 (S [n,p], Z/p) → H 1 (S, Z/p) is an isomorphism. Further,
H 2 (S, Z/p) = 0. By the five-term sequence of cohomology groups [NSW08,
Prop. 1.6.7], trg : H 1 (S (n,p) , Z/p)S → H 2 (S [n,p], Z/p) is an isomorphism.
There is a natural non-degenerate bilinear map
S (n,p) /S (n+1,p) × H 1 (S (n,p) , Z/p)S → Z/p,

(σ̄, ϕ) 7→ ϕ(σ)

(see [EM11, Cor. 2.2]). It induces a bilinear map
(·, ·)n : S (n,p) × H 2 (S [n,p], Z/p) → Z/p,

(σ, α)n = −(trg−1 (α))(σ),

with left kernel S (n+1,p) and trivial right kernel.
Now let w ∈ X ∗ be a word of length 1 ≤ s ≤ n. As in Examples 7.4, we
apply the computations in Section 7 to the group U = Us+1 (Z/pn−s+1 ), the
open normal subgroup N = U(n,p) , the homomorphism ρ = ρw
Z/pn−s+1 : S →
w
[n,p]
[n,p]
U, the induced homomorphism ρ̄ = ρ̄Z/pn−s+1 : S
→ U , and the closed
(n,p)
normal subgroup M = S
of S. We write θw,n , αw,n for θ, α, respectively.

18

IDO EFRAT

Lemma 8.1. For σ ∈ S (n,p) and a word w ∈ X ∗ of length 1 ≤ s ≤ n one
has
(σ, αw,n )n = ιn,s (ǫw,Z/pn−s+1 (σ)).
Proof. By (7.6) and (7.5),
(σ, αw,n )n = θw,n (σ) = ιUn,s (ρw
Z/pn−s+1 (σ)) = ιn,s (ǫw,Z/pn−s+1 (σ)).



This and (6.1) give:
Corollary 8.2. Let w, w ′ be words in X ∗ of lengths 1 ≤ s, s′ ≤ n, respectively, with w Lyndon. Then
n−s

(τwp

, αw′ ,n )n = hw, w ′in .

Proposition 6.4(a)(b) now gives:
Corollary 8.3. Let Lyn≤n (X) be totally ordered by . The matrix
 n−|w|

(τwp
, αw′ ,n )n ,
where w, w ′ ∈ Lyn≤n (X), is upper-triangular unipotent.

In general the above matrix need not be the identity matrix – see e.g.,
Proposition 11.2 below. Next we observe the following general fact:
Lemma 8.4. Let R be a commutative ring and let (·, ·) : A × B → R be
a non-degenerate bilinear map of R-modules. Let (L, ≤) be a finite totally
ordered set, and for every w ∈ L let aw ∈ A, bw ∈ B. Suppose that
the matrix (aw , bw′ ) w,w′∈L is invertible, and that aw , w ∈ L, generate A.
Then aw , w ∈ L, is an R-linear basis of A, and bw , w ∈ L, is an R-linear
basis of B.
Proof. Let b ∈ B, and consider rw′ ∈ R, with w ′ ∈ L. The assumptions
P
P
imply that b = w′ rw′ bw′ if and only if (aw , b − w′ rw′ bw′ ) = 0 for every
P
w. Equivalently, the rw′ solve the linear system w′ (aw , bw′ )Xw′ = (aw , b),
for w ∈ L. By the invertibility, the latter system has a unique solution.
This shows that bw , w ∈ L, is an R-linear basis of B.
By reversing the roles of aw , bw , we conclude that the aw , w ∈ L, form
an R-linear basis of A.

Theorem 8.5.
(a) The cohomology elements αw,n , where w ∈ Lyn≤n (X),
form a Z/p-linear basis of H 2 (S [n,p], Z/p).
n−s
(b) When X is finite, the cosets of the powers τwp , w ∈ Lyn≤n (X),
form a basis of the Z/p-module S (n,p) /S (n+1,p) .

COHOMOLOGY AND LYNDON WORDS

19

Proof. When X is finite, the set Lyn≤n (X) is also finite. By Theorem
n−|w|
5.3, the cosets aw of τwp
, where w ∈ Lyn≤n (X), generate the Z/p(n,p)
(n+1,p)
module S
/S
. We apply Lemma 8.4 with the Z/p-modules A =
(n,p)
(n+1,p)
S
/S
and B = H 2 (S [n,p], Z/p), the non-degenerate bilinear map
A × B → Z/p induced by (·, ·)n , the generators aw of A, and the elements
bw = αw,n of B.
Corollary 8.3 implies that the matrix (aw , bw′ ) is invertible. Therefore
Lemma 8.4 gives both assertions in the finite case.
The general case of (a) follows from the finite case by a standard limit
argument.

