A pro-2 group with full normal Hausdorff spectra
aa r X i v : . [ m a t h . G R ] F e b A PRO- GROUP WITH FULL NORMAL HAUSDORFFSPECTRA
IKER DE LAS HERAS AND ANITHA THILLAISUNDARAM
Abstract.
We construct a 2-generated pro-2 group with full normal Hausdorffspectrum [0 , Introduction
Let Γ be a countably based infinite profinite group and consider a filtrationseries S of Γ, that is, a descending chain Γ = Γ ≥ Γ ≥ · · · of open normalsubgroups Γ i E o Γ such that T i Γ i = 1. These open normal subgroups form a baseof neighbourhoods of the identity and induce a translation-invariant metric on Γgiven by d S ( x, y ) = inf {| Γ : Γ i | − | x ≡ y (mod Γ i ) } , for x, y ∈ Γ. This yields, fora subset U ⊆ Γ, the
Hausdorff dimension hdim S Γ ( U ) ∈ [0 ,
1] with respect to thefiltration series S .Over the past twenty years, there have been interesting applications of Hausdorffdimension to profinite groups, starting with the pioneering work of Abercrom-bie [1], Barnea and Shalev [2]; see [8] for a good overview. Barnea and Shalev [2]established the following algebraic formula of the Hausdorff dimension of a closedsubgroup H of Γ as a logarithmic density:hdim S Γ ( 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 ishspec S (Γ) = { hdim S Γ ( H ) | H ≤ c Γ } ⊆ [0 , , where H runs through all closed subgroups of Γ. Shalev [9, § normal Hausdorff spectrum of Γ, with respect to S , that is,hspec S E (Γ) = { hdim S Γ ( H ) | H E 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 Date : February 4, 2021.2010
Mathematics Subject Classification.
Primary 20E18; Secondary 28A78.
Key words and phrases.
Pro- p groups, Hausdorff dimension, normal Hausdorff spectrum.The first author is supported by the Spanish Government, grant MTM2017-86802-P, partlywith FEDER funds, and by the Basque Government, grant IT974-16. He is also supported by apostdoctoral grant of the Basque Government. The second author acknowledges the support fromEPSRC, grant EP/T005068/1. Both authors thank the Heinrich-Heine-Universit¨at D¨usseldorf,where a large part of this research was carried out. generated pro- p groups. Indeed, early examples of normal Hausdorff spectra wereall finite (see [9, § p group, for every prime p , with infinitenormal 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 Frattiniseries F , and the dimension subgroup series D ; we refer the reader to Section 2for the definitions. The normal Hausdorff spectra of the groups in [7] consistof an interval [0 , ξ ], for ξ ≤ / , with one or two isolated points. The questionwas raised in [7] whether a finitely generated pro- p group, for p any prime, couldbe constructed with full normal Hausdorff spectra [0 , 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 amodification of the construction in [3]. Our group G is a 2-generated extension ofan elementary abelian pro-2 group by the pro-2 wreath product W = C ˆ ≀ Z =lim ←− k ∈ N C ≀ C k . We follow the strategy in [3] to prove the result. Theorem 1.1.
The pro- group G satisfies hspec S E ( G ) = hspec S ( G ) = [0 , , 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 seriescoincides with the Frattini series.It was asked in [2, Prob. 5], whether there is a finitely generated pro- p group Γwith hspec S (Γ) = [0 , C p ˆ ≀ Z p has full Hausdorff spectrawith 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 nothave full Hausdorff spectrum with respect to the lower p -series L . The work ofKlopsch 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 workof Garaialde Oca˜na, Garrido and Klopsch [5] provide many examples of finitelygenerated 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 ofa finitely generated pro-2 group with full Hausdorff spectrum with respect to L . Organisation . Section 2 contains preliminary results. In Section 3 we give apresentation 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 normalHausdorff spectra of G with respect to M , L , D , P and F . Notation.
All subgroups of profinite groups are generally taken to be closedsubgroups. We use the notation ≤ o and ≤ c to denote open and closed sub-groups 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 thefour natural filtration series on Γ. The p -power series of Γ is given by P : Γ p i = h x p i | x ∈ Γ i , i ∈ N , PRO-2 GROUP WITH FULL NORMAL HAUSDORFF SPECTRA 3 where N = N ∪ { } . The lower p -series (or lower p -central series) of Γ is givenrecursively by L : P (Γ) = Γ , and P i (Γ) = P i − (Γ) p [ P i − (Γ) , Γ] for i ∈ N ≥ ,and the Frattini series of Γ is given recursively by F : Φ (Γ) = Γ , and Φ i (Γ) = Φ i − (Γ) p [Φ i − (Γ) , Φ i − (Γ)] for i ∈ N .The (modular) dimension subgroup series (or Jennings series or Zassenhaus series)of Γ is defined recursively by D : D (Γ) = Γ , and D i (Γ) = D ⌈ i/p ⌉ (Γ) p Y ≤ j
Lemma 2.1.
