A pro-2 group with full normal Hausdorff spectra

Iker de las Heras; Anitha Thillaisundaram

doi:10.1515/jgth-2021-0017

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A pro-2 group with full normal Hausdorff spectra

Iker de las Heras and Anitha Thillaisundaram

Published/Copyright: March 3, 2022

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From the journal Journal of Group Theory Volume 25 Issue 5

Abstract

We construct a 2-generated pro-2 group with full normal Hausdorff spectrum [ 0 , 1 ] , with respect to each of the four standard filtration series: the 2-power series, the lower 2-series, the Frattini series, and the dimension subgroup series. This answers a question of Klopsch and the second author, for the even prime case; the odd prime case was settled by the first author and Klopsch. Also, our construction gives the first example of a finitely generated pro-2 group with full Hausdorff spectrum with respect to the lower 2-series.

1 Introduction

Let Γ be a countably based infinite profinite group, and consider a filtration series𝒮 of Γ, that is, a descending chain Γ = Γ 0 ≥ Γ 1 ≥ ⋯ of open normal subgroups Γ i ⁢ ⊴ o ⁢ Γ such that ⋂ i Γ i = 1 . These open normal subgroups form a base of neighbourhoods of the identity and induce a translation-invariant metric on Γ given by d S ( x , y ) = inf { | Γ : Γ i | - 1 ∣ x ≡ y ( mod Γ i ) } for x , y ∈ Γ . This yields, for a subset U ⊆ Γ , the Hausdorff dimension hdim Γ S ⁡ ( U ) ∈ [ 0 , 1 ] with respect to the filtration series 𝒮.

Over the past twenty years, there have been interesting applications of Hausdorff dimension to profinite groups, starting with the pioneering work of Abercrombie [1], Barnea and Shalev [2]; see [8] for a good overview. Barnea and Shalev [2] established the following algebraic formula for the Hausdorff dimension of a closed subgroup 𝐻 of Γ as a logarithmic density:

hdim Γ S ⁡ ( H ) = lim ¯ i → ∞ ⁡ log ⁡ | H ⁢ Γ i : Γ i | log ⁡ | Γ : Γ i | ,

where lim ¯ i → ∞ ⁡ a i is the lower limit of a sequence ( a i ) i ∈ N in ℝ.

The Hausdorff spectrum of Γ with respect to 𝒮 is

hspec S ⁡ ( Γ ) = { hdim Γ S ⁡ ( H ) ∣ H ≤ c Γ } ⊆ [ 0 , 1 ] ,

where 𝐻 runs through all closed subgroups of Γ. Shalev [9, § 4.7] was the first to consider the normal Hausdorff spectrum of Γ, with respect to 𝒮, that is,

hspec ⊴ S ⁡ ( Γ ) = { hdim Γ S ⁡ ( H ) ∣ H ⁢ ⊴ c ⁢ Γ } ,

which reflects the spread of Hausdorff dimensions of closed normal subgroups in Γ. Until very recently, little was known about normal Hausdorff spectra of finitely generated pro-𝑝 groups. Indeed, early examples of normal Hausdorff spectra were all finite (see [9, § 4.7]), until Klopsch and the second author [7] constructed the first example of a finitely generated pro-𝑝 group, for every prime 𝑝, with infinite normal Hausdorff spectra with respect to the five standard filtration series: the 𝑝-power series 𝒫, the iterated 𝑝-power series ℐ, the lower 𝑝-series ℒ, the Frattini series ℱ, and the dimension subgroup series 𝒟; we refer the reader to Section 2 for the definitions. The normal Hausdorff spectra of the groups in [7] consist of an interval [ 0 , ξ ] , for ξ ≤ 1 3 , with one or two isolated points. The question was raised in [7] of whether or not a finitely generated pro-𝑝 group, for 𝑝 any prime, could be constructed with full normal Hausdorff spectra [ 0 , 1 ] . This was answered, for every odd prime 𝑝, by the first author and Klopsch [3]. Their constructed group, however, does not produce the desired result for the case p = 2 .

In this paper, we settle the aforementioned question for p = 2 , by considering a modification of the construction in [3]. Our group 𝐺 is a 2-generated extension of an elementary abelian pro-2 group by the pro-2 wreath product

W = C 2 ≀ ^ Z 2 = lim ← k ∈ N ⁡ C 2 ≀ C 2 k .

We follow the strategy in [3] to prove the result.

Theorem 1.1

The pro-2 group 𝐺 satisfies hspec ⊴ S ⁡ ( G ) = hspec S ⁡ ( G ) = [ 0 , 1 ] for S ∈ { M , L , D , P , F } .

Here ℳ denotes a natural filtration series that arises from the construction of 𝐺; see Sections 3 and 4 for details. Note, for a pro-2 group, the iterated 2-power series coincides with the Frattini series.

It was asked in [2, Problem 5] whether or not there is a finitely generated pro-𝑝 group Γ with hspec P ⁡ ( Γ ) = [ 0 , 1 ] . This was answered in the affirmative independently by Levai (see [9, § 4.2]) and Klopsch (see [6]). In the latter case, it was shown that C p ≀ ^ Z p has full Hausdorff spectra with respect to the 𝑝-power series 𝒫 (which equals the iterated 𝑝-power series ℐ), the Frattini series ℱ, and the dimension subgroup series 𝒟, but that it does not have full Hausdorff spectrum with respect to the lower 𝑝-series ℒ. The work of Klopsch and the second author [7] give further examples of finitely generated pro-𝑝 groups with full Hausdorff spectra with respect to 𝒫, ℐ, ℱ and 𝒟, and the work of Garaialde Ocaña, Garrido and Klopsch [5] provide many examples of finitely generated pro-𝑝 groups with full Hausdorff spectra with respect to ℱ and 𝒟.

As was the case for the groups in [3], our group here yields the first example of a finitely generated pro-2 group with full Hausdorff spectrum with respect to ℒ.

Organisation

Section 2 contains preliminary results. In Section 3, we give a presentation of the pro-2 group 𝐺 and describe a series of finite quotients G k , k ∈ N , such that G = lim ← ⁡ G k . Lastly, in Section 4, we compute the normal Hausdorff spectra of 𝐺 with respect to ℳ, ℒ, 𝒟, 𝒫 and ℱ.

Notation

All subgroups of profinite groups are generally taken to be closed subgroups. We use the notation ≤ o and ≤ c to denote open and closed subgroups respectively. Throughout, we use left-normed commutators, for example,

[ x , y , z ] = [ [ x , y ] , z ] .