We call αw,n , w ∈ Lyn≤n (X), the Lyndon basis of H 2 (S [n,p], Z/p).
Recall that the number of relations in a minimal presentation of a pro-p
group G is given by dim H 2 (G, Z/p) [NSW08, Cor. 3.9.5]. In view of (2.2),
Theorem 8.5 gives this number for G = S [n,p]:
Corollary 8.6. One has
2

dimFp H (S

[n,p]

, Z/p) = dimFp (S

(n,p)

/S

(n+1,p)

)=

n
X

ϕs (|X|),

s=1

where ϕs is the necklace map.
9. The shuffle relations

We recall the following constructions from [CFL58], [Reu93, pp. 134–
135]. Let u1 , . . . , ut ∈ X ∗ be words of lengths s1 , . . . , st , respectively. We
say that a word w ∈ X ∗ of length 1 ≤ n ≤ s1 + · · ·+ st is an infiltration of
u1 , . . . , ut , if there exist sets I1 , . . . , It of respective cardinalities s1 , . . . , st
such that {1, 2, . . . , n} = I1 ∪ · · · ∪ It and the restriction of w to the index
set Ij is uj , j = 1, 2, . . . , t. We then write w = w(I1 , . . . , It , u1 , . . . , ut).
We write Infil(u1 , . . . , ut ) for the set of all infiltrations of u1 , . . . , ut . The
P
infiltration product u1 ↓ · · · ↓ ut of u1 , . . . , ut is the polynomial
w in
ZhXi, where the sum is over all such infiltrations, taken with multiplicity.
If in the above setting, the sets I1 , . . . , It are pairwise disjoint, then
w(I1, . . . , It , u1, . . . , ut ) is called a shuffle of u1 , . . . , ut. We write Sh(u1 , . . . , ut )
for the set of all shuffles of u1, . . . , ut . It consists of the words in Infil(u1 , . . . , ut )
of length s1 + · · · + st . The shuffle product u1 x · · · xut is the polynoP
mial
w(I1 , . . . , It , u1 , . . . , ut ) in ZhXi, where the sum is over all shuffles
of u1 , . . . , ut , taken with multiplicity. Thus u1 x · · · xut is the homogenous

20

IDO EFRAT

part of u1 ↓ · · · ↓ ut of (maximal) degree s1 + · · · + st . For instance
(xy) ↓ (xz) = (xyxz) + 2(xxyz) + 2(xxzy) + (xzxy) + (xyz) + (xzy),
(xy)x(xz) = (xyxz) + 2(xxyz) + 2(xxzy) + (xzxy)
(x) ↓ (x) = 2(xx) + (x),

(x)x(x) = 2(xx).

We may view infiltration and shuffle products also as elements of Zp hXi.
Let Shuffles(X) be the Z-submodule of ZhXi generated by all shuffle products uxv, with ∅ =
6 u, v ∈ X ∗ . Let Shufflesn (X) be its homogenous component of degree n.
Examples 9.1. Shuffles1 (X) = {0},
Shuffles2 (X) = h(xy) + (yx) | x, y ∈ Xi,
Shuffles3 (X) = h(xyz) + (xzy) + (zxy) | x, y, z ∈ Xi.
Let (·, ·) be the pairing of (3.1) for the ring R = Zp . As before, S = SX
is the free profinite group on the set X. The following fact is due to Chen,
Fox, and Lyndon in discrete case [CFL58, Th. 3.6] (see also [Mor12, Prop.
8.6], [Reu93, Lemma 6.7]), as well as [Vog05, Prop. 2.25] in the profinite
case.
Proposition 9.2. For every ∅ =
6 u, v ∈ X ∗ and every σ ∈ S one has
ǫu,Zp (σ)ǫv,Zp (σ) = (ΛZp (σ), u ↓ v).
Corollary 9.3. Let u, v be nonempty words in X ∗ with s = |u| + |v| ≤ n.
For every σ ∈ S (n,p) one has (ΛZp (σ), uxv) ∈ pn−s+1 Zp .
Proof. If w is a nonempty word of length |w| < s, then by Proposition
4.1(b), ǫw,Zp (σ) ∈ pn−|w| Zp ⊆ pn−s+1 Zp . In particular, this is the case
for w = u, w = v, and when w ∈ Infil(u, v) \ Sh(u, v). It follows from

