The pro-$p$ group of upper unitriangular matrices

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by  Nadia Mazza

THE PRO-p GROUP OF UPPER UNITRIANGULAR MATRICES
NADIA MAZZA
7 Abstract. We study the pro-p group G whose finite quotients give the prototypical Sylow
1 p-subgroup of the general linear groups over a finite field of prime characteristic p. In this
0 article, we extend the known results on the subgroup structure of G. In particular, we give an
2
explicit embeddingof theNottinghamgroup asasubgroup andshowthatit isselfnormalising.
n Holubowski ([13, 14, 15]) studies a free product Cp∗Cp as a (discrete) subgroup of G and we
a provethat its closure is selfnormalising of infiniteindex in thesubgroup of 2-periodic elements
J of G. We also discuss change of rings: field extensions and a variant for the p-adic integers,

1 this latter linking G with some well known p-adic analytic groups. Finally, we calculate the
1 Hausdorff dimensions of some closed subgroups of G and show that the Hausdorff spectrum of
G is thewhole interval[0,1] which is obtained by considering partition subgroupsonly.
]
R
G
. MSC: Primary 20E18; Secondary 20H25
h
t Keywords: pro-p group; infinite unitriangular matrix group; Nottingham group; Hausdorff di-
a
m mension
[

1 1. Introduction
v
4 In this paper we investigate the pro-p group whose finite quotients give the prototypical
2 Sylow p-subgroup of the general linear groups over a finite field of prime characteristic p. For
0
convenience, we will consider an odd prime p throughout the paper.
3
0 Sylowp-subgroupsoffinitegenerallineargroupsGLn(q)forq apowerofphavebeenminutely
. analysed by Weir in the 50s ([23]). His findings have subsequently been exploited by many; in
1
0 particular Bier ([4, 5]) who extended some of Weir’s results to the pro-p group G(q) of upper
7 unitriangular matrices with coefficients in the field F with q elements. By upper unitriangular
q
1
matrix, we mean an upper triangular matrix with all diagonal coefficients equal to 1. In the
:
v
first part of this article, we will elaborate on Weir, Bier’s and Holubowski’s results ([13, 14, 15]),
i
X and we will focus on the subgroup structure of G(q), revisiting the notion of partition subgroups
r considered by Weir. We will also discuss the embeddings of a free product of the form Cp ∗Cp
a
as a discrete subgroup of G(p) and of the Nottingham group N(q) ([6, 7, 8]) as a closed pro-p
subgroup of G(q). Then we will discuss how we can relate G(q) and G(p) for a field extension
F /F . In Section 7 we present a p-adic version of the group G(q) and briefly relate this group
q p
to some well-known p-adic analytic groups ([9]). In thelast section of the paperwe calculate the
Hausdorff dimensions ([1, 2, 3, 10]) of the closed subgroups presented in the preceding sections.
AshortappendixincludesbackgroundabouttheautomorphismgroupsoftheSylowp-subgroups
of the general linear groups GL (q) and about the Hausdorff dimension for profinite groups.
n
Definition 1.1. Let q = pf be a power of an odd prime number p, with f ≥ 1. For each n ∈ N,
let G (q) be the Sylow p-subgroup of GL (q) formed by the upper triangular matrices with
n n
diagonal coefficients all equal to 1. Let V (q) = Fn be the set of column vectors of size n with
n q
coefficients in F .
q
Date: January 12, 2017.
1
2 NADIAMAZZA
Note that for each n > 1, we have
G (q) V (q)
G (q) = n−1 n−1 ∼= V (q)⋊G (q),
n 0 1 n−1 n−1
1×(n−1)
(cid:18) (cid:19)
where G (q) acts on V (q) by left multiplication in the obvious way. Thus, the natural
n−1 n−1
projections
(1) ... // G (q)πn+1 // G (q) πn // ... // G (q) π2// G (q)= 1
n+1 n 2 1
form an inverse system.
Definition 1.2. Let G(q) = limG (q) be the inverse limit of (1). The group G(q) is a pro-
←− n
n
p group, which we will call the pro-p group of upper unitriangular matrices over F . If the
q