Let
Γ = h a, b i 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 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 weightat least in v for ≤ s ≤ r . Then ( ab ) p r ≡ K ( a,b ) a p r b p r [ b, a ]( pr )[ b, a, a ]( pr ) · · · [ b, a, p r − . . . , a ]( prpr − )[ b, a, p r − . . . , a ] , (2.1)[ a p r , b ] ≡ K ( a, [ a,b ]) [ a, b ] p r [ a, b, a ]( pr ) · · · [ a, b, a, p r − . . . , a ]( prpr − )[ a, b, a, p r − . . . , a ] . (2.2)We also recall the following definition from [7]: for a countably based infinitepro- p group Γ, equipped with a filtration series S : Γ = Γ ≥ Γ ≥ · · · , and aclosed subgroup H ≤ c Γ, we say that H has strong Hausdorff dimension in Γ withrespect to S if hdim S Γ ( 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 normalHausdorff spectra in the sequel. Proposition 2.2. [3, Prop. 2.4]
Let Γ be a countably based pro- p group with aninfinite abelian normal subgroup Z E c Γ such that Γ /C Γ ( Z ) is procyclic. Let S : Γ = S ≥ S ≥ · · · be a filtration series of Γ and consider the induced filtrationseries S | Z : Z = S ∩ Z ≥ S ∩ Z ≥ · · · of Z ; for i ∈ N , let p e i be the exponent of Z/ ( S i ∩ Z ) . Suppose that, for every i ∈ N , there exist n i ∈ N and K i ≤ c Γ suchthat 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 . I. DE LAS HERAS AND A. THILLAISUNDARAM If Z has strong Hausdorff dimension ξ = hdim S G ( Z ) ∈ [0 , , then [0 , ξ ] ⊆ hspec S E (Γ) . The pro- group G For k ∈ N , let h ˙ x k i ∼ = C k and h ˙ y k i ∼ = C . Define W k = h ˙ y k i ≀ h ˙ x k i ∼ = B k ⋊ h ˙ x k i where B k = Q k − i =0 h ˙ y ˙ x ik k i ∼ = C k . The structural results for the finite wreathproducts W k transfer naturally to the inverse limit W ∼ = lim ←− k W k , i.e. the pro-2wreath product W = h ˙ x, ˙ y i = B ⋊ h ˙ x i ∼ = C ˆ ≀ Z with top group h ˙ x i ∼ = Z and base group B = Q i ∈ Z h ˙ y ˙ x i i ∼ = C ℵ ; see [7, § F = h a, b i be the free pro-2 group on two generators and let k ∈ N . Thereexists a closed normal subgroup R E F , respectively R k E F , such that F /R ∼ = W, respectively F /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 such that Y /R is the pre-imageof B in F /R , and let Y k ≥ R k be the closed normal subgroup of F such that Y k /R k is the pre-image of B k in F /R k .Consider now N = [ R, Y ] R , respectively N k = [ R k , Y k ] R k h a k i F , and define G = F /N, respectively G k = F /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 , respectivelyin G k , by x, y , respectively x k , y k , so that G = h x, y i and G k = h x k , y k i .Note that the groups G k are finite for all k ∈ N and that they naturally form aninverse 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 | G k | = k + 2 k +1 + (cid:18) k (cid:19) = k + 2 k − + 2 k +1 − k − . Proof.
