A pro-p group with infinite normal Hausdorff spectra

TL;DR

This paper constructs a finitely generated pro-p group with an infinite normal Hausdorff spectrum, answering a question of Shalev and analyzing its shape across various filtration series.

Contribution

It introduces a new pro-p group with an infinite normal Hausdorff spectrum and explores its spectral properties across multiple filtration series.

Findings

Normal Hausdorff spectrum contains an infinite interval

Spectra with respect to various filtration series have similar shapes

The spectrum for the lower p-series exhibits surprising features

Abstract

Using wreath products, we construct a finitely generated pro-p group G with infinite normal Hausdorff spectrum with respect to the p-power series. More precisely, we show that this normal Hausdorff spectrum contains an infinite interval; this settles a question of Shalev. Furthermore, we prove that the normal Hausdorff spectra of G with respect to other filtration series have a similar shape. In particular, our analysis applies to standard filtration series such as the Frattini series, the lower p-series and the modular dimension subgroup series. Lastly, we pin down the ordinary Hausdorff spectra of G with respect to the standard filtration series. The spectrum of G for the lower p-series displays surprising new features.

Equations

$${\mathrm{hspec}}_{⊴}^{\mathcal{P}}(G)=\{{\mathrm{hdim}}_G^{\mathcal{P}}(H)\mid H{⊴}_{\mathrm{c}}G\};$$

$${\mathrm{hdim}}_G^{\mathcal{S}}(H)=\underset{i\to \infty }{\underline{\lim }}\frac{\log |H{G}_i:G_i|}{\log |G:G_i|}.$$

$${\mathrm{hspec}}^{\mathcal{S}}(G)=\{{\mathrm{hdim}}_G^{\mathcal{S}}(H)\mid H{\le }_{\mathrm{c}}G\}\subseteq [0,1],$$

$${\mathrm{hspec}}_{⊴}^{\mathcal{S}}(G)=\{{\mathrm{hdim}}_G^{\mathcal{S}}(H)\mid H{⊴}_{\mathrm{c}}G\}$$

$$G=F/N,\text{where}N=[R,F]R^p{⊴}_{\mathrm{c}}F.$$

$${\mathrm{hspec}}_{⊴}^{\mathcal{P}}(G)$$

$${\mathrm{hspec}}_{⊴}^{\mathcal{F}}(G)$$

$${\mathrm{hspec}}_{⊴}^{\mathcal{L}}(G)$$

$${\mathrm{hspec}}^{\mathcal{P}}(G)$$

$${\mathrm{hspec}}^{\mathcal{L}}(G)$$

$$\mathcal{P}:G^{p^i}=⟨x^{p^i}\mid x\in G⟩,i\in {\mathbb{N}}_0.$$

$$\mathcal{L}:P_1(G)=G,$$

$$\mathcal{F}:{\Phi }_0(G)=G,$$

$$\mathcal{D}:D_1(G)=G,$$

$${\mathrm{hdim}}_G^{\mathcal{S}}(H)=\underset{i\to \infty }{\lim }\frac{{\log }_p|H{G}_i:G_i|}{{\log }_p|G:G_i|}$$

$${\mathrm{hdim}}_G^{\mathcal{S}}(K)={\mathrm{hdim}}_G^{\mathcal{S}}(H)\cdot {\mathrm{hdim}}_H^{\mathcal{S}{|}_H}(K).$$

$$\mathcal{S}{|}_N:G_i\cap N,i\in {\mathbb{N}}_0,\text{and}\mathcal{S}{|}_{G/N}:G_{i}N/N,i\in {\mathbb{N}}_0.$$

$${\mathrm{hdim}}_G^{\mathcal{S}}(H)$$

$$\ge (1-\xi ){\mathrm{hdim}}_{G/N}^{\mathcal{S}{|}_{G/N}}(HN/N)+\xi {\mathrm{hdim}}_N^{\mathcal{S}{|}_N}(H\cap N).$$

$${\mathrm{hdim}}_G^{\mathcal{S}}(H)$$

$$\displaystyle\qquad\qquad+\underbrace{\frac{\log_{p}\lvert NG_{i}:G_{i}\rvert}{\log_{p}\lvert G:G_{i}\rvert}}_{\to\,\xi\text{ as }i\to\infty}\frac{\log_{p}\lvert HG_{i}\cap NG_{i}:G_{i}\rvert}{\log_{p}\lvert NG_{i}:G_{i}\rvert}\;\Bigg{)}$$

$$\frac{{\log }_p|H{G}_i\cap N{G}_i:G_i|}{{\log }_p|N{G}_i:G_i|}=\frac{{\log }_p|H{G}_i\cap N:G_i\cap N|}{{\log }_p|N:G_i\cap N|}.$$

$${\log }_p|H{G}_i\cap N:G_i\cap N|\ge {\log }_p|(H\cap N)(G_i\cap N):G_i\cap N|$$

$$e_1+\dots +e_i=\lceil \eta (d_1+\dots +d_i)\rceil .$$

$$\lceil \eta (d_1+\dots +d_{i-1})\rceil \le \lceil \eta (d_1+\dots +d_i)\rceil \le \lceil \eta (d_1+\dots +d_{i-1})\rceil +d_i$$