prime power q is clear from the context, we write simply G and G instead of G(q) and G (q)
n n
respectively.
For each n ∈ N, let θ : G→ G be the universal map, i.e. such that
n n
θ = π ···π θ : G → G for all 1 ≤ m < n.
m m+1 n n m
Let also N =ker(θ ). Thus N is the normal subgroup of G formed by all the matrices whose
n n n
upper left n×n diagonal block is the identity matrix.
From [24, Section 1.2], a filter base for the topology on G is the set of open normal subgroups
B = {θ−1(X) = XN | X E G , n ∈N}
n n n
while the set U = {θ−1(X) = XN | X ⊆ G , n ∈ N} forms a fundamental system of open
n n n
neighbourhoods of the identity in G ([12, p. 26-27]). In particular, G is countably based (cf. [24,
Proposition 4.1.3]).
For any i,j ∈ N let e denote the infinite elementary square matrix whose unique nonzero
ij
coefficient is (i,j) and is equal to 1, i.e. (e ) = δ δ for all i,j,k,l ∈ N. So, if a ,...,a ∈ F×
ij kl ik jl 1 f q
generate F as F -vector space, then G = h1+a e | 1 ≤ i < j , 1 ≤ c ≤ fi, where we write
q p c ij
1 for the identity element of G and F× for the multiplicative group of nonzero elements of F .
q q
The set
E = {1+e | i∈ N}
i,i+1
generates G topologically and converges to 1. Indeed, it generates a dense subgroup of G,
because G = h1+e | 1 ≤ i< ni for all n ∈ N; moreover, any open subgroup of G contains
n i,i+1
all but a finite number of such elements ([21, Section 2.4]).
The metric on G is defined as follows. Let x,y ∈ G and fix a number ǫ ∈ (0,1), for instance
ǫ = p−1. Then
(2) d(x,y) = ǫk where k = max{n | y−1x ∈ N }.
n
Then d( , ) is an ultrametric, i.e. subject to the axioms
• d(x,y) ≥ 0 with equality if and only if x = y;
• d(x,y) = d(y,x);
• d(x,z) ≤ max{d(x,y),d(y,z)} for all x,y,z ∈ G (the ultrametric axiom).
For x ∈ G and k ≥ 0, the open ball of centre x and radius ǫk is the set B(x,ǫk) = {y ∈
G | d(x,y) < ǫk} of all the elements y of G such that y−1x ∈ N for some integer l > k. It
l
differs from its closure only if k is an integer, in which case B(x,ǫk) = {y ∈ G | y−1x ∈ N } =
k
B(x,ρ) = B(x,ǫk−1), for any ǫk−1 ≥ ρ > ǫk. In particular,
N = {y ∈ G | y ∈ N } = B(1,ǫn) = B(1,ǫn−1) for all n ∈ N.
n n
THE PRO-p GROUP OF UPPER UNITRIANGULAR MATRICES 3
2. Partition subgroups of G
Let γ (G) denote the n-th term in the lower central series of G, starting with γ (G) = G and
n 1
γ (G) = [G,G], the derived subgroup of G. The notation [A,B] for subsets A,B of a group
2
denotes the subgroup spanned by the commutators [a,b] = a−1ab with a ∈ A and b ∈ B, where
we write
xy = y−1xy and yx = yxy−1.
At the basis of each computation, lays the ubiquitous commutator relation
(3) [1+ae ,1+be ]= 1+δ abe −δ abe for all i,j,k,l ∈ N and all a,b ∈ F .
ij kl jk il il kj q
Extending work of Weir, Bier investigated a sub-family of the subgroups that Weir called
partition subgroups of G. The results she proves are only for these partition subgroups, but it
is easy to see that they extend to all partition subgroups. For convenience, we introduce the
following definition.
Definition 2.1. A partition diagram is a subset
µ = {(r ,c ) ∈N2 | r < c , i ∈ N}
i i i i
such that
(4) whenever (i,j) ∈ µ and (j,k) ∈ µ , then (i,k) ∈ µ .
That is, a partition diagram is a collection of pairs of distinct positive integers, which should be
regarded as the coordinates of the nondiagonal squares (or coefficients) in the matrices of G:
(r,c) ∈ µ ⇐⇒ 1+e ∈ G.
rc
The corresponding partition subgroup of G is the subgroup
P = {x ∈ G | x = 0 , ∀ (i,j) ∈/ µ}.
µ ij
So, the constraint (4) on the elements of µ reflects the multiplication of the corresponding
matrices (what Weir called “completing the rectangle”) in P , namely
µ
(1+ae )(1+be ) = 1+ae +be +abe .
ij jk ij jk ik
If µ is such that for each j ≥ 2, if (i,j) ∈µ, then all the pairs (k,j) ∈ µ for all k ≤ i, then we
call µ a partition and write it as
µ = (µ ,µ ,...) where µ = max{l | (l,j) ∈ µ} for all j ≥ 2.
2 3 j
Then
P = {x ∈ G | x = 0 , ∀µ < i< j}