We proceed as in [7, Lem. 5.1]. Since G k /Z k ∼ = W k ∼ = C ≀ C k , we havelog | G k | = log | G k /Z k | + log | Z k | = k + 2 k + log | Z k | . By construction, the subgroup Z k is elementary abelian, so we obtain Z k = h{ ( y x ik k ) | ≤ i ≤ k − } ∪ { [ y x ik k , y x jk k ] | ≤ i < j ≤ k − }i . PRO-2 GROUP WITH FULL NORMAL HAUSDORFF SPECTRA 5
Therefore log | G k | ≤ k + 2 k +1 + (cid:0) k (cid:1) .For the converse, we will construct a factor group e G k of G k whose logarithmicorder is | e G k | = k + 2 k +1 + (cid:0) k (cid:1) .We consider the finite 2-group M = h e y , . . . , e y k − i = E/ [Φ( E ) , E ]Φ( E ) , where E is the free group on 2 k generators. The images of e y , . . . , e y k − gener-ate independently the elementary abelian quotient M/ Φ( M ), and the elements e y , . . . , e y k − together with the commutators [ e y i , e y j ] for 0 ≤ i < j ≤ k − M ). The latter can beverified by considering homomorphisms from M onto groups of the form C k − × C and C k − × Heis( F ), where Heis( F ) denotes the group of upper unitriangular3 × F . Next consider the faithful action of the cyclic group X ∼ = h e x i ∼ = C k induced by e y e xi = (e y i +1 if 0 ≤ i ≤ k − , e y if i = 2 k − . We define e G k = X ⋉ M and observe that log | e G k | = k + 2 k +1 + (cid:0) k (cid:1) . Furthermore,it is easy to see that e G k / Φ( M ) ∼ = W k . Thus, there is an epimorphism ǫ : G k → e G k with ǫ ( x k ) = e x and ǫ ( y k ) = e y , and since | G k | ≤ | e G k | we conclude that G k ∼ = e G k . (cid:3) Remark 3.2.
The proof of Proposition 3.1 shows that H k = Z k and, consequently,that H = 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 , andtherefore of G . For that purpose, we recall the convenient notation introducedin [3], which will be used frequently in the paper. We will denote c = y and c i = [ y, x, i − . . ., x ] for i ∈ N ≥ . Similarly, we set c i,j = [ c i , y, x, j − . . ., x ] for every j ∈ N . For the sake of simplicity, for k ∈ N , we will use the same notation forthe corresponding elements in the group G k . Thus, we will also write c = y k , c i = [ y k , x k , i − . . ., x k ], and c i,j = [ c i , y k , x k , j − . . ., x k ] for every i ∈ N ≥ and j ∈ N , whenworking in the groups G k , for k ∈ N . It will be clear from the context whetherour considerations apply to G or G k . Lemma 3.3.
In the group G k , for k ∈ N , (i) for i ≥ k − + 1 , we have c i ∈ γ i +1 ( G k ) ; (ii) for i ≥ k + 1 , we have c i = 1 ; (iii) for i ≥ k + 2 k − + 1 , we have c i ∈ γ i +1 ( G k ) .Proof. (i) As H k has exponent 4 and [ H k , H k ] ≤ Z k exponent 2, it follows from (2.2)that 1 = [ y k , x k k ] ≡ [ y k , x k , k − . . . , x k ] [ y k , x k , k . . ., x k ] (mod γ k +2 ( G k )) . (3.1)Therefore c k − +1 ≡ c k +1 (mod γ k +2 ( G k )).For i ≥ k − + 2, notice that c i = [ y k , x k , i − . . ., x k ] ≡ [ c k − +1 , x k , i − k − − . . . , x k ] (mod γ i +1 ( G k )) ≡ [ c k +1 , x k , i − k − − . . . , x k ] (mod γ i +1 ( G k )) . I. DE LAS HERAS AND A. THILLAISUNDARAM
Hence the result follows.(ii) This follows immediately from the fact that c i ∈ Z k for i ≥ k +1; compare [7,Prop. 2.6(1)].(iii) It suffices to prove the result for i = 2 k + 2 k − + 1. From (3.1), we obtain c i = [ c k +1 , x k , k − . . . , x k ] ≡ [ c k − +1 , x k , k − . . . , x k ] (mod γ i +1 ( G k )) . As [ c k − +1 , x k , k − . . . , x k ] ≡ c k +1 (mod γ i +1 ( G k ))we have c i ≡ c k +1 (mod γ i +1 ( G k )) , and by (ii), it follows that c i ∈ γ i +1 ( G k ), asrequired. (cid:3) Let us write z i,j = [ c i , c j ] for every i, j ∈ N . From [7, Prop. 2.6(1)] we have H = h c n | n ∈ N i , and from Remark 3.2 we deduce that Z = h c n , z i,j | n, i, j ∈ N with j < i i . We recall the following result from [3, Lem. 4.3], which although was writtenin the setting of the pro- p group constructed in [3], for p an odd prime, the sameproof holds for our group G . Lemma 3.4.
In the group G , for i, j ∈ N we have [ z i,j , x, k . . ., x ] = k Y s =0 s Y n =0 z ( ks )( sn ) i + k − n,j + k − s + n ! for every k ∈ N . Corollary 3.5.
In the group G , for i, j ∈ N we have [ z i,j , x k ] = z i +2 k ,j z i,j +2 k z i +2 k ,j +2 k for every k ∈ N . Proof.
As was reasoned in [3], this result follows directly from (2.2) and Lemma 3.4. (cid:3)
Lemma 3.6.