$$e_i=\lceil \eta (d_1+\dots +d_i)\rceil -(e_1+\dots +e_{i-1}),$$

$$\eta \le \frac{e_1+\dots +e_i}{d_1+\dots +d_i}\le \eta +\frac{1}{d_1+\dots +d_i}.$$

$${\log }_p|K{Z}_i:Z_i|=e_1+\dots +e_i.$$

$$Z=⟨z_{1,1},\dots ,z_{1,d_1},z_{2,1},\dots ,z_{2,d_2},\dots ,z_{i,1},\dots ,z_{i,d_i},\dots ⟩$$

$$K=⟨z_{1,1},\dots ,z_{1,e_1},z_{2,1},\dots ,z_{2,e_2},\dots ,z_{i,1},\dots ,z_{i,e_i},\dots ⟩.\text{qed}$$

$$(ab{)}^{p^r}$$

$$[a^{p^r},b]$$

$${\gamma }_2(G{)}^p=1\text{and}[{\gamma }_2(G),{\gamma }_2(G)]\subseteq Z(G),$$

$$(ab{)}^{p^r}{\equiv }_{L(a,b)}{a}^{p^r}{b}^{p^r}[b,a,\stackrel{p^r-1}{\dots },a]\text{and}[a^{p^r},b]{\equiv }_{M(a,b)}[a,b,a,\stackrel{p^r-1}{\dots },a],$$

Full text

[TABLE]

here $\operatorname{hdim}_{G}^{\mathcal{P}}\colon\{X\mid X\subseteq G\}\to[0,1]$ denotes the Hausdorff dimension function associated to the $p$-power series $\mathcal{P}\colon G^{p^{i}}$, $i\in\mathbb{N}_{0}$. More precisely, we show that $\operatorname{hspec}_{\trianglelefteq}^{\mathcal{P}}(G)=[0,\nicefrac{{1}}{{3}}]\cup\{1\}$ contains an infinite interval; this settles a question of Shalev. Furthermore, we prove that the normal Hausdorff spectra $\operatorname{hspec}_{\trianglelefteq}^{\mathcal{S}}(G)$ with respect to other filtration series $\mathcal{S}$ have a similar shape. In particular, our analysis applies to standard filtration series such as the Frattini series, the lower $p$-series and the modular dimension subgroup series.

Lastly, we pin down the ordinary Hausdorff spectra $\operatorname{hspec}^{\mathcal{S}}(G)=\{\operatorname{hdim}_{G}^{\mathcal{S}}(H)\mid H\leq_{\mathrm{c}}G\}$ with respect to the standard filtration series $\mathcal{S}$. The spectrum $\operatorname{hspec}^{\mathcal{L}}(G)$ for the lower $p$-series $\mathcal{L}$ displays surprising new features.

The second author acknowledges the support from the Alexander von Humboldt Foundation and from the Forscher-Alumni-Programm of the Heinrich-Heine-Universität Düsseldorf (HHU); she thanks HHU for its hospitality.

1. Introduction

The concept of Hausdorff dimension has led to interesting applications in the context of profinite groups; see [4] and the references given therein. Let $G$ be a countably based infinite profinite group and consider a filtration series $\mathcal{S}$ of $G$, that is, a descending chain $G=G_{0}\supseteq G_{1}\supseteq\ldots$ of open normal subgroups $G_{i}\trianglelefteq_{\mathrm{o}}G$ such that $\bigcap_{i}G_{i}=1$. These open normal subgroups form a base of neighbourhoods of the identity and induce a translation-invariant metric on $G$ given by $d^{\mathcal{S}}(x,y)=\inf\left\{\lvert G:G_{i}\rvert^{-1}\mid x\equiv y\pmod{G_{i}}\right\}$, for $x,y\in G$. This, in turn, supplies the Hausdorff dimension $\operatorname{hdim}_{G}^{\mathcal{S}}(U)\in[0,1]$ of any subset $U\subseteq G$, with respect to the filtration series $\mathcal{S}$.

Barnea and Shalev [1] established the following ‘group-theoretic’ interpretation of the Hausdorff dimension of a closed subgroup $H$ of $G$ as a logarithmic density:

[TABLE]

The Hausdorff spectrum of $G$, with respect to $\mathcal{S}$, is

[TABLE]

where $H$ runs through all closed subgroups of $G$. As indicated by Shalev in [7, §4.7], it is also natural to consider the normal Hausdorff spectrum of $G$, with respect to $\mathcal{S}$, namely

[TABLE]

which reflects the range of Hausdorff dimensions of closed normal subgroups. Apart from the observations in [7, §4.7], very little appears to be known about normal Hausdorff spectra of profinite groups.

Throughout we will be concerned with pro-$p$ groups, where $p$ denotes an odd prime; in Appendix A we indicate how our results extend to $p=2$. We recall that even for well structured groups, such as $p$-adic analytic pro-$p$ groups $G$, the Hausdorff dimension function and the Hausdorff spectrum of $G$ are known to be sensitive to the choice of $\mathcal{S}$; compare [4]. However, for a finitely generated pro-$p$ group $G$ there are natural choices for $\mathcal{S}$, such as the $p$-power series $\mathcal{P}$, the Frattini series $\mathcal{F}$, the lower $p$-series $\mathcal{L}$ and the modular dimension subgroup series $\mathcal{D}$; see Section 2.