µ ij j
is formed by all the elements of G whose jth column has (j−1−µ ) zeroes above the diagonal.
j
A partition of the form µ = (0c−1,c,c+1,c+2,...) defines the partition subgroup N , for any
c
c ≥ 1 (and if c = 1, then N = G). An exponent “λs” in a partition µ means λ repeated s times.
1
Given a partition diagram µ, we denote |µ| its shape, i.e. the set of all squares on an infinite
chessboard N2 which consist of the possible nonzero squares (i,j) in P .
µ
A square (i,j) covers (k,l) if (i,j) 6= (k,l) and if k ≤ i and l ≥ j. We say that (i,j) avoids
|µ| if (i,j) covers a square outside of |µ|.
Remark 2.2. In[4,5],Bieronlyconsiderspartitions. Moreover, shetakesthe“complementary”
definition of a partition than the one we take here. That is, the parts in a partition denote the
numberofzeroesabovethediagonal. Instead,wehavechosentousethesameconventionasWeir
in [23], in order to include the more general partition subgroups defined by partition diagrams.
4 NADIAMAZZA
If µ ⊆ N2 is a partition diagram, we call a subpartition (diagram) of µ a subset of µ which
is a partition (diagram) on its own. So a partition diagram is a lattice. That is ([11, Sec-
tion 8.2]), given any two subpartitions diagrams µ and µ of µ, their union and intersection are
1 2
also subpartition diagrams of µ. The union of two partition diagrams is the smallest partition
diagram which contains them (i.e. obtained by “completing the rectangles” in Weir’s terminol-
ogy), whilst their intersection is their set intersection. In particular, if µ = (u ,u ,...) and
1 2 3
µ = (v ,v ,...) are subpartitions of µ, then
2 2 3
µ ∪µ = (m ,m ,...) where m = max{u ,v } and
1 2 2 3 j j j
µ ∩µ = (n ,n ,...) where n = min{u ,v }.
1 2 2 3 j j j
It follows that each partition diagram has a unique maximal subpartition
µ = P where P = {µ′ ⊆ µ | µ′ is a subpartition of µ }.
max µ µ
We say that a partitio[n diagram µ converges to a partition if there exists n ≥ 2 such that for
any (r,c) ∈ µ with c ≥ n, then (i,c) ∈ µ for all 1 ≤ i ≤ r. That is, µ becomes a partition for n
large enough. The trivial partition is the partition (0ℵ0), where ℵ is the cardinality of N.
0
[23,Theorem2]describesthepartitiondiagramsµwhichdefinenormalsubgroupsP : namely
µ
µ is a partition and the boundaryof |µ| should move monotonically downward to the right. The
point is that if P E G and (r,c) ∈ µ, then conjugation by any 1+e and 1+e implies that
µ ir cj
(i,c) and (r,j) must also be in µ for all i ≤ r and all j ≥ c, i.e. µ contains all the squares
covered by (r,c). So µ must be a partition, and its “boundary”, determined by all the squares
(i ,c) with i = max{r | (r,c) ∈µ}, must give an increasing sequence i ≤ i ≤ i ....
c c 2 3 4
Arectangular partition subgroupP is anormalsubgroupof Gforµof theformµ = (0c,dℵ0),
µ
where 0 < d ≤ c for some c ∈ N (we could extend to d = 0 by admitting the trivial subgroup
of G as a rectangular partition subgroup). The shape of such |µ| explains the terminology. If
p > 2, then the maximal abelian (and characteristic) subgroups of G have this form, with d= c
([23, Theorem 6]). That is,
*
I
µ = (0c,cℵ0) and P = c+1
µ
0 I
∞
(cid:18) (cid:19)
*
where the coefficients in the block can take any value in F .
q
Extending Pavlov ([20]) and Weir’s ([23]) results, Bier proves that the automorphism group
(cid:0) (cid:1)
of G is generated by three types of continuous automorphisms: inner, diagonal (i.e. conjugation
by an infinite diagonal matrix), and those induced by field automorphisms. Furthermore, shifts
are surjective group homomorphism, where for d ∈ N, the dth shift of x ∈ G is the matrix x[d]
obtained by deleting the first d rows and columns of x.
Definition 2.3. We call a matrix x ∈ G periodic (of period d) if there exists d ∈ N such that
x = x[d]. A subgroup H ≤ G is periodic (of period d) if every element of H is periodic (of
period d).