Let k ∈ N . In the group G k , for m ∈ N even, we have c m, k ∈ γ k + m +1 ( G k ) . Proof.
Note that c m, k = [ z m, , x k , k − . . . , 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, , x k , k − . . . , x k ] ≡ k − Y n =0 z ( k − n ) m +2 k − − n, n (mod γ k + m +1 ( G k )) . In addition, the exponent of Z k is 2 by construction, so since all the binomialnumbers (cid:0) k − n (cid:1) are odd, we get[ z m, , x k , k − . . . , x k ] ≡ k − Y n =0 z m +2 k − − n, n (mod γ k + m +1 ( G k )) . PRO-2 GROUP WITH FULL NORMAL HAUSDORFF SPECTRA 7
Recall that c i ∈ Z k for every i ≥ k + 1 by [7, Prop. 2.6(1)], so z m +2 k − − n, n = 1for all n ≤ m −
2. Thus,[ z m, , x k , k − . . . , x k ] ≡ k − Y n = m − z m +2 k − − n, n (mod γ k + m +1 ( G k )) ≡ k − m Y n =0 z k − n,m + n (mod γ 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, , x k , k − . . . , x k ] ≡ γ k + m +1 ( G k )) , as required. (cid:3) Proposition 3.7.
For k ∈ N , the nilpotency class of G k is k +1 − and the lowercentral series of G k satisfies: • γ ( G k ) = G k = h x k , y k i γ ( G k ) with γ ( G k ) /γ ( G k ) ∼ = C k × C . • If ≤ i ≤ k , then γ i ( G k ) = ( h c i , c ,i − , c ,i − , . . . , c i − , i γ i +1 ( G k ) if i ≡ , h c i , c ,i − , c ,i − , . . . , c i − , i γ i +1 ( G k ) if i ≡ , with γ i ( G k ) /γ i +1 ( G k ) ∼ = C × C × ( i − / . . . × C if ≤ i ≤ k − and i ≡ ,C × C × ( i − / . . . × C if ≤ i ≤ k − and i ≡ ,C × i/ . . . × C if k − + 1 ≤ i ≤ k and i ≡ ,C × ( i +1) / . . . × C if k − + 1 ≤ i ≤ k and i ≡ . (3.2) • If k + 1 ≤ i ≤ k + 2 k − , then γ i ( G k ) = ( h c i , c i − k +2 , k − , c i − k +4 , k − , . . . , c k − ,i − k +2 , c k ,i − k i γ i +1 ( G k ) if i ≡ , h c i , c i − k +1 , k − , c i − k +3 , k − , . . . , c k − ,i − k +2 , c k ,i − k i γ i +1 ( G k ) if i ≡ , with γ i ( G k ) /γ i +1 ( G k ) ∼ = ( C × (2 k +1 − i +2) / . . . × C if i ≡ ,C × (2 k +1 − i +3) / . . . × C if i ≡ . (3.3) • If k + 2 k − + 1 ≤ i ≤ k +1 , then γ i ( G k ) = ( h c i − k +2 , k − , c i − k +4 , k − , . . . , c k − ,i − k +2 , c k ,i − k i γ i +1 ( G k ) if i ≡ , h c i − k +1 , k − , c i − k +3 , k − , . . . , c k − ,i − k +2 , c k ,i − k i γ i +1 ( G k ) if i ≡ , with γ i ( G k ) /γ i +1 ( G k ) ∼ = ( C × (2 k +1 − i ) / . . . × C , if i ≡ ,C × (2 k +1 − i +1) / . . . × C if i ≡ . (3.4) I. DE LAS HERAS AND A. THILLAISUNDARAM
Proof.