In this paper, we are interested in a particular group $G$ constructed as follows. The pro-$p$ wreath product $W=C_{p}\mathrel{\hat{\wr}}\mathbb{Z}_{p}$ is the inverse limit $\varprojlim_{k\in{\mathbb{N}}}C_{p}\mathrel{\wr}C_{p^{k}}$ of the finite standard wreath products of cyclic groups with respect to the natural projections; clearly, $W$ is $2$-generated as a topological group. Let $F$ be the free pro-$p$ group on two generators and let $R\trianglelefteq_{\mathrm{c}}F$ be the kernel of a presentation $\pi\colon F\to W$. We are interested in the pro-$p$ group

[TABLE]

Up to isomorphism, the group $G$ does not depend on the particular choice of $\pi$, as can be verified using Gaschütz’ Lemma; see [6, Prop. 2.2]. Indeed, $G$ can be described as the universal covering group for $2$-generated central extensions of elementary abelian pro-$p$ groups by $W$, i.e., for $2$-generated pro-$p$ groups $E$ admitting a central elementary abelian subgroup $A$ such that $E/A\cong W$.

Theorem 1.1**.**

For $p>2$, the normal Hausdorff spectra of the pro-$p$ group $G$ constructed above, with respect to the standard filtration series $\mathcal{P}$, $\mathcal{D}$, $\mathcal{F}$ and $\mathcal{L}$ respectively, satisfy:

[TABLE]

In particular, they each contain an infinite real interval.

This solves a problem posed by Shalev [7, Problem 16]. We observe that the normal Hausdorff spectrum of $G$ is sensitive to changes in filtration and that the normal Hausdorff spectrum of $G$ with respect to the Frattini series varies with $p$.

In Section 4 we show that finite direct powers $G\times\ldots\times G$ of the group $G$ provide examples of normal Hausdorff spectra consisting of multiple intervals. Furthermore, the sequence $G\times\overset{m}{\ldots}\times G$, $m\in\mathbb{N}$, has normal Hausdorff spectra ‘converging’ to $[0,1]$; compare Corollary 4.5. We highlight three natural problems.

Problem 1.2**.**

Does there exist a finitely generated pro-$p$ group $H$

1. (a)

with countably infinite normal Hausdorff spectrum $\operatorname{hspec}_{\trianglelefteq}^{\mathcal{S}}(H)$, 2. (b)

with full normal Hausdorff spectrum $\operatorname{hspec}_{\trianglelefteq}^{\mathcal{S}}(H)=[0,1]$, 3. (c)

such that $1$ is not an isolated point in $\operatorname{hspec}_{\trianglelefteq}^{\mathcal{S}}(H)$,

for one or several of the standard series $\mathcal{S}\in\{\mathcal{P},\mathcal{D},\mathcal{F},\mathcal{L}\}$?

We also compute the entire Hausdorff spectra of $G$ with respect to the four standard filtration series, answering en route a question raised in [3, VIII.7.2].

Theorem 1.3**.**

For $p>2$, the Hausdorff spectra of the pro-$p$ group $G$ constructed above, with respect to the standard filtration series, satisfy:

[TABLE]

The qualitative shape of the spectrum $\operatorname{hspec}^{\mathcal{L}}(G)$, i.e., its decomposition into a continuous and a non-continuous, but dense part, is unprecedented and of considerable interest; in Corollary 2.11 we show that already the wreath product $W=C_{p}\mathrel{\hat{\wr}}\mathbb{Z}_{p}$ has a similar Hausdorff spectrum with respect to the lower $p$-series.

Organisation. Section 2 contains preliminary results. In Section 3 we give an explicit presentation of the pro-$p$ group $G$ and describe a series of finite quotients $G_{k}$, $k\in\mathbb{N}$, such that $G=\varprojlim G_{k}$. In Section 4 we provide a general description of the normal Hausdorff spectrum of $G$ and, with respect to certain induced filtration series, we generalise this to finite direct powers of $G$. In Section 5 we compute the normal Hausdorff spectrum of $G$ with respect to the $p$-power series $\mathcal{P}$, and in Section 6 we compute the normal Hausdorff spectra of $G$ with respect to the other three standard filtration series $\mathcal{D},\mathcal{F},\mathcal{L}$. In Section 7 we compute the entire Hausdorff spectra of $G$. Finally, in Appendix A we indicate how our results extend to the case $p=2$.

Notation. Throughout, $p$ denotes an odd prime, although some results hold also for $p=2$, possibly with minor modifications; only in Appendix A we discuss the analogous pro-$2$ groups. We denote by $\varliminf_{i\to\infty}a_{i}$ the lower limit (limes inferior) of a sequence $(a_{i})_{i\in\mathbb{N}}$ in $\mathbb{R}\cup\{\pm\infty\}$. Tacitly, subgroups of profinite groups are generally understood to be closed subgroups. Subscripts are used to emphasise that a subgroup is closed respectively open, as in $H\leq_{\mathrm{c}}G$ respectively $H\leq_{\mathrm{o}}G$. We use left-normed commutators, e.g., $[x,y,z]=[[x,y],z]$.

Acknowledgement. Discussions between Amaia Zugadi-Reizabal and the first author several years ago led to the initial idea that the pro-$p$ group $G$ constructed in this paper should have an infinite normal Hausdorff spectrum with respect to the $p$-power filtration series. At the time Zugadi-Reizabal was supported by the Spanish Government, grant MTM2011-28229-C02-02, partly with FEDER funds, and by the Basque Government, grant IT-460-10. We also thank Yiftach Barnea, Dan Segal and Matteo Vannacci for useful conversations, and the referee for suggesting several improvements to the exposition.