Here is a summary of Bier and Weir’s results as they apply to G = G(q).
Proposition 2.4.
(1) Partition subgroups are closed.
(2) A partition subgroup P is open if and only if the partition diagram µ is such that there
µ
exists N ∈ N for which (i,j) ∈ µ for all 1≤ i < j and for all j ≥ N.
(3) Let H be a closed subgroup of G. The following are equivalent.
(a) H is a normal subgroup of G.
(b) H is a normal partition subgroup of G.
(c) H is a characteristic subgroup of G.
THE PRO-p GROUP OF UPPER UNITRIANGULAR MATRICES 5
(d) H is a partition subgroup defined by an increasing partition µ, i.e. such that µ ≤
j
µ .
j+1
If H satisfies these equivalent conditions, we call H a normal partition subgroup.
(4) Given a partition diagram µ, the normal core ∩ gP of P is the partition subgroup
g µ µ
∩ggPµ = Pµ′ where µ′ = (µ′2,µ′3,...) with µ′j = min{i | (i,k) ∈ µ , ∀ k ≥ j}
for all j ≥ 2. In particular, if the maximal subpartition of µ converges to the trivial
partition, then Pµ′ = {1}.
(5) The normal closure h(P )Gi of a partition subgroup P of G is the partition subgroup
µ µ
Pµ′′, where µ′′ = (µ′2′,µ′3′,...) is the partition with µ′j′ = max{i | (i,k) ∈ µ , ∀ k ≤ j}.
It is clear that partition subgroups are closed, since any sequence of elements in a partition
subgroup P which converges in G must converge in P . The other statements are routine.
µ µ
Weir obtained specific results for normal partition subgroupsof the finite quotients G (q), for
n
all n,q, and these also apply to G.
Proposition 2.5. [23, Theorem 3] Given a normal partition subgroup P , then
µ
(1) [Pµ,G] =Pµ′, where |µ′| are all the squares covered by |µ|.
(2) Let P∗ be the preimage of Z(G/P ) in G, then P∗ = P is the normal partition subgroup
µ µˆ
where |µˆ| are all the squares which do not avoid µ.
Thus to get the commutator subgroup [P ,G] we “delete” the squares at the corners of |µ|,
µ
i.e. if (i,j) and (i+1,j+1) are both outside |µ| but (i,j+1) is in |µ|, then we delete (i,j+1)
in |µ′|. On the other hand, if (i,j) and (i+1,j +1) are both in |µ| but (i+1,j) is not in |µ|,
then we add (i+1,j) to |µ| to get |µˆ|.
As an example for Proposition 2.5, we get the subgroups γ (G) in the lower central series of
d
G by deleting successive super diagonals, where the dth super diagonal is the set of all squares
(i,i+d) ∈N2. Thus
γd(G) = P(0d−1,1,2,3,...) for all d ∈N, starting with γ1(G) = G.
Similar considerations allow us to calculate the derived subgroups G(d) = [G(d−1),G(d−1)] of G,
starting with
G(2) = [G,G] = γ (G) = {1+ a e ∈G}.
2 ij ij
j≥i+2
X
Thus elementary commutators calculations (Equation (3)), with d≥ 2, give
1+ a e ,1+ b e = 1+ c e .
ij ij ij ij ik ik
(cid:2) j≥iX+2d−1 j≥iX+2d−1 (cid:3) k≥Xi+2d
Hence, as partition subgroup,
G(d) = P(0(−1+2d−1),1,2,3,...) for all d≥ 2.
In particular, G is not soluble, because its derived series does not converge.
Partition subgroups can also be used to show that G is not hereditarily just infinite. A
profinite group G is hereditarily just infinite if every every open subgroup is just infinite ([17,
Definition I.3]). That is, every nontrivial closed normal subgroup of any open subgroup of G
has finite index. By the above discussion, the open subgroups of G have infinitely many closed
normal subgroups of infinite index (e.g. the subgroups in the lower central series).
Fromthebasiccommutatorformula(3),weobtainthestructureofthecentralisersofpartition
subgroups of G.
6 NADIAMAZZA
Definition 2.6. Let µ be a partition diagram. The orthogonal partition diagram of µ is the
partition diagram
(5) µ⊥ = {(k,l) ∈ N2 | k <l and∀ (i,j) ∈ µ then k 6= j andl 6=i}.
The centre of µ is the subpartition diagram
(6) ζ = µ∩µ⊥ of µ.
µ
For instance, if µ = {(3,4)}, then
µ⊥ = {(k,l) ∈N2 | k 6= 4 , l 6= 3} and ζ = µ , i.e.,
µ
1 ∗ 0 ∗ ∗ ∗ ∗
1 0 0 0 0 0 0
1 0 ∗ ∗ ∗ ∗
  1 0 0 0 0 0
1 ∗ ∗ ∗ ∗  
1 ∗ 0 0 0
Pµ⊥ =  1 0 0 0  and Pζµ =Pµ = .
   1 0 0 0 
 1 ∗ ···∗  
 ... ...   ... ... ...
   