2 Preliminaries

Let 𝑝 be a prime. For Γ a finitely generated pro-𝑝 group, we define below the four natural filtration series on Γ. The 𝑝-power series of Γ is given by

P : Γ p i = ⟨ x p i ∣ x ∈ Γ ⟩ , i ∈ N 0 ,

where N 0 = N ∪ { 0 } . The lower 𝑝-series (or lower 𝑝-central series) of Γ is given recursively by

L : P 1 ⁢ ( Γ ) = Γ and P i ⁢ ( Γ ) = P i - 1 ⁢ ( Γ ) p ⁢ [ P i - 1 ⁢ ( Γ ) , Γ ] for ⁢ i ∈ N ≥ 2 ,

and the Frattini series of Γ is given recursively by

F : Φ 0 ⁢ ( Γ ) = Γ and Φ i ⁢ ( Γ ) = Φ i - 1 ⁢ ( Γ ) p ⁢ [ Φ i - 1 ⁢ ( Γ ) , Φ i - 1 ⁢ ( Γ ) ] for ⁢ i ∈ N .

The (modular) dimension subgroup series (or Jennings series or Zassenhaus series) of Γ is defined recursively by

D : D 1 ⁢ ( Γ ) = Γ , D i ⁢ ( Γ ) = D ⌈ i / p ⌉ ⁢ ( Γ ) p ⁢ ∏ 1 ≤ j < i [ D j ⁢ ( Γ ) , D i - j ⁢ ( Γ ) ] for ⁢ i ∈ N ≥ 2 .

In addition, we set P 0 ⁢ ( Γ ) = D 0 ⁢ ( Γ ) = Γ .

Often, an additional natural filtration series is considered: the iterated 𝑝-power series of Γ, which is defined by

I : I 0 ⁢ ( Γ ) = Γ and I j ⁢ ( Γ ) = I j - 1 ⁢ ( Γ ) p for ⁢ j ∈ N .

Recall, for a pro-2 group Γ, the iterated 2-power series coincides with the Frattini series.

Next, for convenience, we recall the following standard commutator identities.

Lemma 2.1

Let Γ = ⟨ a , b ⟩ be a group, let 𝑝 be any prime, and let r ∈ N . For u , v ∈ Γ , let K ⁢ ( u , v ) denote the normal closure in Γ of (i) all commutators in { u , v } of weight at least p r that have weight at least 2 in 𝑣, together with (ii) the p r - s + 1 th powers of all commutators in { u , v } of weight less than p s and of weight at least 2 in 𝑣 for 1 ≤ s ≤ r . Then

(2.1) ( a ⁢ b ) p r ≡ K ⁢ ( a , b ) a p r ⁢ b p r ⁢ [ b , a ] ( p r 2 ) ⁢ [ b , a , a ] ( p r 3 ) ⁢ ⋯ [ b , a , … p r - 2 , a ] ( p r p r - 1 ) ⁢ [ b , a , … p r - 1 , a ] ,

(2.2) [ a p r , b ] ≡ K ⁢ ( a , [ a , b ] ) [ a , b ] p r ⁢ [ a , b , a ] ( p r 2 ) ⁢ ⋯ [ a , b , a , … p r - 2 , a ] ( p r p r - 1 ) ⁢ [ a , b , a , … p r - 1 , a ] .

We also recall the following definition from [7]: for a countably based infinite pro-𝑝 group Γ, equipped with a filtration series S : Γ = Γ 0 ≥ Γ 1 ≥ ⋯ , and a closed subgroup H ≤ c Γ , we say that 𝐻 has strong Hausdorff dimension in Γ with respect to 𝒮 if

hdim Γ S ⁡ ( H ) = lim i → ∞ ⁡ log p ⁡ | H ⁢ Γ i : Γ i | log p ⁡ | Γ : Γ i |

is given by a proper limit.

Lastly, we note here a result that will be useful for the computation of normal Hausdorff spectra in what follows.

Proposition 2.2

Proposition 2.2 ([3, Proposition 2.4])

Let Γ be a countably based pro-𝑝 group with an infinite abelian normal subgroup Z ⁢ ⊴ c ⁢ Γ such that Γ / C Γ ⁢ ( Z ) is procyclic. Let S : Γ = S 0 ≥ S 1 ≥ ⋯ be a filtration series of Γ, and consider the induced filtration series S | Z : Z = S 0 ∩ Z ≥ S 1 ∩ Z ≥ ⋯ of 𝑍; for i ∈ N 0 , let p e i be the exponent of Z / ( S i ∩ Z ) . Suppose that, for every i ∈ N 0 , there exist n i ∈ N and K i ≤ c Γ such that

γ n i + 1 ⁢ ( Γ ) ∩ Z ≤ S i ∩ Z ≤ K i and lim ¯ i → ∞ ⁡ e i ⁢ n i log p ⁡ | Z : K i ∩ Z | = 0 .

If 𝑍 has strong Hausdorff dimension ξ = hdim G S ⁡ ( Z ) ∈ [ 0 , 1 ] , then

[ 0 , ξ ] ⊆ hspec ⊴ S ⁡ ( Γ ) .

3 The pro-2 group 𝐺

For k ∈ N , let ⟨ x ˙ k ⟩ ≅ C 2 k and ⟨ y ˙ k ⟩ ≅ C 2 . Define

W k = ⟨ y ˙ k ⟩ ≀ ⟨ x ˙ k ⟩ ≅ B k ⋊ ⟨ x ˙ k ⟩ ,

where B k = ∏ i = 0 2 k - 1 ⟨ y ˙ k x ˙ k i ⟩ ≅ C 2 2 k . The structural results for the finite wreath products W k transfer naturally to the inverse limit W ≅ lim ← k ⁡ W k , i.e. the pro-2 wreath product

W = ⟨ x ˙ , y ˙ ⟩ = B ⋊ ⟨ x ˙ ⟩ ≅ C 2 ⁢ ≀ ^ ⁢ Z 2

with top group ⟨ x ˙ ⟩ ≅ Z 2 and base group B = ⟨ y ˙ x ˙ i ∣ i ∈ Z ⟩ ≅ C 2 ℵ 0 ; see [7, § 2.4] for further results.

Let F 2 = ⟨ a , b ⟩ be the free pro-2 group on two generators, and let k ∈ N . There exists a closed normal subgroup R ⁢ ⊴ ⁢ F 2 , respectively R k ⁢ ⊴ ⁢ F 2 , such that

F 2 / R ≅ W , respectively F 2 / R k ≅ W k ,

with 𝑎 corresponding to x ˙ , respectively x ˙ k , and 𝑏 corresponding to y ˙ , respectively y ˙ k .

Let Y ≥ R be the closed normal subgroup of F 2 such that Y / R is the pre-image of 𝐵 in F 2 / R , and let Y k ≥ R k be the closed normal subgroup of F 2 such that Y k / R k is the pre-image of B k in F 2 / R k .

Consider now

N = [ R , Y ] ⁢ R 2 , respectively N k = [ R k , Y k ] ⁢ R k 2 ⁢ ⟨ a 2 k ⟩ F 2 ,

and define

G = F 2 / N , respectively G k = F 2 / N k .

We denote by 𝐻 and 𝑍 the closed normal subgroups of 𝐺 corresponding to Y / N and R / N , and we denote by H k and Z k the closed normal subgroups of G k corresponding to Y k / N k and R k / N k . We denote the images of a , b in 𝐺, respectively in G k , by x , y , respectively x k , y k , so that G = ⟨ x , y ⟩ and G k = ⟨ x k , y k ⟩ .