Proposition 9.2 that (ΛZp (σ), uxv) ∈ pn−s+1 Zp .
We obtain the following shuffle relations (see also [Vog04, Cor. 1.2.10]
and [FS84, Th. 6.8]). We write X s for the set of words in X ∗ of length s.
Theorem 9.4. For every ∅ =
6 u, v ∈ X ∗ with s = |u| + |v| ≤ n one has
X
(uxv)w αw,n = 0.
w∈X s

Proof. For σ ∈ S (n,p) , Corollary 9.3 gives
X
X
(uxv)w ǫw,Zp (σ) = (ΛZp (σ), uxv) ∈ pn−s+1 Zp .
(uxv)w ǫw,Zp (σ) =
w∈X s

w∈X ∗

COHOMOLOGY AND LYNDON WORDS

21

Therefore, by Lemma 6.2,
X
X
X
(uxv)w ιn,s (ǫw,Z/pn−s+1 (σ))
(uxv)w (σ, αw,n )n =
(uxv)w αw,n )n =
(σ,
w∈X s

w∈X s

w∈X s

= ιn,s

X

w∈X s


(uxv)w ǫw,Z/pn−s+1 (σ) = 0.

Now use the fact that (·, ·)n : S (n,p) × H 2 (S [n,p], Z/p) → Z/p has a trivial
right kernel.

Corollary 9.5. There is a canonical epimorphism
n  M

M

Z / Shuffless (X) ⊗ (Z/p) → H 2 (S [n,p], Z/p)
s=1

w∈X s

(r̄w )w 7→

X

rw αw,n .

w

Proof. By Theorem 9.4 this homomorphism is well defined. By Theorem
8.5(a), it is surjective.

Remark 9.6. In view of Lemma 6.2, the epimorphism of Corollary 9.5
and the canonical pairing (·, ·)n induce a bilinear map
n  M

M

(n,p)
Z / Shuffless (X) → Z/p,
S
×
s=1

w∈X s

(σ, (rw )w ) =

X

rw ιn,s (ǫw,Z/pn−s+1 (σ))

w

with left kernel S (n+1,p) .

Example 9.7. We show that for every x1 , x2 , . . . , xk ∈ X one has
(x1 x2 · · · xk ) + (−1)k−1 (xk · · · x2 x1 ) ∈ Shufflesn (X).
We may assume that x1 , x2 , . . . , xk are distinct. For 1 ≤ l ≤ k − 1 let
u = (x · · · x x ) and vl = (xl+1 · · · xk ). We consider the polynomial
Pl k−1 l l−1 2 1
ul xvl in ZhXi. It is homogenous of degree k. If w ∈ Sh(ul , vl ),
l=1 (−1)
then either:
(1) xl appears before xl+1 in w, and then w appears with an opposite
sign also in Sh(ul−1 , vl−1 ); or
(2) xl+1 appears before xl in w, and then w appears with opposite sign
also in Sh(ul+1 , vl+1 ).

22

IDO EFRAT

The only exceptions are w = (x1 x2 · · · xk ) ∈ Sh(u1 , v1 ) and w = (xk · · · x2 x1 ) ∈
Sh(uk−1, vk−1 ). This shows that
k

(x1 x2 · · · xk ) + (−1) (xk · · · x2 x1 ) =

k−1
X

(−1)l−1 ul xvl .

l=1

For 1 ≤ k ≤ n Corollary 9.5 therefore implies that
α(x1 x2 ···xk ),n = (−1)k−1 α(xk ···x2 x1 ),n .
10. Example: The case n = 2
Our results in this case are fairly well known, and are brought here in
order to illustrate the general theory.
As before let S = SX with X totally ordered. Here S (2,p) = S p [S, S]
and S̄ = S [2,p] is the maximal elementary p-abelian quotient of S. We
L
may identify H 1 (S, Z/p) = H 1 (S̄, Z/p) ∼
= x∈X Z/p. Let χx,Z/p = ǫ(x),Z/p ,
x ∈ X, be the basis of H 1 (S, Z/p) dual to X (see Remark 6.2).
The Lyndon words w of length ≤ 2 are (x), where x ∈ X, and (xy), where
x, y ∈ X and x < y. For these words we have τ(x) = x and τ(xy) = [x, y].
By Examples 7.4(1)(3),
α(x),2 = Bockp,S̄ (χx,Z/p ),

α(xy),2 = χx,Z/p ∪ χy,Z/p .