 
By definition C (P ) = C (1 + a e ) and each C (1 + a e ) = C (1 + e ) is a
G µ G ij ij G ij ij G ij
(i,j)∈µ
\
aij∈Fq
partition subgroup of G. Now 1+ae ∈ C (1+e ) for each (i,j) ∈ µ and a ∈ F if and only if
kl G ij q
1 = [1+e ,1+e ] =1+δ e −δ e .
kl ij il kj jk il
So l 6= i and k 6= j for each (i,j) ∈ µ. In other words,
1+e ∈ C (P ) ⇐⇒(k,l) ∈ µ⊥.
kl G µ
Which leads to the following conclusion.
Proposition 2.7. Let µ be a partition diagram. Then
CG(Pµ)= Pµ⊥ and Pζµ is the centre of Pµ.
In particular, C (P ) = {1} for any open partition subgroup.
G µ
3. Examples of torsion subgroups
The direct limit limG of the G ’s is a discrete torsion group, and so not a subgroup of G.
−→ n n
n
Here limG is the group formed by all the square matrices x such that there exists m ∈ N for
−→ n
n
which x ∈ G . Now, each element of limG can be regarded as a torsion element in G in the
m −→ n
n
obvious way, by taking each x ∈ limG to xN /N ) ∈ G. This mapping, let’s call it ρ, is an
−→ n n n n∈N
n
injective homomorphism of abstract group(cid:0)s, which takes limG into G and with the property
−→ n
n∈N
that its image is dense in G, i.e. im(ρ)N /N = G. Note that G contains “many” torsion
n n
n∈N
\
elements which are not in im(ρ) (take for instance 1+ e ).
1j
1<j
X
In [14], Holubowski studies string subgroups, which form a large class of torsion discrete
subgroups of G. In particular string subgroups cannot contain any open subgroup of G.
THE PRO-p GROUP OF UPPER UNITRIANGULAR MATRICES 7
Definition 3.1. A matrix a ∈ G is a string if a is in the image of some injective group homo-
morphism
G ֒→ G of pairwise diagonal commuting block matrices of size greater than 1.
ni
nYi>1
Thus a has finite order and a−1 is a string with the same block structure as a. A string subgroup
of G is a subgroup of G formed by strings.
So a string subgroup Q of G is isomorphic to a subgroup of a partition subgroup of G of the
form G for some non-negative integers n , for i ∈N.
ni i
nYi>1
The equality
G ∼= G/P where µ = (0n1,nn2,(n +n )n3,...,( n )nj+1,...)
ni µ 1 1 2 i
i∈N 1≤i≤j
Y X
shows that, regarded as abstract groups (as in [14]), string subgroups are the complements of
normal partition subgroups. That is,
G = P · G , P ∩ G = {1} and P E G.
µ ni µ ni µ
i∈N (cid:18)i∈N (cid:19)
Y Y
4. Free subgroups of G
In this section, we let q = p, and G= limG (p).
←− n
n∈N
We discuss a particular discrete subgroup of G investigated by Holubowski (cf. [14]), and we
also look at its closure in G. This subgroup of G is the product of two string subgroups, but is
not a string subgroup itself.
Definition 4.1. Let
1 1
1
 