Note that the groups G k are finite for all k ∈ N and that they naturally form an inverse system so that lim ← k ⁡ G k = G . Furthermore, we have [ H , Z ] = 1 , respectively [ H k , Z k ] = 1 .

3.1 Properties of G k

Proposition 3.1

For k ∈ N , the logarithmic order of G k is

log 2 ⁡ | G k | = k + 2 k + 1 + ( 2 k 2 ) = k + 2 2 ⁢ k - 1 + 2 k + 1 - 2 k - 1 .

Proof

We proceed as in [7, Lemma 5.1]. Since G k / Z k ≅ W k ≅ C 2 ≀ C 2 k , we have

log 2 ⁡ | G k | = log 2 ⁡ | G k / Z k | + log 2 ⁡ | Z k | = k + 2 k + log 2 ⁡ | Z k | .

By construction, the subgroup Z k is elementary abelian, so we obtain

Z k = ⟨ { ( y k x k i ) 2 ∣ 0 ≤ i ≤ 2 k - 1 } ∪ { [ y k x k i , y k x k j ] ∣ 0 ≤ i < j ≤ 2 k - 1 } ⟩ .

Therefore log 2 ⁡ | G k | ≤ k + 2 k + 1 + ( 2 k 2 ) .

For the converse, we will construct a factor group G ~ k of G k whose logarithmic order is | G ~ k | = k + 2 k + 1 + ( 2 k 2 ) .

We consider the finite 2-group

M = ⟨ y ~ 0 , … , y ~ 2 k - 1 ⟩ = E / [ Φ ⁢ ( E ) , E ] ⁢ Φ ⁢ ( E ) 2 ,

where 𝐸 is the free group on 2 k generators. The images of y ~ 0 , … , y ~ 2 k - 1 generate independently the elementary abelian quotient M / Φ ⁢ ( M ) , and the elements y ~ 0 2 , … , y ~ 2 k - 1 2 together with the commutators [ y ~ i , y ~ j ] for 0 ≤ i < j ≤ 2 k - 1 generate independently the elementary abelian subgroup Φ ⁢ ( M ) . The latter can be verified by considering homomorphisms from 𝑀 onto groups of the form C 2 2 k - 1 × C 4 and C 2 2 k - 2 × Heis ⁢ ( F 2 ) , where Heis ⁢ ( F 2 ) denotes the group of upper unitriangular 3 × 3 matrices over F 2 . Next consider the faithful action of the cyclic group X ≅ ⟨ x ~ ⟩ ≅ C 2 k induced by

y ~ i x ~ = { y ~ i + 1 if ⁢ 0 ≤ i ≤ 2 k - 2 , y ~ 0 if ⁢ i = 2 k - 1 .

We define G ~ k = X ⋉ M and observe that log 2 ⁡ | G ~ k | = k + 2 k + 1 + ( 2 k 2 ) . Furthermore, it is easy to see that G ~ k / Φ ⁢ ( M ) ≅ W k . Thus, there is an epimorphism ϵ : G k → G ~ k with ϵ ⁢ ( x k ) = x ~ and ϵ ⁢ ( y k ) = y ~ 0 , and since | G k | ≤ | G ~ k | , we conclude that G k ≅ G ~ k . ∎

Remark 3.2

The proof of Proposition 3.1 shows that H k 2 = Z k and, consequently, that H 2 = Z . In particular, the exponent of H k , and of 𝐻, is 4.

Our next aim is to compute the lower central series of the groups G k , and therefore of 𝐺. For that purpose, we recall the convenient notation introduced in [3], which will be used frequently in the paper. We will denote

c 1 = y and c i = [ y , x , … i - 1 , x ] for ⁢ i ∈ N ≥ 2 .

Similarly, we set

c i , j = [ c i , y , x , … j - 1 , x ] for every ⁢ j ∈ N .

For the sake of simplicity, for k ∈ N , we will use the same notation for the corresponding elements in the group G k . Thus, we will also write

c 1 = y k , c i = [ y k , x k , … i - 1 , x k ] , and c i , j = [ c i , y k , x k , … j - 1 , x k ]

for every i ∈ N ≥ 2 and j ∈ N , when working in the groups G k , for k ∈ N . It will be clear from the context whether our considerations apply to 𝐺 or G k .

Lemma 3.3

In the group G k , for k ∈ N ,

for i ≥ 2 k - 1 + 1 , we have c i 2 ∈ γ i + 1 ⁢ ( G k ) ;
for i ≥ 2 k + 1 , we have c i 2 = 1 ;
for i ≥ 2 k + 2 k - 1 + 1 , we have c i ∈ γ i + 1 ⁢ ( G k ) .

Proof

(i) Observe that ( 2 k 2 k - 1 ) ≡ 4 2 and that ( 2 k j ) ≡ 4 0 for any 1 ≤ j ≤ 2 k - 1 with j ≠ 2 k - 1 . As H k has exponent 4 and [ H k , H k ] ≤ Z k exponent 2, it follows from (2.2) that

(3.1) 1 = [ y k , x k 2 k ] ≡ [ y k , x k , … 2 k - 1 , x k ] 2 ⁢ [ y k , x k , … 2 k , x k ] ( mod γ 2 k + 2 ⁢ ( G k ) ) .

Therefore c 2 k - 1 + 1 2 ≡ c 2 k + 1 ( mod γ 2 k + 2 ⁢ ( G k ) ) .

For i ≥ 2 k - 1 + 2 , note that

c i 2 = [ y k , x k , … i - 1 , x k ] 2 ≡ [ c 2 k - 1 + 1 2 , x k , … i - 2 k - 1 - 1 , x k ] ( mod γ i + 1 ⁢ ( G k ) ) ≡ [ c 2 k + 1 , x k , … i - 2 k - 1 - 1 , x k ] ( mod γ i + 1 ⁢ ( G k ) ) .

Hence, the result follows.

(ii) This follows immediately from the fact that c i ∈ Z k for i ≥ 2 k + 1 ; compare [7, Proposition 2.6 (1)].

(iii) It suffices to prove the result for i = 2 k + 2 k - 1 + 1 . From (3.1), we obtain

c i = [ c 2 k + 1 , x k , … 2 k - 1 , x k ] ≡ [ c 2 k - 1 + 1 2 , x k , … 2 k - 1 , x k ] ( mod γ i + 1 ⁢ ( G k ) ) .

[ c 2 k - 1 + 1 2 , x k , … 2 k - 1 , x k ] ≡ c 2 k + 1 2 ( mod γ i + 1 ⁢ ( G k ) )

we have c i ≡ c 2 k + 1 2 ( mod γ i + 1 ⁢ ( G k ) ) , and by (ii), it follows that c i ∈ γ i + 1 ⁢ ( G k ) , as required. ∎

Let us write z i , j = [ c i , c j ] for every i , j ∈ N . From [7, Proposition 2.6(1)], we have H = ⟨ c n ∣ n ∈ N ⟩ , and from Remark 3.2, we deduce that

Z = ⟨ c n 2 , z i , j ∣ ⁢ n , i , j ∈ N ⁢ with ⁢ j ⁢ < i ⟩ .