2. Preliminaries

2.1.

Let $G$ be a finitely generated pro-$p$ group. We consider four natural filtration series on $G$. The $p$-power series of $G$ is given by

[TABLE]

The lower $p$-series (or lower $p$-central series) of $G$ is given recursively by

[TABLE]

As a default we set $P_{0}(G)=D_{0}(G)=G$.

2.2.

Next, we collect auxiliary results to detect Hausdorff dimensions of closed subgroups of pro-$p$ groups. For a countably based infinite pro-$p$ group $G$, equipped with a filtration series $\mathcal{S}\colon G=G_{0}\supseteq G_{1}\supseteq\ldots$, and a closed subgroup $H\leq_{\mathrm{c}}G$ we say that $H$ has strong Hausdorff dimension in $G$ with respect to $\mathcal{S}$ if

[TABLE]

is given by a proper limit.

The first lemma is an easy variation of [4, Lem. 5.3] and we omit the proof.

Lemma 2.1**.**

Let $G$ be a countably based infinite pro-$p$ group with closed subgroups $K\leq_{\mathrm{c}}H\leq_{\mathrm{c}}G$. Let $\mathcal{S}\colon G=G_{0}\supseteq G_{1}\supseteq\ldots$ be a filtration series of $G$ and write $\mathcal{S}|_{H}\colon H=H_{0}\supseteq H_{1}\supseteq\ldots$, with $H_{i}=H\cap G_{i}$ for $i\in\mathbb{N}_{0}$, for the induced filtration series of $H$. If $K$ has strong Hausdorff dimension in $H$ with respect to $\mathcal{S}|_{H}$, then

[TABLE]

Lemma 2.2**.**

Let $G$ be a countably based infinite pro-$p$ group with closed subgroups $N\trianglelefteq_{\mathrm{c}}G$ and $H\leq_{\mathrm{c}}G$. Let $\mathcal{S}\colon G=G_{0}\supseteq G_{1}\supseteq\ldots$ be a filtration series of $G$, and consider the induced filtration series of $N$ and $G/N$ defined by

[TABLE]

Suppose that $N$ has strong Hausdorff dimension $\xi=\operatorname{hdim}_{G}^{\mathcal{S}}(N)$ in $G$, with respect to $\mathcal{S}$. Then we have

[TABLE]

Moreover, equality holds in ($\ast$ ‣ 2.2), if $HN/N$ has strong Hausdorff dimension in $G/N$ with respect to $\mathcal{S}|_{G/N}$ or if the lower limit on the right-hand side is actually a limit. Similarly, equality holds in ($\ast\ast$ ‣ 2.2) if

1. (i)

$H\cap N\leq_{\mathrm{o}}N$* is an open subgroup or* 2. (ii)

$G_{i}N=(G_{i}\cap H)N$, for all sufficiently large $i\in\mathbb{N}$.

Proof.

We observe that

[TABLE]

and that, for each $i\in\mathbb{N}_{0}$,

[TABLE]

Finally,

[TABLE]

and, if condition (i) or (ii) holds, the difference between the two terms is bounded by a constant that is independent of $i\in\mathbb{N}_{0}$. ∎

Lemma 2.3**.**

Let $Z\cong C_{p}^{\,\aleph_{0}}$ be a countably based infinite elementary abelian pro-$p$ group, equipped with a filtration series $\mathcal{S}$. Then, for every $\eta\in[0,1]$, there exists a closed subgroup $K\leq_{\mathrm{c}}Z$ with strong Hausdorff dimension $\eta$ in $Z$ with respect to $\mathcal{S}$.

Proof.

Write $\mathcal{S}\colon Z=Z_{0}\supseteq Z_{1}\supseteq Z_{2}\supseteq\ldots$ and let $\eta\in[0,1]$. For $i\in\mathbb{N}$, we have $Z_{i-1}/Z_{i}\cong C_{p}^{\,d_{i}}$ for non-negative integers $d_{i}$.

Claim: There exist non-negative integers $e_{1},e_{2},\ldots$ such that, for each $i\in\mathbb{N}$, we have $0\leq e_{i}\leq d_{i}$ and

[TABLE]

Indeed, with $e_{1}=\lceil\eta d_{1}\rceil$ the statement holds true for $i=1$. Now, let $i\geq 2$ and suppose that $e_{1}+\ldots+e_{i-1}=\lceil\eta(d_{1}+\ldots+d_{i-1})\rceil$. Then

[TABLE]

and thus we may set

[TABLE]

to satisfy the statement for $i$. The claim is proved.