The first assertion is obvious. To prove the other ones, we start by showingthat γ i ( G k ) = ( h c i , c ,i − , c ,i − , . . . , c i − , i γ i +1 ( G k ) if i ≡ , h c i , c ,i − , c ,i − , . . . , c i − , i γ i +1 ( G k ) if i ≡ . We proceed by induction on i . For i = 2, we have γ ( G k ) = h c i γ ( G k ) and for i = 3 we have γ ( G k ) = h c , c , i γ ( G k ). Assume then that i >
3, that i is even,and that γ i − ( G k ) = h c i − , c ,i − , c ,i − , . . . , c i − , i γ i − ( G k )and γ i − ( G k ) = h c i − , c ,i − , c ,i − , . . . , c i − , i 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 ) = h c i , c i − , , c ,i − , c ,i − , . . . , c i − , i γ i +1 ( G k ) . For convenience, we write M = h c i , c ,i − , c ,i − , . . . , c i − , i γ i +1 ( G k ) , and we have to check that c i − , ∈ M . Notice that c i − , = [ c i − , x k , y k ], and theHall-Witt identity yields[ c i − , x k , y k ][ x k , y k , c i − ][ y k , c i − , x k ] ≡ M ) . Note also that [ y k , c i − , x k ] ≡ c − i − , ≡ M ), so c i − , ≡ [ c i − , c ] − (mod M ) . Let us prove that [ c m , c n ] ≡ M ) for all n ≥ 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. Nowsuppose that m − n >
0, which, since i is even, implies that m − n ≥
2. Observethat [ c m , c n ] = [ c m − , x k , c n ], so again by the Hall-Witt identity,[ c m − , x k , c n ][ x k , c n , c m − ][ c n , c m − , x k ] ≡ M ) . Since 0 ≤ ( m − − ( n + 1) = m − n − < m − n , we have[ x k , c n , c m − ] ≡ [ c m − , c n +1 ] ≡ M )by induction. Hence [ c m , c n ] ≡ [ c n , c m − , x k ] − (mod M ). As[ c n , c m − ] ∈ γ i − ( G k ) ∩ Z k , it follows that [ c n , c m − ] ≡ c n i − c n ,i − c n ,i − · · · c n i − i − , (mod γ i ( G k ))for some n , n , . . . , n i − ∈ Z . Hence[ c m , c n ] ≡ [ c n , c m − , x k ] ≡ c n i c n ,i − c n ,i − · · · c n i − i − , ≡ M ) , as we wanted to prove. This, in particular, implies that c i − , ∈ M , as claimed.Hence, γ i ( G k ) = h c i , c ,i − , c ,i − , . . . , c i − , i γ i +1 ( G k )and, again, since [ c n,m , y k ] = 1, we have γ i +1 ( G k ) = h c i +1 , c ,i − , c ,i − , . . . , c i − , , c i, i γ i +2 ( G k ) . Now, for 2 k + 1 ≤ i ≤ k +1 , we add x k and y k to the commutators that induc-tively generate γ i − ( G k ) modulo γ i ( G k ). Removing the unnecessary generatorsaccording to Lemmata 3.3 and 3.6, one can deduce that the generators of γ i ( G )modulo γ i +1 ( G ) are precisely the stated ones. PRO-2 GROUP WITH FULL NORMAL HAUSDORFF SPECTRA 9
Finally, it suffices to check that the isomorphisms (3.2), (3.3) and (3.4) aresatisfied. 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 ofthe lower central series is actually isomorphic to a quotient of the correspondingabelian group in the statement. In particular, we obtain upper bounds for thelogarithmic orders of the quotients of two consecutive terms of the lower centralseries. In fact, all these upper bounds sum to the logarithmic order of G k . Indeed,we have k +1 X i =1 log | γ i ( G k ) : γ i +1 ( G k ) | = ( k + 2) + (cid:16) k − − k X i =2 ⌈ i/ ⌉ (cid:17) + (cid:16) k − + k +1 X i =2 k +1 ⌈ (2 k +1 − i ) / ⌉ (cid:17) = k + 2 k +1 + (cid:18) k (cid:19) , which is precisely, by Proposition 3.1 the logarithmic order of G k . Hence, isomor-phisms (3.2), (3.3) and (3.4) are satisfied and the proof is finished. (cid:3) Remark 3.8.