We recall the following result from [3, Lemma 4.3], which although this was written in the setting of the pro-𝑝 group constructed in [3], for 𝑝 an odd prime, the same proof holds for our group 𝐺.

Lemma 3.4

In the group 𝐺, for i , j ∈ N , we have

[ z i , j , x , … k , x ] = ∏ s = 0 k ( ∏ n = 0 s z i + k - n , j + k - s + n ( k s ) ⁢ ( s n ) ) for every ⁢ k ∈ N 0 .

Corollary 3.5

In the group 𝐺, for i , j ∈ N , we have

[ z i , j , x 2 k ] = z i + 2 k , j ⁢ z i , j + 2 k ⁢ z i + 2 k , j + 2 k for every ⁢ k ∈ N 0 .

Proof

As was reasoned in [3], this follows directly from (2.2) and Lemma 3.4. ∎

Lemma 3.6

Let k ∈ N . In the group G k , for m ∈ N even, we have

c m , 2 k ∈ γ 2 k + m + 1 ⁢ ( G k ) .

Proof

Note that

c m , 2 k = [ z m , 1 , x k , … 2 k - 1 , x k ] ,

and since z i , j ∈ γ i + j ⁢ ( G k ) for every i , j ∈ N , we have by Lemma 3.4 that

[ z m , 1 , x k , … 2 k - 1 , x k ] ≡ ∏ n = 0 2 k - 1 z m + 2 k - 1 - n , 1 + n ( 2 k - 1 n ) ( mod γ 2 k + m + 1 ⁢ ( G k ) ) .

In addition, the exponent of Z k is 2 by construction, so since all the binomial numbers ( 2 k - 1 n ) are odd, we get

[ z m , 1 , x k , … 2 k - 1 , x k ] ≡ ∏ n = 0 2 k - 1 z m + 2 k - 1 - n , 1 + n ( mod γ 2 k + m + 1 ⁢ ( G k ) ) .

Recall that c i ∈ Z k for every i ≥ 2 k + 1 by [7, Proposition 2.6 (1)], so

z m + 2 k - 1 - n , 1 + n = 1 for all ⁢ n ≤ m - 2 .

Thus,

[ z m , 1 , x k , … 2 k - 1 , x k ] ≡ ∏ n = m - 1 2 k - 1 z m + 2 k - 1 - n , 1 + n ( mod γ 2 k + m + 1 ⁢ ( G k ) ) ≡ ∏ n = 0 2 k - m z 2 k - n , m + n ( mod γ 2 k + m + 1 ⁢ ( G k ) ) .

As z i , j = z j , i for all i , j ∈ N and since 𝑚 is even, we finally obtain

[ z m , 1 , x k , … 2 k - 1 , x k ] ≡ 1 ( mod γ 2 k + m + 1 ⁢ ( G k ) ) ,

as required. ∎

Proposition 3.7

For k ∈ N , the nilpotency class of G k is 2 k + 1 - 1 , and the lower central series of G k satisfies

γ 1 ⁢ ( G k ) = G k = ⟨ x k , y k ⟩ ⁢ γ 2 ⁢ ( G k ) with γ 1 ⁢ ( G k ) / γ 2 ⁢ ( G k ) ≅ C 2 k × C 4 .
If 2 ≤ i ≤ 2 k , then
γ i ⁢ ( G k ) = { ⟨ c i , c 2 , i - 2 , c 4 , i - 4 , … , c i - 2 , 2 ⟩ ⁢ γ i + 1 ⁢ ( G k ) if ⁢ i ≡ 2 0 , ⟨ c i , c 2 , i - 2 , c 4 , i - 4 , … , c i - 1 , 1 ⟩ ⁢ γ i + 1 ⁢ ( G k ) if ⁢ i ≡ 2 1 ,
with
(3.2) γ i ⁢ ( G k ) / γ i + 1 ⁢ ( G k ) ≅ { C 4 × C 2 × ⋯ ( i - 2 ) / 2 × C 2 if ⁢ 2 ≤ i ≤ 2 k - 1 ⁢ and ⁢ i ≡ 2 0 , C 4 × C 2 × ⋯ ( i - 1 ) / 2 × C 2 if ⁢ 2 ≤ i ≤ 2 k - 1 ⁢ and ⁢ i ≡ 2 1 , C 2 × ⋯ i / 2 × C 2 if ⁢ 2 k - 1 + 1 ≤ i ≤ 2 k ⁢ and ⁢ i ≡ 2 0 , C 2 × ⋯ ( i + 1 ) / 2 × C 2 if ⁢ 2 k - 1 + 1 ≤ i ≤ 2 k ⁢ and ⁢ i ≡ 2 1 .
If 2 k + 1 ≤ i ≤ 2 k + 2 k - 1 , then
γ i ⁢ ( G k ) = { ⟨ c i , c i - 2 k + 2 , 2 k - 2 , c i - 2 k + 4 , 2 k - 4 , … , c 2 k - 2 , i - 2 k + 2 , c 2 k , i - 2 k ⟩ γ i + 1 ( G k ) if ⁢ i ≡ 2 0 , ⟨ c i , c i - 2 k + 1 , 2 k - 1 , c i - 2 k + 3 , 2 k - 3 , … , c 2 k - 2 , i - 2 k + 2 , c 2 k , i - 2 k ⟩ γ i + 1 ( G k ) if ⁢ i ≡ 2 1 ,
with
(3.3) γ i ⁢ ( G k ) / γ i + 1 ⁢ ( G k ) ≅ { C 2 × ⋯ ( 2 k + 1 - i + 2 ) / 2 × C 2 if ⁢ i ≡ 2 0 , C 2 × ⋯ ( 2 k + 1 - i + 3 ) / 2 × C 2 if ⁢ i ≡ 2 1 .
If 2 k + 2 k - 1 + 1 ≤ i ≤ 2 k + 1 , then
γ i ⁢ ( G k ) = { ⟨ c i - 2 k + 2 , 2 k - 2 , c i - 2 k + 4 , 2 k - 4 , … , c 2 k - 2 , i - 2 k + 2 , c 2 k , i - 2 k ⟩ γ i + 1 ( G k ) if ⁢ i ≡ 2 0 , ⟨ c i - 2 k + 1 , 2 k - 1 , c i - 2 k + 3 , 2 k - 3 , … , c 2 k - 2 , i - 2 k + 2 , c 2 k , i - 2 k ⟩ γ i + 1 ( G k ) if ⁢ i ≡ 2 1 ,
with
(3.4) γ i ⁢ ( G k ) / γ i + 1 ⁢ ( G k ) ≅ { C 2 × ⋯ ( 2 k + 1 - i ) / 2 × C 2 if ⁢ i ≡ 2 0 , C 2 × ⋯ ( 2 k + 1 - i + 1 ) / 2 × C 2 if ⁢ i ≡ 2 1 .