1 1
s = 1+ e = and
2n−1,2n
 1 
nX∈N  .. 
 .
 
 
1
1 1
 
1
t = 1+ e =
2n,2n+1  1 1 
 
nX∈N  1 
 
 ...
 
 
and regard s and t as elements of G.
Let F = hxi∗hyi be the free product of two groups of order p.
Holubowski ([14, Theorem 1]) defines a function ϕ : F → G, by
ϕ(x) = s = 1+ e and ϕ(y) = t = 1+ e
2n−1,2n 2n,2n+1
n∈N n∈N
X X
and proves that ϕ is an injective group homomorphism. [14, Theorem 1] shows that the image
is contained in the intersection of the subgroup of so-called banded matrices with the subgroup
of periodic matrices of G. A banded matrix is a matrix (a ) ∈ G for which there exists d ∈ N
ij
8 NADIAMAZZA
such that a = 0 whenever j >i+d. In particular, im(ϕ) is not a closed subgroup of G because
ij
im(ϕ) < im(ϕ)N /N . We let Q = ϕ(F) be the closure of ϕ(F) in G.
n n n
A word w(x,y) = xa1yb1···xalybl is mapped to
T
ϕ(w(x,y)) = w(s,t) = sa1tb1···saltbl
= 1+ A e + B e
j 2n−1,2n+j j 2n,2n+j
 
n∈N 0≤j<2l 1≤j<2l
X X X
 
where the coefficients A ,B are monomials in a ,...,a ,b ,...,b of degrees at most j +1 and
j j 1 l 1 l
j respectively. For instance, for any a,b,c,d ∈ F ,
p
satbsctd =
1+ (a+c)e +(ab+cd+ad)e +
2n−1,2n 2n−1,2n+1
n∈N(cid:18)
X
+abce +abcde +
2n−1,2n+2 2n−1,2n+3
(7) +(b+d)e2n,2n+1+bce2n,2n+2+bcde2n,2n+3
(cid:19)
1 a+c ab+cd+ad abc abcd 0 ...
0 1 b+d bc bcd 0 ...
=   .
0 0 1 a+c ab+cd+ad abc ...
 ... ... 
 