For all sufficiently large $i\in\mathbb{N}$ we have $d_{1}+\ldots+d_{i}>0$ and

[TABLE]

With these preparations, it suffices to display a subgroup $K\leq_{\mathrm{c}}Z$ such that

[TABLE]

For this purpose, we write

[TABLE]

such that $Z_{i-1}=\langle z_{i,1},\ldots,z_{i,d_{i}}\rangle Z_{i}$ for each $i\in\mathbb{N}$. Then we set

[TABLE]

Corollary 2.4**.**

Let $G$ be a countably based pro-$p$ group, equipped with a filtration series $\mathcal{S}$, and let $N\trianglelefteq_{\mathrm{c}}H\leq_{\mathrm{c}}G$ such that $H/N\cong C_{p}^{\,\aleph_{0}}$. Set $\xi=\operatorname{hdim}_{G}^{\mathcal{S}}(N)$ and $\eta=\operatorname{hdim}_{G}^{\mathcal{S}}(H)$. If $N$ or $H$ has strong Hausdorff dimension in $G$ with respect to $\mathcal{S}$, then $[\xi,\eta]\subseteq\operatorname{hspec}^{\mathcal{S}}(G)$.

Proof.

If $N$ has strong Hausdorff dimension, we apply Lemmata 2.1, 2.2 and 2.3. If $H$ has strong Hausdorff dimension the claim follows from [4, Thm. 5.4]. ∎

2.3.

For convenience we recall two standard commutator collection formulae.

Proposition 2.5**.**

Let $G=\langle a,b\rangle$ be a finite $p$-group, and let $r\in{\mathbb{N}}$. For $u,v\in G$ let $K(u,v)$ denote the normal closure in $G$ 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 $1\leq s\leq r$. Then

[TABLE]

Remark.* Under the standing assumption $p\geq 3$ and the extra assumptions*

[TABLE]

the congruences (2.1) and (2.2) simplify to

[TABLE]

where $L(a,b)$ denotes the normal closure in $G$ of all commutators in $\{a,b\}$ of weight at least $p^{r}$ that have weight at least $2$ in $b$ and $M(a,b)$ denotes the normal closure in $G$ of all commutators $[[b,a,\overset{i}{\ldots},a],[b,a,\overset{j}{\ldots},a]]$ with $i+j\geq p^{r}$.

The general result is recorded (in a slighter stronger form) in [5, Prop. 1.1.32]; we remark that (2.2) follows directly from (2.1), due to the identity $[a^{p^{r}},b]=a^{-p^{r}}(a[a,b])^{p^{r}}$. The first congruence in (2.3) follows directly from (2.1); the second congruence in (2.3) is derived from (2.2) by standard commutator manipulations.

2.4.

Now we describe, for $k\in\mathbb{N}$, the lower central series, the lower $p$-series and the Frattini series of the finite wreath product

[TABLE]

with top group $\langle x\rangle\cong C_{p^{k}}$ and base group $\langle y,y^{x},\ldots,y^{x^{p^{k}-1}}\rangle\cong C_{p}^{\,p^{k}}$.

Proposition 2.6**.**

For $k\in\mathbb{N}$, the finite wreath product $W_{k}$ defined above is nilpotent of class $p^{k}$ and $W_{k}^{\,p^{k}}=\langle yy^{x}y^{x^{2}}\cdots y^{x^{p^{k}-1}}\rangle\cong C_{p}$.

1. (1)

The lower central series of $W_{k}$ satisfies

[TABLE] 2. (2)

The lower $p$-series of $W_{k}$ has length $p^{k}$; it satisfies, for $1\leq i\leq k$,

[TABLE] 3. (3)

The Frattini series of $W_{k}$ has length $k+1$; it satisfies, for $0\leq i<k$,

[TABLE] 4. (4)

The dimension subgroup series of $W_{k}$ has length $p^{k}$; in particular, it satisfies, for $p^{k-1}+1\leq i\leq p^{k}$,

[TABLE]

Proof.

The assertions are well known and easy to verify from the concrete realisation of $W_{k}$ as a semidirect product

[TABLE]

in terms of polynomials over the finite field $\mathbb{F}_{p}$: here $y^{x^{i}}$ corresponds to $(1+t)^{i}$ modulo $t^{p^{k}}\mathbb{F}_{p}[t]$, and it is easy to describe all normal subgroups. In particular the normal subgroups of $W_{k}$ contained in the base group form a descending chain, corresponding to the groups $t^{i-1}\mathbb{F}_{p}[t]/t^{p^{k}}\mathbb{F}_{p}[t]$, $1\leq i\leq p^{k}+1$.

For $0\leq m<k$ and $z\in\langle y,y^{x},\ldots,y^{x^{p^{k}-1}}\rangle$ the element

[TABLE]

corresponds in $\mathbb{F}_{p}[t]/t^{p^{k}}\mathbb{F}_{p}[t]$ to a multiple of

[TABLE]

this shows that $W_{k}^{\,p^{k}}=\langle yy^{x}y^{x^{2}}\cdots y^{x^{p^{k}-1}}\rangle\cong C_{p}$.

Clearly, $\gamma_{1}(W_{k})=W_{k}$. For $2\leq i\leq p^{k}+1$, the group $\gamma_{i}(W_{k})$ corresponds to the subgroup $t^{i-1}\mathbb{F}_{p}[t]/t^{p^{k}}\mathbb{F}_{p}[t]$ of the base group. In particular, $W_{k}$ has nilpotency class $p^{k}$. For $1\leq i\leq k$, we have $P_{i}(W_{k})=\langle x^{p^{i-1}}\rangle\gamma_{i}(W_{k})$, while for $k<i\leq p^{k}$ we get $P_{i}(W_{k})=\gamma_{i}(W_{k})$. For $0\leq i\leq k$ a simple induction shows that the group $\Phi_{i}(W_{k})$ is the normal closure in $W_{k}$ of the two elements