Proof

The first assertion is obvious. To prove the other ones, we start by showing that

γ i ⁢ ( G k ) = { ⟨ c i , c 2 , i - 2 , c 4 , i - 4 , … , c i - 2 , 2 ⟩ ⁢ γ i + 1 ⁢ ( G k ) if ⁢ i ≡ 2 0 , ⟨ c i , c 2 , i - 2 , c 4 , i - 4 , … , c i - 1 , 1 ⟩ ⁢ γ i + 1 ⁢ ( G k ) if ⁢ i ≡ 2 1 .

We proceed by induction on 𝑖. For i = 2 , we have γ 2 ⁢ ( G k ) = ⟨ c 2 ⟩ ⁢ γ 3 ⁢ ( G k ) , and for i = 3 , we have γ 3 ⁢ ( G k ) = ⟨ c 3 , c 2 , 1 ⟩ ⁢ γ 4 ⁢ ( G k ) . Assume then that i > 3 , that 𝑖 is even and that

γ i - 2 ⁢ ( G k ) = ⟨ c i - 2 , c 2 , i - 4 , c 4 , i - 6 , … , c i - 4 , 2 ⟩ ⁢ γ i - 1 ⁢ ( G k ) , γ i - 1 ⁢ ( G k ) = ⟨ c i - 1 , c 2 , i - 3 , c 4 , i - 5 , … , c i - 2 , 1 ⟩ ⁢ γ i ⁢ ( G k ) .

Note that c n , m ∈ [ H k , H k ] ≤ Z k for all n , m ∈ N , so [ c n , m , y k ] = 1 . Thus,

γ i ⁢ ( G k ) = ⟨ c i , c i - 1 , 1 , c 2 , i - 2 , c 4 , i - 4 , … , c i - 2 , 2 ⟩ ⁢ γ i + 1 ⁢ ( G k ) .

For convenience, we write

M = ⟨ c i , c 2 , i - 2 , c 4 , i - 4 , … , c i - 2 , 2 ⟩ ⁢ γ i + 1 ⁢ ( G k ) ,

and we have to check that c i - 1 , 1 ∈ M . Notice that c i - 1 , 1 = [ c i - 2 , x k , y k ] , and the Hall–Witt identity yields

[ c i - 2 , x k , y k ] ⁢ [ x k , y k , c i - 2 ] ⁢ [ y k , c i - 2 , x k ] ≡ 1 ( mod M ) .

Note also that [ y k , c i - 2 , x k ] ≡ c i - 2 , 2 - 1 ≡ 1 ( mod M ) , so

c i - 1 , 1 ≡ [ c i - 2 , c 2 ] - 1 ( mod M ) .

Let us prove that [ c m , c n ] ≡ 1 ( mod M ) for all n ≥ 2 with n ≤ m and n + m = i . We argue by induction on m - n . If m - n = 0 , then n = m and [ c m , c n ] = 1 . Now suppose that m - n > 0 , which, since 𝑖 is even, implies that m - n ≥ 2 . Observe that [ c m , c n ] = [ c m - 1 , x k , c n ] , so again by the Hall–Witt identity,

[ c m - 1 , x k , c n ] ⁢ [ x k , c n , c m - 1 ] ⁢ [ c n , c m - 1 , x k ] ≡ 1 ( mod M ) .

Since 0 ≤ ( m - 1 ) - ( n + 1 ) = m - n - 2 < m - n , we have

[ x k , c n , c m - 1 ] ≡ [ c m - 1 , c n + 1 ] ≡ 1 ( mod M )

by induction. Hence, [ c m , c n ] ≡ [ c n , c m - 1 , x k ] - 1 ( mod M ) . As

[ c n , c m - 1 ] ∈ γ i - 1 ⁢ ( G k ) ∩ Z k ,

it follows that

[ c n , c m - 1 ] ≡ c i - 1 n 0 ⁢ c 2 , i - 3 n 2 ⁢ c 4 , i - 5 n 4 ⁢ ⋯ ⁢ c i - 2 , 1 n i - 2 ( mod γ i ⁢ ( G k ) )

for some n 0 , n 2 , … , n i - 2 ∈ Z . Hence,

[ c m , c n ] ≡ [ c n , c m - 1 , x k ] ≡ c i n 0 ⁢ c 2 , i - 2 n 2 ⁢ c 4 , i - 4 n 4 ⁢ ⋯ ⁢ c i - 2 , 2 n i - 2 ≡ 1 ( mod M ) ,

as we wanted to prove. This, in particular, implies that c i - 1 , 1 ∈ M , as claimed. Hence,

γ i ⁢ ( G k ) = ⟨ c i , c 2 , i - 2 , c 4 , i - 4 , … , c i - 2 , 2 ⟩ ⁢ γ i + 1 ⁢ ( G k ) ,

and again, since [ c n , m , y k ] = 1 , we have

γ i + 1 ⁢ ( G k ) = ⟨ c i + 1 , c 2 , i - 1 , c 4 , i - 3 , … , c i - 2 , 3 , c i , 1 ⟩ ⁢ γ i + 2 ⁢ ( G k ) .

Now, for 2 k + 1 ≤ i ≤ 2 k + 1 , we add x k and y k to the commutators that inductively generate γ i - 1 ⁢ ( G k ) modulo γ i ⁢ ( G k ) . Removing the unnecessary generators according to Lemmas 3.3 and 3.6, one can deduce that the generators of γ i ⁢ ( G k ) modulo γ i + 1 ⁢ ( G k ) are precisely the stated ones.

Finally, it suffices to check that the isomorphisms (3.2), (3.3) and (3.4) are satisfied. The generators we have found for the terms of the lower central series, together with Lemma 3.3, show that any quotient of two consecutive terms of the lower central series is actually isomorphic to a quotient of the corresponding abelian group in the statement. In particular, we obtain upper bounds for the logarithmic orders of the quotients of two consecutive terms of the lower central series. In fact, all these upper bounds sum to the logarithmic order of G k . Indeed, we have

∑ i = 1 2 k + 1 log 2 ⁡ | γ i ⁢ ( G k ) : γ i + 1 ⁢ ( G k ) | = ( k + 2 ) + ( 2 k - 1 - 1 + ∑ i = 2 2 k ⌈ i / 2 ⌉ ) + ( 2 k - 1 + ∑ i = 2 k + 1 2 k + 1 ⌈ ( 2 k + 1 - i ) / 2 ⌉ ) = k + 2 k + 1 + ( 2 k 2 ) ,

which is precisely, by Proposition 3.1, the logarithmic order of G k . Hence, isomorphisms (3.2), (3.3) and (3.4) are satisfied, and the proof is finished. ∎

Remark 3.8

From Proposition 3.7, we can deduce that the logarithmic order of Z / ( γ i ⁢ ( G ) ∩ Z ) is

2 ⁢ ( 1 + 2 + ⋯ + i - 1 2 ) = 2 ⁢ ( ( i + 1 ) / 2 2 ) if ⁢ i ⁢ is odd or 2 ⁢ ( 1 + 2 + ⋯ + i - 2 2 ) + i 2 = 2 ⁢ ( i / 2 2 ) + i 2 if ⁢ i ⁢ is even .