 
Recall from Definition 2.3 that X[2] is the matrix obtained from X ∈ G by deleting the first
2 rows and columns. From Equation (7) and elaborating by induction on it, we record the
following.
Proposition 4.2. For any w(x,y) ∈ F, we have
ϕ(w(x,y)) = w(s,t) = w(s,t)[2].
Moreover, the length of w(x,y) can be read from the last nonzero squares in the first two rows
of w(s,t) = xa1yb1···xalybl. Namely,
(i) if a b 6= 0, i.e. a ,b 6= 0 for all 1 ≤ j ≤l, then the last nonzero squares in the first two
1 l j j
rows of w(s,t) are (1,2l+1) and (2,2l+1) respectively;
(ii) if a = 0 6= b , (i.e. a is the only zero exponent) then the last nonzero squares in the
1 l 1
first two rows of w(s,t) are (1,2l−1) and (2,2l+1);
(iii) if a = b = 0 and no other exponent is zero, then the last nonzero squares in the first
1 l
two rows of w(s,t) are (1,2(l−1)) and (2,2l).
Inparticular, weobtain2-periodic elementsofGwhose lastnonzerosquaresinanytwosuccessive
rows (i,j),(i+1,k) are such that k−j ∈ {0,2}. Therefore
Q < {X ∈ G | X = X[2]}
is a closed subgroup of infinite index in the subgroup of 2-periodic elements of G. Furthermore,
ϕ−1(γ (G)) = γ (F) for all d ∈N.
d d
A counting exercise gives the indices |PN : N | = p2n−3 and |QN : N | = pξn, where P
n n n n
n+1
is the subgroup of 2-periodic elements of G and ξ = n − 2 + ⌊ ⌋, this latter obtained
n
2
inductively on n.
Remark 4.3.
THE PRO-p GROUP OF UPPER UNITRIANGULAR MATRICES 9
(1) It is important to emphasise that Holubowski regards ϕ(F) (as most of the subgroups
he investigates in [13, 14, 15]) as a discrete group, and this can be seen from the fact
that im(ϕ) is not closed in G. Recall that F is a pro-p group for the topology defined
by taking the set F of all the subgroups of F of finite p-power index (cf. [12, Example
(iii), p. 29]). Letting R run through all the normal subgroups of F of finite p-power
index, we obtain all the 2-generated finite p-groups as the quotients F/R. For instance
γ (F) ∈ F and F/γ (F) ∼= C ×C .
2 2 p p
(2) Let us also mention the description of F from [22, p. 28], i.e. that of the fundamental
1
group of a tree with fundamental domain a segment hxi hyi .
5. Nottingham group
As pointed out in the concluding remark of [13], the Nottingham group can be seen as a
subgroup of G. We make this embedding of topological groups explicit in this section. There
are equivalent definitions of the Nottingham group. We follow [7].
Definition 5.1. The Nottingham group N = N is the group of algebra automorphisms of
q
F [[t]] of the form
q
t 7→ t+ a tj a ∈ F .
j j q
j≥2
X
R. Camina investigated the subgroups of N and proved that N contains every countably
based pro-p group as a closed subgroup.
Now the tantalising fact that G = limG is countably based as pro-p group, implies by
←− n
n
Camina’s result that G embeds into N as a closed subgroup. On the other hand, by linear
algebra, the elements of N can be expressed as infinite unitriangular matrices, i.e. elements of
G, and therefore N is a subset of G; but it certainly cannot be the whole of G, because N only
consists of algebra automorphisms.
The convention is that N acts on the right of F [[t]]. So, we can identify the nonconstant
q
elements of F [[t]] as infinite row vectors
q
a tj ∈ F [[t]] ←→ (a ,a ,a ,...)∈ Fℵ0.
j q 1 2 3 q
j≥1
X
Matrix multiplication induces a linear transformation of Fℵ0,
q
(a ,a ,a ,...) 7→ (a ,a ,a ,...)x = (a ,a +x a ,...,a + x a ,...) ,
1 2 3 1 2 3 1 2 12 1 j ij i
1≤i<j
X
for all (x ) ∈ G. which translates as function on F [[t]] as follows:
ij q
a tj 7→ a t+(a +x a )t2+···+ a + x a tj +...
j 1 2 12 1 j ij i
 