Hence, by Theorem 8.5, Bockp,S̄ (χx,Z/p ) and χx,Z/p ∪ χy,Z/p , where x < y,
form a Z/p-linear basis of H 2 (S̄, Z/p). Furthermore, when X is finite, the
elements of the form xp and [x, y] with x, y ∈ X, x < y, form a Z/p-linear
basis of S (2,p) /S (3,p) . In view of Examples 9.1 and Corollary 9.5, the map
P
(r̄w ) 7→ w rw αw,2 induces an epimorphism
 M 

M
Z/p ⊕
Z /h(xy) + (yx) | x, y ∈ Xi ⊗ (Z/p) → H 2 (S̄, Z/p).
x∈X

x,y∈X

It coincides with the map

H 1 (S̄, Z/p) ⊕

V2

H 1 (S̄, Z/p) → H 2 (S̄, Z/p)

which is Bockp,S̄ on the first component and ∪ on the second component. When p 6= 2 the direct sum is a free Z/p-module on Lyn≤2 (X),
and by comparing dimensions we see that the epimorphism is in fact an
isomorphism (compare [EM11, Cor. 2.9(a)]). However when p = 2 one has
Bock2,S̄ (χ) = χ ∪ χ [EM11, Lemma 2.4], so the above epimorphism is not
injective.

COHOMOLOGY AND LYNDON WORDS

23

Next, Proposition 6.4 shows that the matrix (hw, w ′i2 ), where w, w ′ ∈
Lyn≤2 (X), is the identity matrix. In view of Corollary 8.2, it coincides

2−|w|
with the matrix (τwp
, αw′,2 )2 . Thus
(xp , Bockp,S̄ (χx,Z/p ))2 = 1 for every x ∈ X,

(xp , Bockp,S̄ (χy,Z/p ))2 = 0 for every x, y ∈ X, x 6= y,
(xp , χy,Z/p ∪ χz,Z/p )2 = 0 for every x, y, z ∈ X,
([x, y], Bockp,S̄ (χz,Z/p ))2 = 0 for every x, y, z ∈ X,
([x, y], χz,Z/p ∪ χt,Z/p )2 = 0 for every x, y, z, t ∈ X, (xy) 6= (zt), (tz),
([x, y], χx,Z/p ∪ χy,Z/p )2 = 1 for every x, y ∈ X with x < y.
This recovers well known facts from [Lab66, §2.3], [Koc02, §7.8] and [NSW08,
Th. 3.9.13 and Prop. 3.9.14]
11. Example: The case n = 3.
2

Here S (3,p) = S p [S, S]p [S, [S, S]]. We abbreviate S̄ = S [3,p] . Recall that
Lyn≤3 (X) consists of the words
(x) for x ∈ X,
(xy), (xxy), (xyy) for x, y ∈ X with x < y,
(xyz), (xzy) for x, y, z ∈ X with x < y < z.
For these words
τ(x) = x, τ(xy) = [x, y],

τ(xxy) = [x, [x, y]],

τ(xyz) = [x, [y, z]],

τ(xyy) = [[x, y], y],

τ(xzy) = [[x, z], y].

By Theorem 5.3, the cosets of
3

xp , [x, y]p , [x, [x, y]], [[x, y], y], [x, [y, z]], [[x, z], y],
with x, y, z as above, generate S (3,p) /S (4,p) . When X is finite, they form
a linear basis of S (3,p) /S (4,p) over Z/p (Theorem 8.5(b)). Furthermore,
Theorem 8.5(a) gives:
Theorem 11.1. The following cohomology elements form a Z/p-linear basis of H 2 (S̄, Z/p):
α(x),3 , α(xy),3 , α(xxy),3 , α(xyy),3 , α(xyz),3 , α(xzy),3 ,
where x, y, z ∈ X and we assume that x < y < z.

24

IDO EFRAT

By Examples 7.4, α(x),3 = Bockp2 ,S̄ (χx,Z/p2 ), and for every x, y, z ∈
X, α(xyz),3 belongs to the triple Massey product hχx,Z/p , χy,Z/p , χz,Z/p i ⊆
H 2 (S̄, Z/p).
(xy)
We further recall that α(xy),3 is the pullback to H 2 (S̄, Z/p) under ρ̄Z/p2 : S̄ →
U3 (Z/p2 )[3,p] of the cohomology element in H 2 (U3 (Z/p2 )[3,p], Z/p) corresponding to the central extension
0 → Z/p → U3 (Z/p2 ) → U3 (Z/p2 )[3,p] → 1.
Alternatively, it has the following explicit description: By Proposition
∼
6.3(a), U3 (Z/p2 )(3,p) = I3 + ZpE13 , and let ι = ιU3,2 : U3 (Z/p2 )(3,p) −
→ Z/p
be the natural isomorphism. By (7.5) and (7.6), α(xy),3 = − trg(θ), where
∼
(xy)
θ = ι ◦ (ρZ/p2 |S (3,p) ) and trg : H 1 (S (3,p) , Z/p)S −
→ H 2 (S̄, Z/p) is the transgression isomorphism.
3−|w|
, αw′ ,3 )3 ) = (hw, w ′i3 ), where w, w ′ ∈
Next we compute the matrix ((τwp
Lyn≤3 (X):
Proposition 11.2. For w, w ′ ∈ Lyn≤3 (X) one has