We include a result which highlights a further difference between the group 𝐺 and the group constructed in [3].

Lemma 3.9

For k ∈ N , the group G k has exponent 2 k + 2 .

Proof

First we show that x k ⁢ y k has order 2 k + 2 . Consider ( x k ⁢ y k ) 2 k , and observe that K ⁢ ( x k , y k ) ≤ [ H k , H k ] in (2.1) has exponent 2. Therefore,

( x k ⁢ y k ) 2 k ≡ K ⁢ ( x k , y k ) [ y k , x k , … 2 k - 1 - 1 , x k ] 2 ⁢ [ y k , x k , … 2 k - 1 , x k ]

yields ( x k ⁢ y k ) 2 k + 1 = c 2 k 2 , which is non-trivial by the proof of Proposition 3.7. Hence, the result follows.

For a general element g = x k i ⁢ h for some 0 ≤ i ≤ 2 k - 1 and h ∈ H k , it similarly follows that g 2 k + 2 = 1 . ∎

The following two results will be needed for computing the normal Hausdorff spectra of 𝐺 with respect to the series ℒ and 𝒟 in the next section.

Proposition 3.10

For k ∈ N , the length of the lower 2-series of G k is 2 k + 1 - 1 and

P 1 ⁢ ( G k ) = G k , P 2 ⁢ ( G k ) = ⟨ x k 2 , y k 2 ⟩ ⁢ γ 2 ⁢ ( G k ) , P i ⁢ ( G k ) = { ⟨ x k 2 i - 1 , c i - 1 2 ⟩ ⁢ γ i ⁢ ( G k ) for ⁢ 3 ≤ i ≤ 2 k - 1 + 1 , ⟨ x k 2 i - 1 ⟩ ⁢ γ i ⁢ ( G k ) for ⁢ 2 k - 1 + 2 ≤ i ≤ 2 k + 1 .

Proof

If i = 1 or 2, the results are obvious, so consider i = 3 . As

[ ⟨ x k 2 , y k 2 ⟩ , G k ] ≤ γ 2 ⁢ ( G k ) 2 ⁢ γ 3 ⁢ ( G k ) ,

it suffices to show that ⟨ x k 2 , y k 2 ⟩ 2 ≤ ⟨ x k 4 ⟩ ⁢ γ 3 ⁢ ( G k ) and γ 2 ⁢ ( G k ) 2 ≤ ⟨ c 2 2 ⟩ ⁢ γ 3 ⁢ ( G k ) . Note that [ y k 2 , x k 2 ] ≡ [ y k , x k ] 4 = 1 ( mod γ 3 ⁢ ( G k ) ) , and since x k 4 , y k 4 ∈ ⟨ x k 4 ⟩ ⁢ γ 3 ⁢ ( G k ) , the first inclusion holds. For the second statement, it follows from Proposition 3.7, as [ y k , x k ] is the only generator of γ 2 ⁢ ( G k ) modulo γ 3 ⁢ ( G k ) .

Now let 4 ≤ i ≤ 2 k - 1 + 1 , and assume by induction that

P i - 1 ⁢ ( G k ) = ⟨ x k 2 i - 2 , c i - 2 2 ⟩ ⁢ γ i - 1 ⁢ ( G k ) .

On the one hand,

[ P i - 1 ⁢ ( G k ) , G k ] = [ ⟨ x k 2 i - 2 , c i - 2 2 ⟩ ⁢ γ i - 1 ⁢ ( G k ) , G k ] = [ ⟨ x k 2 i - 2 , c i - 2 2 ⟩ , G k ] ⁢ γ i ⁢ ( G k ) ,

and Proposition 3.7 and (2.2) yield

[ ⟨ x k 2 i - 2 , c i - 2 2 ⟩ , G k ] ⁢ γ i ⁢ ( G k ) = [ ⟨ c i - 2 2 ⟩ , G k ] ⁢ γ i ⁢ ( G k ) .

Then, by arguments similar to above, one deduces that

[ c i - 2 2 , G k ] ⁢ γ i ⁢ ( G k ) = ⟨ c i - 1 2 ⟩ ⁢ γ i ⁢ ( G k ) .

On the other hand,

P i - 1 ⁢ ( G k ) 2 ≡ ⟨ x k 2 i - 2 ⟩ 2 ⁢ γ i - 1 ⁢ ( G k ) 2 ≡ ⟨ x k 2 i - 1 , c i - 1 2 ⟩ ( mod γ i ⁢ ( G k ) ) ,

so we conclude that

P i ⁢ ( G k ) = ⟨ x k 2 i - 1 , c i - 1 2 ⟩ ⁢ γ i ⁢ ( G k ) ,

as asserted. The case 2 k - 1 + 2 ≤ i ≤ 2 k + 1 follows similarly, using Lemma 3.3 (ii). ∎

Proposition 3.11

For k ∈ N , the length of the dimension subgroup series of G k is 2 k + 1 and

D i ⁢ ( G k ) = ⟨ x k 2 l ⁢ ( i ) ⟩ ⁢ γ ⌈ i / 2 ⌉ ⁢ ( G k ) 2 ⁢ γ i ⁢ ( G k ) for ⁢ 1 ≤ i ≤ 2 k + 1 ,

where l ⁢ ( i ) = ⌈ log 2 ⁡ i ⌉ .

Proof

By [4, Theorem 11.2], we have

D i ⁢ ( G k ) = ∏ n ⋅ 2 m ≥ i γ n ⁢ ( G k ) 2 m

for every i ∈ N , and since exp ⁡ ( γ 2 ⁢ ( G k ) ) = 4 , we obtain

D i ⁢ ( G k ) = G k 2 l ⁢ ( i ) ⁢ γ ⌈ i / 2 ⌉ ⁢ ( G k ) 2 ⁢ γ i ⁢ ( G k ) .

The result is clear for i = 1 , 2 , so we assume i ≥ 3 . By (2.1), for every a , b ∈ G k , it follows that

( a ⁢ b ) 2 l ⁢ ( i ) = a 2 l ⁢ ( i ) ⁢ b 2 l ⁢ ( i ) ⁢ [ b , a , … 2 l ⁢ ( i ) - 1 - 1 , a ] ( 2 l ⁢ ( i ) 2 l ⁢ ( i ) - 1 ) ⁢ c

with c ∈ γ 2 l ⁢ ( i ) ⁢ ( G k ) . Since

[ b , a , … 2 l ⁢ ( i ) - 1 - 1 , a ] ( 2 l ⁢ ( i ) 2 l ⁢ ( i ) - 1 ) ∈ γ ⌈ i / 2 ⌉ ⁢ ( G k ) 2

and γ 2 l ⁢ ( i ) ⁢ ( G k ) ≤ γ i ⁢ ( G k ) , we get

D i ⁢ ( G k ) = ⟨ x k 2 l ⁢ ( i ) ⟩ ⁢ γ ⌈ i / 2 ⌉ ⁢ ( G k ) 2 ⁢ γ i ⁢ ( G k ) ,

as required. ∎

4 The normal Hausdorff spectra of 𝐺

In this section, we compute the normal Hausdorff spectra of 𝐺 with respect to the filtration series ℳ, ℒ, 𝒟, 𝒫 and ℱ. Here M : M 0 ≥ M 1 ≥ ⋯ stands for the natural filtration series of 𝐺 where each M i is the subgroup of 𝐺 corresponding to N i / N , where here N 0 = F 2 .