j≥1 1≤i<j
X X
Given that N = he [α ],e [α ] | 1 ≤ c ≤ fi, whereα ,...,α ∈ F generate F as F -vector
1 c 2 c 1 f q q p
space, and
e [α ] : t 7→ t+α tr+1 ∈ N , we have
r c c
j
( a tj)e [α ] = a (t+α tr+1)j = a αitri+j .
j r c j c j i c
j j j (cid:18)0≤i≤j(cid:18) (cid:19) (cid:19)
X X X X
This suggests the following mapping σ : N → G, defined on the generators of N by
i j−i i j−i
def
(8) σ(e [α ]) = g [α ] = α r e = 1+ α r e ,
r c r c j−i c ij j−i c ij
1≤i≤j(cid:18) r (cid:19) 1≤i<j(cid:18) r (cid:19)
X X
10 NADIAMAZZA
where the sums run over all the positive integers i ≤ j, resp. i < j, such that
j −i
∈ Z and j ≤ 2i
r
and all the other coefficients are zero. Note that in the first sum g [α] = 1 for all i ∈N.
r ii
In particular, for α = 1, if we write e = e [1] and g = g [1], then the ith row of g contains
1 r r r r r
theithrowofPascal’s trianglestartingfromthediagonal1,andspacedby(r−1)zeroesbetween
each coefficient in a row. Note that g [α ]∈ γ (G) for all r ≥ 1.
r c r
For example,
1 1 0 ...
1 2 1 0 ...
g =   and
1 1 3 3 1 0 ...
 ... ... ... 
 
 
1 0 1 0 ...
1 0 2 0 1 0 ...
g =  
2 1 0 3 0 3 0 1 0...
 ... ... ... 
 
 
Matrix multiplication yields
(a ,a ,...)g = (a ,a +a ,2a +a ,a +3a +a ,...)
1 2 1 1 1 2 2 3 2 3 4
which corresponds to
a t+(a +a )t2+(2a +a )t3+(a +3a +a )t4+··· ∈ F [[t]]
1 1 2 2 3 2 3 4 q
and so gives in particular (1,0,0,...)g = (1,1,0,...), i.e. te = t+t2. Accordingly, for any
1 1
“canonical” vector ti ∈ F [[t]] the corresponding “canonical” row vector v ∈ Fℵ0 has a unique
q i q
nonzero coefficient equal to 1 in the ith coordinate, so that v g [α ] is the ith row of g [α ],
i r c r c
i
(0i−1,1,0r−1,iα ,0r−1, α2,0r−1,...,0r−1,αi,0r−1,...)
c 2 c c
(cid:18) (cid:19)
which corresponds to the element
i
ti+iα ti+r +α2 ti+2r +···+αiti(r+1) ∈ F [[t]].
c c 2 c q
(cid:18) (cid:19)
Routine computations give
i k
(g2) = (g ) (g ) =
1 ij 1 ik 1 kj k−i j −k
k≥1 i≤k≤2i(cid:18) (cid:19)(cid:18) (cid:19)
X X
for i≤ j ≤ 4i and (g2) = 0 otherwise. That is, a “row-palindrome” matrix
1 ij
1 2 2 1 ...
1 4 8 10 8 4 1... .
 
...
 