From Proposition 3.7 we deduce that the logarithmic order of Z/ ( γ i ( G ) ∩ Z ) is 2 (cid:16) · · · + i − (cid:17) = 2 (cid:18) ( i +1) / (cid:19) if i is odd or 2 (cid:16) · · · + i − (cid:17) + i (cid:18) i / (cid:19) + i 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 k +2 .Proof. First we show that x k y k has order 2 k +2 . Consider ( x k y k ) k and observethat K ( x k , y k ) ≤ [ H k , H k ] in (2.1) has exponent 2. Therefore, ( x k y k ) k ≡ K ( x k ,y k ) [ y k , x k , k − − . . . , x k ] [ y k , x k , k − . . . , x k ] yields ( x k y k ) k +1 = c k , which is non-trivial bythe proof of Proposition 3.7. Hence the result.For a general element g = x ik h for some 0 ≤ i ≤ k − h ∈ H k , it similarlyfollows that g k +2 = 1. (cid:3) The following two results will be needed for computing the normal Hausdorffspectra 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 -series of G k is k +1 − and P ( G k ) = G k ,P ( G k ) = h x k , y k i γ ( G k ) , and P i ( G k ) = ( h x i − k , c i − i γ i ( G k ) for ≤ i ≤ k − + 1 , h x i − k i γ i ( G k ) for k − + 2 ≤ i ≤ k +1 . Proof. If i = 1 or 2, the results are obvious, so consider i = 3. As[ h x k , y k i , G k ] ≤ γ ( G k ) γ ( G k ) , it suffices to show that h x k , y k i ≤ h x k i γ ( G k ) and γ ( G k ) ≤ h c i γ ( G k ). Firstconsider ( x k y k ) . Note that[ y k , x k ] ≡ [ y k , x k ] = 1 (mod γ ( G k )) , and since x k , y k ∈ h x k i γ ( G k ), the first inclusion holds. For the second state-ment, it follows from Proposition 3.7, as [ y k , x k ] is the only generator of γ ( G k )modulo γ ( G ).Now let 4 ≤ i ≤ k − and assume by induction that P i − ( G k ) = h x i − k , c i − i γ i − ( G k ) . On the one hand,[ P i − ( G k ) , G k ] = [ h x i − k , c i − i γ i − ( G k ) , G k ] = [ h x i − k , c i − i , G k ] γ i ( G k )and Proposition 3.7 and (2.2) yield[ h x i − k , c i − i , G k ] γ i ( G k ) = [ h c i − i , G k ] γ i ( G k ) . Then by similar arguments as above, one deduces that[ c i − , G k ] γ i ( G k ) = h c i − i γ i ( G k ) . On the other hand, P i − ( G k ) ≡ h x i − k i γ i − ( G k ) ≡ h x i − k , c i − i (mod γ i ( G k )) , so we conclude that P i ( G k ) = h x i − k , c i − i γ i ( G k ) , as asserted. The case 2 k − + 2 ≤ i ≤ k +1 follows similarly, using Lemma 3.3(ii). (cid:3) Proposition 3.11.
For k ∈ N , the length of the dimension subgroup series of G k is k +1 and D i ( G k ) = h x l ( i ) k i γ ⌈ i / ⌉ ( G k ) γ i ( G k ) for ≤ i ≤ k +1 , where l ( i ) = ⌈ log i ⌉ .Proof. By [4, Thm. 11.2], we have D i ( G k ) = Y n · m ≥ i γ n ( G k ) m for every i ∈ N , and since exp( γ ( G k )) = 4, we obtain D i ( G k ) = G l ( i ) k γ ⌈ i / ⌉ ( G k ) 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 ( ab ) l ( i ) = a l ( i ) b l ( i ) [ b, a, l ( i ) − − . . . , a ]( l ( i )2 l ( i ) − ) c with c ∈ γ l ( i ) ( G k ). Since [ b, a, l ( i ) − − . . . , a ]( l ( i )2 l ( i ) − ) ∈ γ ⌈ i / ⌉ ( G k ) and γ l ( i ) ( G k ) ≤ γ i ( G k ), we get D i ( G k ) = h x l ( i ) k i γ ⌈ i / ⌉ ( G k ) γ i ( G k ) , as required. (cid:3) PRO-2 GROUP WITH FULL NORMAL HAUSDORFF SPECTRA 11 The normal Hausdorff spectra of G In this section we compute the normal Hausdorff spectra of G with respect tothe filtration series M , L , D , P , and F . Here M : M ≥ M ≥ · · · stands for thenatural filtration series of G where each M i is the subgroup of G correspondingto N i /N , where here N = F .As an easy illustration of our methods we start computing the normal Hausdorffspectrum of G with respect to M . Theorem 4.1.
The pro- group G satisfies hspec M E ( G ) = [0 , and Z has strong Hausdorff dimension in G with respect to M .Proof. By Proposition 3.7 we know that γ k +1 ( G ) ≤ M k for all k ∈ N . On theother hand, by the proof of Proposition 3.1 we havelog | Z : M k ∩ Z | = log | ZM k : M k | = 2 k + (cid:18) k (cid:19) . In particular lim k →∞ k +1 log | Z : M k ∩ Z | = 0 . Moreover,hdim M G ( Z ) = lim k →∞ log | ZM k : M k | log | G : M k | = lim k →∞ k + (cid:0) k (cid:1) k + 2 k +1 + (cid:0) k (cid:1) = 1 . Therefore, the second statement follows and, by Proposition 2.2, we conclude thathspec M E ( G ) = [0 , . (cid:3) Theorem 4.2.