As an easy illustration of our methods, we start computing the normal Hausdorff spectrum of 𝐺 with respect to ℳ.

Theorem 4.1

The pro-2 group 𝐺 satisfies hspec ⊴ M ⁡ ( G ) = [ 0 , 1 ] , and 𝑍 has strong Hausdorff dimension 1 in 𝐺 with respect to ℳ.

Proof

By Proposition 3.7, we know that γ 2 k + 1 ⁢ ( G ) ≤ M k for all k ∈ N . On the other hand, by the proof of Proposition 3.1, we have

log 2 ⁡ | Z : M k ∩ Z | = log 2 ⁡ | Z ⁢ M k : M k | = 2 k + ( 2 k 2 ) .

In particular,

lim ¯ k → ∞ ⁡ 2 k + 1 log 2 ⁡ | Z : M k ∩ Z | = 0 .

Moreover,

hdim G M ⁡ ( Z ) = lim ¯ k → ∞ ⁡ log 2 ⁡ | Z ⁢ M k : M k | log 2 ⁡ | G : M k | = lim ¯ k → ∞ ⁡ 2 k + ( 2 k 2 ) k + 2 k + 1 + ( 2 k 2 ) = 1 .

Therefore, the second statement follows, and by Proposition 2.2, we conclude that hspec ⊴ M ⁡ ( G ) = [ 0 , 1 ] . ∎

Theorem 4.2

For S ∈ { L , D } , the pro-2 group 𝐺 satisfies hspec ⊴ S ⁡ ( G ) = [ 0 , 1 ] , and 𝑍 has strong Hausdorff dimension 1 in 𝐺 with respect to 𝒮.

Proof

By Remark 3.8, we have

lim ¯ k → ∞ ⁡ k log 2 ⁡ | Z : P k ⁢ ( G ) ∩ Z | = 0 and lim ¯ k → ∞ ⁡ k log 2 ⁡ | Z : D k ⁢ ( G ) ∩ Z | = 0 ,

and they are furthermore given by proper limits. Then it suffices by Proposition 2.2 to show that 𝑍 has strong Hausdorff dimension 1 with respect to 𝒮. Write G = S 0 ≥ S 1 ≥ S 2 ≥ ⋯ for the subgroups of the filtration series 𝒮. Observe that, by [7, Proposition 2.6], we have log 2 ⁡ | G : S k ⁢ Z | = 2 ⁢ k , and so

lim k → ∞ ⁡ log 2 ⁡ | G : S k ⁢ Z | log 2 ⁡ | Z : S k ∩ Z | = 0

is given by a proper limit. Thus,

hdim G S ⁡ ( Z ) = lim ¯ k → ∞ ⁡ ( log 2 ⁡ | G : S k | log 2 ⁡ | S k ⁢ Z : S k | ) - 1

= lim ¯ k → ∞ ⁡ ( log 2 ⁡ | G : S k ⁢ Z | + log 2 ⁡ | S k ⁢ Z : S k | log 2 ⁡ | S k ⁢ Z : S k | ) - 1

= lim ¯ k → ∞ ⁡ ( log 2 ⁡ | G : S k ⁢ Z | log 2 ⁡ | Z : S k ∩ Z | + 1 ) - 1 = 1 ,

and 𝑍 has strong Hausdorff dimension 1, as desired. Thus, the proof is complete. ∎

For all n ∈ N , define Γ n = ⟨ x 2 n ⟩ ⁢ Q n - 1 ⁢ γ 2 n ⁢ ( G ) ⁢ ⊴ ⁢ G , where

Q n - 1 = ⟨ c i 2 ∣ i ≥ 2 n - 1 ⟩ .

Lemma 4.3

For each n ∈ N , we have Γ n 2 ≤ Γ n + 1 .

Proof

We only have to check that Γ n ′ ≤ Γ n + 1 . Clearly, [ Q n - 1 , Q n - 1 ⁢ γ 2 n ⁢ ( G ) ] = 1 and [ γ 2 n ⁢ ( G ) , γ 2 n ⁢ ( G ) ] ≤ γ 2 n + 1 ⁢ ( G ) ≤ Γ n + 1 , so it suffices to prove that

[ ⟨ x 2 n ⟩ , Q n - 1 ⁢ γ 2 n ⁢ ( G ) ] ≤ Γ n + 1 .

On the one hand, (2.2) yields

[ γ 2 n ⁢ ( G ) , x 2 n ] ≤ γ 2 n + 2 n - 1 ⁢ ( G ) 2 ⁢ γ 2 n + 1 ⁢ ( G ) ≤ Γ n + 1 .

On the other hand, for 2 n - 1 ≤ i ≤ 2 n - 1 , again by (2.2), we have

[ c i , x 2 n ] ∈ γ 2 n ⁢ ( G ) 2 ⁢ γ 2 n + 2 n - 1 ⁢ ( G ) ,

so [ c i 2 , x 2 n ] = [ c i , x 2 n ] 2 ⁢ [ c i , x 2 n , c i ] ∈ Γ n + 1 , as required. ∎

Theorem 4.4

The pro-2 group 𝐺 satisfies hspec ⊴ P ⁡ ( G ) = [ 0 , 1 ] , and 𝑍 has strong Hausdorff dimension 1 in 𝐺 with respect to 𝒫.

Proof

An arbitrary element of 𝐺 can be written as x i ⁢ h with h ∈ H and i ∈ Z 2 , and by (2.1), it follows that ( x i ⁢ h ) 2 k ∈ Γ k for k ∈ N . Then G 2 k ≤ Γ k and, in particular, G 2 k ∩ Z ≤ Γ k ∩ Z . It is easy to see that

Γ k ∩ Z = ( γ 2 k - 1 ⁢ ( G ) 2 ⁢ γ 2 k ⁢ ( G ) ) ∩ Z = γ 2 k - 1 ⁢ ( G ) 2 ⁢ ( γ 2 k ⁢ ( G ) ∩ Z ) ,

and since [ γ 2 k - 1 ⁢ ( G ) , γ 2 k - 1 ⁢ ( G ) ] ≤ γ 2 k ⁢ ( G ) ∩ Z , it follows that

γ 2 k - 1 ⁢ ( G ) 2 ≤ ⟨ c 2 k - 1 2 , c 2 k - 1 + 1 2 , … , c 2 k - 1 2 ⟩ ⁢ ( γ 2 k ⁢ ( G ) ∩ Z ) .