[TABLE]

the intersection of $\Phi_{i}(W_{k})$ with the base group corresponds to $t^{\frac{p^{i}-1}{p-1}}\mathbb{F}_{p}[t]/t^{p^{k}}\mathbb{F}_{p}[t]$. Thus $\Phi_{i}(W_{k})=\langle x^{p^{i}}\rangle\gamma_{\frac{p^{i}-1}{p-1}+1}(W_{k})$. In particular, $\Phi_{k}(W_{k})$ is elementary abelian and $\Phi_{i}(W_{k})=1$ for $i>k$. Finally, for $i\geq p^{k-1}+1$, we use [2, Thm. 11.2] to deduce that $D_{i}(W_{k})=\gamma_{i}(W_{k})$. ∎

The structural results for the finite wreath products $W_{k}$ transfer naturally to the inverse limit $W\cong\varprojlim_{k}W_{k}$, i.e., the pro-$p$ wreath product

[TABLE]

with top group $\langle x\rangle\cong\mathbb{Z}_{p}$ and base group $B=\prod_{i\in\mathbb{Z}}\langle y^{x^{i}}\rangle\cong C_{p}^{\,\aleph_{0}}$. Compatible with (2.4), the group $W$ has a concrete realisation as a semidirect product

[TABLE]

in terms of formal power series over the finite field $\mathbb{F}_{p}$. We record the following lemma on closed normal subgroups of $W$.

Lemma 2.7**.**

Let $W=\langle x,y\rangle\cong C_{p}\mathrel{\hat{\wr}}\mathbb{Z}_{p}$ with base group $B$ as above, and let $1\neq K\trianglelefteq_{\mathrm{c}}W$ be a non-trivial closed normal subgroup. Then either $K$ is open in $W$ or $K$ is open in $B$; in particular, $K\cap B\leq_{\mathrm{o}}B$ and $\lvert K\cap B:[K\cap B,W]\rvert=p$.

Proof.

The lower central series of $W$ is well known and easy to compute: $\gamma_{1}(W)=W$ and $\gamma_{i}(W)=B_{i-1}$ for $i\geq 2$, where $B=B_{0}\geq B_{1}\geq B_{2}\geq\ldots$ with $B_{i-1}=\langle[y,x,\overset{i-1}{\ldots},x]\rangle B_{i}$ and $\lvert B_{i-1}:B_{i}\rvert=p$; in other words, $\langle x\rangle$ acts uniserially on $B$; compare Proposition 2.6.

It follows that $1\neq K\cap B=B_{i}$ for some non-negative integer $i$, hence $K\cap B\leq_{\mathrm{o}}B$ and $\lvert K\cap B:[K\cap B,W]\rvert=\lvert B_{i}:B_{i+1}\rvert=p$. Suppose now that $K\not\subseteq B$. Then there exists $x^{m}z\in K$ with $m\in\mathbb{N}$ and $z\in B$. We may assume that $m=p^{k}$ is a $p$-power. Then $\langle x^{p^{k}}z\rangle\ltimes B\cong\langle x\rangle\ltimes(B\times\overset{p^{k}}{\ldots}\times B)$, where $x^{p^{k}}z$ maps to $x$ and, on the right-hand side, $x$ acts diagonally and in each coordinate according to the original action in $W$. Hence we may assume that $x\in K$. Now the description of the lower central series of $W$ yields $\langle x\rangle B_{1}\leq_{\mathrm{c}}K$ and thus $K\leq_{\mathrm{o}}W$. ∎

From Proposition 2.6 and Lemma 2.7 we deduce the following; cf. [3, Ch. VIII.7].

Corollary 2.8**.**

The normal Hausdorff spectrum of the pro-$p$ group $W=C_{p}\mathrel{\hat{\wr}}\mathbb{Z}_{p}$ with respect to the standard filtration series $\mathcal{P}$, $\mathcal{D}$, $\mathcal{F}$ and $\mathcal{L}$ respectively, satisfies:

[TABLE]

The next result is well known (and not difficult to prove directly); compare [9, Cor. 12.5.10]. It gives a first indication that Theorem 1.1 is at least plausible.

Proposition 2.9**.**

The pro-$p$ group $W=C_{p}\mathrel{\hat{\wr}}\mathbb{Z}_{p}$ is not finitely presented.

The final result in this section concerns the finitely generated Hausdorff spectrum of the pro-$p$ group $W=C_{p}\mathrel{\hat{\wr}}\mathbb{Z}_{p}$, with respect to a standard filtration series $\mathcal{S}$; it is defined as

[TABLE]

and reflects the range of Hausdorff dimensions of (topologically) finitely generated subgroups; compare [7, §4.7].

Theorem 2.10**.**

With respect to the standard filtration series $\mathcal{P}$, $\mathcal{D}$, $\mathcal{F}$ and $\mathcal{L}$ respectively, the pro-$p$ group $W=C_{p}\mathrel{\hat{\wr}}\mathbb{Z}_{p}$ satisfies:

[TABLE]

Proof.

As above, let $B$ denote the base group of the wreath product $W=\langle x,y\rangle$. Let $\mathcal{S}\in\{\mathcal{P},\mathcal{D},\mathcal{F},\mathcal{L}\}$, and let $K$ be a finitely generated subgroup of $W$.