For S ∈ { L , D } , the pro- group G satisfies hspec S E ( G ) = [0 , and Z has strong Hausdorff dimension in G with respect to S .Proof. By Remark 3.8 we havelim k →∞ k log | Z : P k ( G ) ∩ Z | = 0 and lim k →∞ k log | Z : D k ( G ) ∩ Z | = 0 , and they are furthermore given by proper limits. Then it suffices by Proposition 2.2to show that Z has strong Hausdorff dimension 1 with respect to S . Write G = S ≥ S ≥ S ≥ · · · for the subgroups of the filtration series S . Observe thatby [7, Prop. 2.6] we have log | G : S k Z | = 2 k, and so lim k →∞ log | G : S k Z | log | Z : S k ∩ Z | = 0 , is given by a proper limit. Thus,hdim S G ( Z ) = lim k →∞ (cid:18) log | G : S k | log | S k Z : S k | (cid:19) − = lim k →∞ (cid:18) log | G : S k Z | + log | S k Z : S k | log | S k Z : S k | (cid:19) − = lim k →∞ (cid:18) log | G : S k Z | log | Z : S k ∩ Z | + 1 (cid:19) − = 1 , and Z has strong Hausdorff dimension 1, as we wanted. Thus, the proof is com-plete. (cid:3) For all n ∈ N define Γ n = h x n i Q n − γ n ( G ) E G where Q n − = h c i | i ≥ n − i . Lemma 4.3.
For each n ∈ N , we have Γ n ≤ Γ n +1 .Proof. We only have to check that Γ ′ n ≤ Γ n +1 . Clearly [ Q n − , Q n − γ n ( G )] = 1and [ γ n ( G ) , γ n ( G )] ≤ γ n +1 ( G ) ≤ Γ n +1 , so it suffices to prove that[ h x n i , Q n − γ n ( G )] ≤ Γ n +1 . On the one hand, (2.2) yields[ γ n ( G ) , x n ] ≤ γ n +2 n − ( G ) γ n +1 ( G ) ≤ Γ n +1 . On the other hand, for 2 n − ≤ i ≤ n −
1, again by (2.2) we have[ c i , x n ] ∈ γ n ( G ) γ n +2 n − ( G ) , so [ c i , x n ] = [ c i , x n ] [ c i , x n , c i ] ∈ Γ n +1 , as required. (cid:3) Theorem 4.4.
The pro- group G satisfies hspec P E ( G ) = [0 , and Z has strong Hausdorff dimension 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 ,and by (2.1), it follows that ( x i h ) k ∈ Γ k for k ∈ N . Then G k ≤ Γ k and, inparticular, G k ∩ Z ≤ Γ k ∩ Z . It is easy to see thatΓ k ∩ Z = ( γ k − ( G ) γ k ( G )) ∩ Z = γ k − ( G ) ( γ k ( G ) ∩ Z ) , and since [ γ k − ( G ) , γ k − ( G )] ≤ γ k ( G ) ∩ Z , it follows that γ k − ( G ) ≤ h c k − , c k − +1 , . . . , c k − i ( γ k ( G ) ∩ Z ) . Thus, by Remark 3.8, we havelog | Z : Γ k ∩ Z | = log | Z : γ k − ( G ) ( γ k ( G ) ∩ Z ) | = 2 (cid:18) k − (cid:19) . On the other hand, from the construction of G and G k it can be deduced easilythat G/G k ∼ = G k /G k k . Indeed, N k = N h a k i F and h a k i F ≤ F k . Hence, byProposition 3.7, we get γ k +1 ( G ) ≤ G k . PRO-2 GROUP WITH FULL NORMAL HAUSDORFF SPECTRA 13
Now, lim k →∞ k +1 log | Z : Γ k ∩ Z | = 0 , so again by Proposition 2.2 it suffices to check that Z has strong Hausdorff di-mension 1 with respect to P . Note thatlog | G : G k Z | = log | G k : G k k Z k | ≤ log | W k | = k + 2 k , and so, lim k →∞ log | G : G k Z | log | Z : G k ∩ Z | ≤ lim k →∞ log | G : G k Z | log | Z : Γ k ∩ Z | = 0is given by a proper limit. Thus, the result follows as in the proof of Theorem 4.2. (cid:3) Theorem 4.5.