Thus, by Remark 3.8, we have

log 2 ⁡ | Z : Γ k ∩ Z | = log 2 ⁡ | Z : γ 2 k - 1 ⁢ ( G ) 2 ⁢ ( γ 2 k ⁢ ( G ) ∩ Z ) | = 2 ⁢ ( 2 k - 1 2 ) .

On the other hand, from the construction of 𝐺 and G k , it can be deduced easily that G / G 2 k ≅ G k / G k 2 k . Indeed, N k = N ⁢ ⟨ a 2 k ⟩ F 2 and ⟨ a 2 k ⟩ F 2 ≤ F 2 2 k . Hence, by Proposition 3.7, we get γ 2 k + 1 ⁢ ( G ) ≤ G 2 k .

Now,

lim k → ∞ ⁡ 2 k + 1 log 2 ⁡ | Z : Γ k ∩ Z | = 0 ,

so again, by Proposition 2.2, it suffices to check that 𝑍 has strong Hausdorff dimension 1 with respect to 𝒫. Note that

log 2 ⁡ | G : G 2 k ⁢ Z | = log 2 ⁡ | G k : G k 2 k ⁢ Z k | ≤ log 2 ⁡ | W k | = k + 2 k ,

and so

is given by a proper limit. Thus, the result follows as in the proof of Theorem 4.2. ∎

Theorem 4.5

The pro-2 group 𝐺 satisfies hspec ⊴ F ⁡ ( G ) = [ 0 , 1 ] , and 𝑍 has strong Hausdorff dimension 1 in 𝐺 with respect to ℱ.

Proof

We claim that

T k ⁢ ( γ 2 k + 2 k - 1 - 1 ⁢ ( G ) ∩ Z ) ≤ Φ k ⁢ ( G ) ≤ Γ k ,

where T k = ⟨ x 2 k , c i 2 , c j ∣ i ≥ 2 k - 1 , j ≥ 2 k ⟩ for all k ∈ N . We will proceed by induction on 𝑘. If k = 1 , the result is clear, so assume k ≥ 2 . On the one hand, it follows from Lemma 4.3 that

Φ k ⁢ ( G ) = Φ k - 1 ⁢ ( G ) 2 ≤ Γ k - 1 2 ≤ Γ k .

Hence, we only need to check that

T k ⁢ ( γ 2 k + 2 k - 1 - 1 ⁢ ( G ) ∩ Z ) ≤ Δ ,

where Δ = Φ ⁢ ( T k - 1 ⁢ ( γ 2 k - 1 + 2 k - 2 - 1 ⁢ ( G ) ∩ Z ) ) . Of course, we have x 2 k , c i 2 ∈ Δ for all i ≥ 2 k - 1 . We also have T k - 1 ′ ≤ Δ , so ⟨ z i , j ⁢ ∣ i > ⁢ j ≥ 2 k - 1 ⟩ ≤ Δ . Let us see that z i , j ∈ Δ whenever i > j , i + j ≥ 2 k + 2 k - 1 - 1 and j ≤ 2 k - 1 - 1 . Consider the element z i - 2 k - 1 , j , and observe that

z i - 2 k - 1 , j ∈ γ 2 k - 1 ⁢ ( G ) ∩ Z as ⁢ i - 2 k - 1 + j ≥ 2 k - 1 .

Therefore, [ z i - 2 k - 1 , j , x 2 k - 1 ] ∈ Δ . By Corollary 3.5, it follows then that

z i , j ⁢ z i - 2 k - 1 , j + 2 k - 1 ⁢ z i , j + 2 k - 1 ∈ Δ .

Now we have i > j + 2 k - 1 and j + 2 k - 1 ≥ 2 k - 1 , so z i , j + 2 k - 1 ∈ Δ . Next, if i - 2 k - 1 > j + 2 k - 1 , then z i - 2 k - 1 , j + 2 k - 1 ∈ Δ , and if i - 2 k - 1 ≤ j + 2 k - 1 , then as i - 2 k - 1 ≥ 2 k - 1 , we have

z i - 2 k - 1 , j + 2 k - 1 = z j + 2 k - 1 , i - 2 k - 1 - 1 ∈ Δ .

Therefore z i , j ∈ Δ and γ 2 k + 2 k - 1 - 1 ⁢ ( G ) ∩ Z ≤ Δ .

Finally, for j ≥ 2 k - 1 , observe that

[ c j , x 2 k - 1 ] ≡ c j + 2 k - 2 2 ⁢ c j + 2 k - 1 ( mod γ 2 ⁢ j + 2 k - 1 ⁢ ( G ) ∩ Z ) ,

and since γ 2 ⁢ j + 2 k - 1 ⁢ ( G ) ∩ Z ≤ Δ and c i 2 ∈ Δ for all i ≥ 2 k - 1 , we have c j ∈ Δ for all j ≥ 2 k . We conclude that

T k ⁢ ( γ 2 k + 2 k - 1 - 1 ⁢ ( G ) ∩ Z ) ≤ Δ ≤ Φ k ⁢ ( G ) ,

as claimed. In particular, we get γ 2 k + 2 k - 1 - 1 ⁢ ( G ) ∩ Z ≤ Φ k ⁢ ( G ) ∩ Z ≤ Γ k ∩ Z . Now, from Remark 3.8, we deduce that

(4.1) lim ¯ k → ∞ ⁡ 2 k + 2 k - 1 - 1 log 2 ⁡ | Z : Γ k ∩ Z | = lim k → ∞ ⁡ 2 k + 2 k - 1 - 1 log 2 ⁡ | Z : Γ k ∩ Z | = 0 .

Hence, by Proposition 2.2, it only remains to show that hdim G F ⁡ ( Z ) = 1 and that it is given by a proper limit. This follows easily since, from [7, Proposition 2.6 (3)] and (4.1), we deduce that

lim k → ∞ ⁡ log 2 ⁡ | G : Φ k ⁢ ( G ) ⁢ Z | log 2 ⁡ | Z : Φ k ⁢ ( G ) ∩ Z | = 0 ,

and so, as done in the proof of Theorem 4.2, we obtain hdim G F ⁡ ( Z ) = 1 . It further follows that 𝑍 has strong Hausdorff dimension with respect to ℱ. ∎

Funding source: Ministerio de Economía, Industria y Competitividad, Gobierno de España

Award Identifier / Grant number: MTM2017-86802-P

Funding source: Eusko Jaurlaritza

Award Identifier / Grant number: IT974-16

Funding source: Engineering and Physical Sciences Research Council

Award Identifier / Grant number: EP/T005068/1

Funding statement: The first author is supported by the Spanish Government, grant MTM2017-86802-P, partly with FEDER funds, and by the Basque Government, grant IT974-16. He is also supported by a postdoctoral grant of the Basque Government. The second author acknowledges the support from EPSRC, grant EP/T005068/1.

Acknowledgements

We thank the Heinrich-Heine-Universität Düsseldorf, where a large part of this research was carried out. We also thank the referee for their helpful comments.

Communicated by: Dessislava Kochloukova

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Received: 2021-02-03

Revised: 2021-12-15

Published Online: 2022-03-03

Published in Print: 2022-09-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2021-0017