If $K\subseteq B$ then $K$ is finite and $\operatorname{hdim}^{\mathcal{S}}_{W}(K)=0$. Now suppose that $K\not\subseteq B$; in the proof below we will no longer use that $K$ is finitely generated, but it will become clear that this is automatically so. Write $K=\langle x^{p^{n}}z\rangle\ltimes M$, where $n\in\mathbb{N}_{0}$, $z\in B$ and $M=K\cap B$. Let $B=B_{0}\geq B_{1}\geq\ldots$ be the filtration corresponding to $\mathbb{F}_{p}[\![t]\!]\geq t\mathbb{F}_{p}[\![t]\!]\geq\ldots$ under  (2.6), as in the proof of Lemma 2.7. We set

[TABLE]

Under the isomorphism (2.6), we may regard $M$ as an $\mathbb{F}_{p}[\![t^{p^{n}}]\!]$-submodule of $\mathbb{F}_{p}[\![t]\!]$. Hence $J+p^{n}\mathbb{N}_{0}\subseteq J$ and

[TABLE]

From Proposition 2.6 it is easily seen that $B$ has strong Hausdorff dimension

[TABLE]

compare Corollary 2.8. Using Lemma 2.2, we deduce that

[TABLE]

lies in the desired range; in fact, the argument even shows that $K$ has strong Hausdorff dimension.

Conversely, our analysis above shows that, for $n\in\mathbb{N}_{0}$ and $0\leq m\leq p^{n}$, the subgroup $K_{n,m}=\langle x^{p^{n}},[y,x],[y,x,x],\ldots,[y,x,\overset{m}{\ldots},x]\rangle$ has Hausdorff dimension

[TABLE]

The next corollary answers a question raised in [3, VIII.7.2]; it was shown there that $[0,\nicefrac{{1}}{{2}}]\subseteq\operatorname{hspec}^{\mathcal{L}}(W)$, while $(\nicefrac{{1}}{{2}},1)\,\cap\,\operatorname{hspec}^{\mathcal{L}}(W)$ remained undetermined.

Corollary 2.11**.**

The Hausdorff spectrum of the pro-$p$ group $W=C_{p}\mathrel{\hat{\wr}}\mathbb{Z}_{p}$ with respect to the lower $p$-series $\mathcal{L}$ is

[TABLE]

Furthermore, every subgroup $K\leq_{\mathrm{c}}W$ with $\operatorname{hdim}^{\mathcal{L}}_{W}(K)>\nicefrac{{1}}{{2}}$ has strong Hausdorff dimension in $W$, with respect to $\mathcal{L}$.

Proof.

The subgroups contained in the base group $B$ of $W$ yield $[0,\nicefrac{{1}}{{2}}]$ as part of the Hausdorff spectrum; cf. Lemma 2.3. The proof of Theorem 2.10 shows that the subgroups not contained in $B$ yield the remaining part of the claimed spectrum and that each of them has strong Hausdorff dimension in $W$. ∎

3. An explicit presentation for the pro-$p$ group $G$ and

a description of its finite quotients $G_{k}$ for $k\in\mathbb{N}$

Recall that $p$ is an odd prime. As indicated in the paragraph before Theorem 1.1, we consider the pro-$p$ group $G=F/N$, where

• •

$F=\langle x,y\rangle$ is a free pro-$p$ group and

• •

$N=[R,F]R^{p}\trianglelefteq_{\mathrm{c}}F$ for the kernel $R\trianglelefteq_{\mathrm{c}}F$ of the presentation $\pi\colon F\to W$ sending $x,y$ to the generators of the same name in (2.5).

By producing generators for $R$ and $N$ as closed normal subgroups of $F$ we obtain explicit presentations for the pro-$p$ groups $W$ and $G$.

It is convenient to write $y_{i}=y^{x^{i}}$ for $i\in\mathbb{Z}$. Setting

[TABLE]

for $k\in\mathbb{N}$, we obtain a descending chain of open normal subgroups

[TABLE]

with quotient groups $F/R_{k}\cong W_{k}\cong C_{p}\mathrel{\wr}C_{p^{k}}$. We put

[TABLE]

and observe that $R_{k}=\langle x^{p^{k}}\rangle^{F}\,R$ for each $k\in\mathbb{N}$. Since $x^{p^{k}}\to 1$ as $k\to\infty$, this yields $R=\bigcap_{k\in\mathbb{N}}R_{k}$ and thus $F/R\cong W\cong C_{p}\mathrel{\hat{\wr}}\mathbb{Z}_{p}$. With hindsight there is no harm in taking $W_{k}=F/R_{k}$ for $k\in\mathbb{N}$ and $W=F/R$.

Setting $N_{k}=[R_{k},F]R_{k}^{\,p}$ for $k\in\mathbb{N}$, we observe that

[TABLE]

and as in (3.2) we obtain a descending chain $F\supseteq N_{1}\supseteq N_{2}\supseteq\ldots$ of open normal subgroups. Moreover, it follows that $\bigcap_{k\in\mathbb{N}}N_{k}\supseteq[R,F]R^{p}=N$. On the other hand, if $z\not\in N$ then there exists an open normal subgroup $K\trianglelefteq_{\mathrm{o}}F$ and $k\in\mathbb{N}$ such that $z\not\in NK=[R_{k},F]R_{k}^{\,p}K$, hence $z\not\in N_{k}$. Thus we conclude that

[TABLE]

Consequently, $G=F/N\cong\varprojlim G_{k}$, where

[TABLE]

for $k\in\mathbb{N}$, and

[TABLE]

is a presentation of $G$ as a pro-$p$ group. Indeed,

[TABLE]

satisfies, for each $k\in\mathbb{N}$,

[TABLE]

where $x^{p^{k+1}},[x^{p^{k}},y]\to 1$ as $k\to\infty$. This yields $\widetilde{N}=\bigcap_{k\in{\mathbb{N}}}N_{k}=N$. To facilitate later use, we have underlined the two relations in (3.3) that do not yet occur in (3.4).