The pro- group G satisfies hspec F E ( G ) = [0 , and Z has strong Hausdorff dimension in G with respect to F .Proof. We claim that T k ( γ k +2 k − − ( G ) ∩ Z ) ≤ Φ k ( G ) ≤ Γ k where T k = h x k , c i , c j | i ≥ k − , j ≥ k i for all k ∈ N . We will proceed by induction on k . If k = 1 the result is clear, soassume k ≥
2. On the one hand, it follows from Lemma 4.3 thatΦ k ( G ) = Φ k − ( G ) ≤ Γ k − ≤ Γ k . Hence, we only need to check that T k ( γ k +2 k − − ( G ) ∩ Z ) ≤ ∆ , where ∆ = Φ (cid:0) T k − ( γ k − +2 k − − ( G ) ∩ Z ) (cid:1) . Of course we have x k , c i ∈ ∆ for all i ≥ k − . We also have T ′ k − ≤ ∆, so h z i,j | i > j ≥ k − i ≤ ∆. Let us see that z i,j ∈ ∆ whenever i > j , i + j ≥ k + 2 k − − j ≤ k − −
1. Consider the element z i − k − ,j and observe that z i − k − ,j ∈ γ k − ( G ) ∩ Z as i − k − + j ≥ k − . Therefore [ z i − k − ,j , x k − ] ∈ ∆ . By Corollary 3.5, it follows then that z i,j z i − k − ,j +2 k − z i,j +2 k − ∈ ∆ . Now we have i > j + 2 k − and j + 2 k − ≥ k − , so z i,j +2 k − ∈ ∆. Next, if i − k − > j + 2 k − , then z i − k − ,j +2 k − ∈ ∆, and if i − k − ≤ j + 2 k − , then as i − k − ≥ k − , we have z i − k − ,j +2 k − = z − j +2 k − ,i − k − ∈ ∆ . Therefore z i,j ∈ ∆ and γ k +2 k − − ( G ) ∩ Z ≤ ∆.Finally, for j ≥ k − , observe that[ c j , x k − ] ≡ c j +2 k − c j +2 k − (mod γ j +2 k − ( G ) ∩ Z ) , and since γ j +2 k − ( G ) ∩ Z ≤ ∆ and c i ∈ ∆ for all i ≥ k − , we have c j ∈ ∆ for all j ≥ k . We conclude that T k (cid:0) γ k +2 k − − ( G ) ∩ Z (cid:1) ≤ ∆ ≤ Φ k ( G ) , as claimed. In particular, we get γ k +2 k − − ( G ) ∩ Z ≤ Φ k ( G ) ∩ Z ≤ Γ k ∩ Z. Now, from Remark 3.8 we deduce thatlim k →∞ k + 2 k − − | Z : Γ k ∩ Z | = lim k →∞ k + 2 k − − | Z : Γ k ∩ Z | = 0 . (4.1)Hence, by Proposition 2.2, it only remains to show that hdim F G ( Z ) = 1 and thatit is given by a proper limit. This follows easily since, from [7, Prop. 2.6(3)] and(4.1), we deduce that lim k →∞ log | G : Φ k ( G ) Z | log | Z : Φ k ( G ) ∩ Z | = 0 , and so, as done in the proof of Theorem 4 .
2, we obtain hdim F G ( Z ) = 1. It furtherfollows that Z has strong Hausdorff dimension with respect to F . (cid:3) References [1] A. G. Abercrombie, Subgroups and subrings of profinite rings,
Math. Proc. Camb. Phil. Soc.
116 (2) (1994), 209–222.[2] Y. Barnea and A. Shalev, Hausdorff dimension, pro- p groups, and Kac-Moody algebras, Trans. Amer. Math. Soc. (1997), 5073–5091.[3] I. de las Heras and B. Klopsch, A pro- p group with full normal Hausdorff spectra, Math.Nachr , to appear.[4] J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal,
Analytic pro- p groups , 2nd edition,Cambridge University Press, 1999.[5] O. Garaialde Oca˜na, A. Garrido and B. Klopsch, Pro- p groups of positive rank gradientand Hausdorff dimension, J. London Math. Soc.
101 (3) (2020), 1008–1040.[6] B. Klopsch,
Substitution Groups, Subgroup Growth and Other Topics , D.Phil. Thesis, Uni-versity of Oxford, 1999.[7] B. Klopch and A. Thillaisundaram, A pro- p group with infinite normal Hausdorff spectra, Pacific J. Math.
303 (2) (2019), 569–603.[8] B. Klopsch, A. Thillaisundaram, and A. Zugadi-Reizabal, Hausdorff dimensions in p -adicanalytic groups, Israel J. Math. (2019), 1–23.[9] A. Shalev, Lie methods in the theory of pro- p groups, in: New horizons in pro- p groups ,Birkh¨auser, Boston 2000. Iker de las Heras: Mathematisches Institut, Heinrich-Heine-Universit¨at, 40225D¨usseldorf, Germany; Department of Mathematics, University of the BasqueCountry UPV/EHU, 48940 Leioa, Spain
Email address : [email protected] Anitha Thillaisundaram: School of Mathematics and Physics, University ofLincoln, Lincoln LN6 7TS, United Kingdom
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