′


 1, if w = w ;
hw, w ′i3 = −1, if w = (xyz), w ′ = (xzy) for some x, y, z ∈ X, x < y < z;


 0, otherwise.

Proof. In view of Proposition 6.4, it is enough to show the assertion when
w 6= w ′ , either |w| = |w ′ | or 2|w| ≤ |w ′|, and the letters of w ′ appear in w.
Furthermore, when |w| = |w ′ | we may assume that w ≤alp w ′ .
Thus when w has one of the forms (x), (xy), (xyy), (xzy) (where x < y <
z) there is nothing more to show.
When w = (xxy) with x < y we need to check only the word w ′ = (xyy).
Then Lemma 4.2 gives
h(xxy), (xyy)i3 = ǫ(xyy),Z/p ([x, [x, y]])
= ǫ(x),Z/p (x) · ǫ(yy),Z/p ([x, y]) − ǫ(y),Z/p (x) · ǫ(xy),Z/p ([x, y])
= 1 · 0 − 0 · 1 = 0.
When w = (xyz) with x < y < z we need to check only the word
w = (xzy). Then Lemma 4.2 gives
′

h(xyz), (xzy)i3 = ǫ(xzy),Z/p ([x, [y, z]])
= ǫ(x),Z/p (x) · ǫ(zy),Z/p ([y, z]) − ǫ(y),Z/p (x) · ǫ(xz),Z/p ([y, z])
= 1 · (−1) − 0 · 0 = −1.
This completes the verification in all cases.



COHOMOLOGY AND LYNDON WORDS

25

In view of Examples 9.1, Corollary 9.5 gives rise to an epimorphism
(11.1)
 M

M

Z/p ⊕
Z(xy) /h(xy) + (yx) | x, y ∈ Xi ⊗ (Z/p)
x∈X

x,y∈X

⊕



M

x,y,z∈X



Z(xyz) /h(xyz) + (xzy) + (zxy) | x, y, z ∈ Xi ⊗ (Z/p)

→ H 2 (S̄, Z/p).

Moreover, for x, y, z ∈ X, x < y < z, we have
(yx) = (x)x(y) − (xy)
2(xx) = (x)x(x)
(xyx) = (x)x(xy) − 2(xxy)
(yxx) = (x)x(yx) − (xx)x(y) + (xxy)
(yxy) = (xy)x(y) − 2(xyy)
(yyx) = (yy)x(x) − (y)x(xy) + (xyy)
(yxz) = (y)x(xz) − (xyz) − (xzy)
(zxy) = (z)x(xy) − (xzy) − (xyz)
(yzx) = (zx)x(y) − (x)x(zy) + (xzy)
(zyx) = (yx)x(z) − (x)x(yz) + (xyz)
3(xxx) = (x)x(xx).
These congruences and Example 2.1 imply that
X
X
Zw ≡
Zw + 2-torsion (mod Shuffles2 (X)),
w∈X 2

X

w∈X 3

w∈Lyn2 (X)

Zw ≡

X

Zw + 3-torsion (mod Shuffles3 (X)).

w∈Lyn3 (X)

Therefore, for p > 3, the direct sum in (11.1) is the free Z/p-module on
the basis Lyn≤3 (X). Thus the epimorphism (11.1) maps the Z/p-linear
basis 1w, w ∈ Lyn≤3 (X), bijectively onto the Z/p-linear basis αw,3, w ∈
Lyn≤3 (X) (see Theorem 8.5). Consequently we have:
Theorem 11.3. For n = 3 and p > 3, (11.1) is an isomorphism. Thus
all relations in H 2 (S [3,p] , Z/p) are consequences of the shuffle relations of
Theorem 9.4.

26

IDO EFRAT

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Department of Mathematics, Ben-Gurion University of the Negev, P.O.
Box 653, Be’er-Sheva 84105, Israel
E-mail address: efrat@math.bgu.ac.il