To summarise and supplement some of the notation introduced above, we define

[TABLE]

Diagrammatically, we have:

[TABLE]

[TABLE]

Lemma 3.1**.**

The centre of $G$ is $Z(G)=Z$, and $Z_{k}\leq Z(G_{k})$ for $k\in\mathbb{N}$.

Proof.

By construction, $Z\leq Z(G)$ and $Z_{k}\leq Z(G_{k})$ for $k\in\mathbb{N}$. From (2.6) we see that $G/Z\cong W$ has trivial centre. Therefore $Z=Z(G)$. ∎

In fact, $Z_{k}\lneqq Z(G_{k})$ for $k\in\mathbb{N}$; see Lemma 5.3 below.

4. General description of the normal Hausdorff spectrum of the

pro-$p$ group $G$ and its finite direct powers

We continue to use the notation set up in Section 3 to study the pro-$p$ group $G$ and its finite direct powers.

Proposition 4.1**.**

Let $K\trianglelefteq_{\mathrm{c}}G$ be a closed normal subgroup such that $K\not\subseteq Z$. Then either $K$ is open in $H$ or $K$ is open in $G$; in particular, $K\cap H\leq_{\mathrm{o}}H$. Furthermore, $[K\cap H,G]\leq_{\mathrm{o}}H$.

Proof.

Lemma 2.7 shows: $KZ\cap H\leq_{\mathrm{o}}H$; hence it suffices to prove $K\cap Z\leq_{\mathrm{o}}Z$. Choose $\hat{y}_{1},\hat{y}_{2},\ldots\in H$, converging to $1$ modulo $Z$, and $m\in\mathbb{N}$ such that (the images of) $\hat{y}_{1},\hat{y}_{2},\ldots$ (modulo $Z$) yield a basis for the elementary abelian pro-$p$ group $H/Z$ and $\hat{y}_{m+1},\hat{y}_{m+2},\ldots$ generate $KZ\cap H$ modulo $Z$.

Recall that $Z$ is central in $G$ and of exponent $p$. Thus $K\cap Z$ contains $\hat{y}_{i}^{\,p}$ and $[\hat{y}_{i},\hat{y}_{j}]$ for all $i,j\in\mathbb{N}$ with $i>m$. Hence the finite set

[TABLE]

generates the elementary abelian group $Z$ modulo $K\cap Z$, and $K\cap Z\leq_{\mathrm{o}}Z$.

Finally, Lemma 2.7 implies that $[K\cap H,G]\not\subseteq Z$. Hence $[K\cap H,G]\leq_{\mathrm{o}}H$. ∎

From Proposition 4.1, Lemma 3.1 and Lemmata 2.1 and 2.3 we deduce the general shape of the normal Hausdorff spectrum of $G$.

Corollary 4.2**.**

Let $\mathcal{S}$ be an arbitrary filtration series of $G$. Then the normal Hausdorff spectrum of $G$ has the form

[TABLE]

where $\xi=\operatorname{hdim}^{\mathcal{S}}_{G}(Z)$ and $\eta=\operatorname{hdim}^{\mathcal{S}}_{G}(H)$.

More generally we obtain a description of the normal Hausdorff spectrum of finite direct powers $G^{(m)}=G\times\overset{m}{\ldots}\times G$ of $G$, with respect to suitable ‘product filtration series’. For any filtration series $\mathcal{S}\colon G=S_{0}\supseteq S_{1}\ldots$ of $G$ we consider the naturally induced product filtration series on $G^{(m)}$ given by

[TABLE]

For a standard filtration series $\mathcal{S}\in\{\mathcal{P},\mathcal{L},\mathcal{F},\mathcal{D}\}$ on $G$ the product filtration series $\mathcal{S}^{(m)}$ is actually the corresponding standard filtration series on $G^{(m)}$.

Corollary 4.3**.**

Let $m\in\mathbb{N}$, and let $K\trianglelefteq_{\mathrm{c}}G^{(m)}$. For $1\leq j\leq m$, let $\pi_{j}\colon G^{(m)}\to G$ be the canonical projection onto the $j$th factor and set

[TABLE]

Then $K\leq\prod_{j=1}^{m}\overline{K}(j)$ and $K$ contains an open normal subgroup of $\prod_{j=1}^{m}\underline{K}(j)$.

Proof.

Observe that

[TABLE]

Thus $K$ is contained in $\prod_{j=1}^{m}\overline{K}(j)$, and it suffices to show that $[K\pi_{j}\cap H,G]\leq_{\mathrm{o}}H$ for each $j$ with $K\pi_{j}\not\subseteq Z$. This follows by Proposition 4.1. ∎

Corollary 4.4**.**

Let $m\in\mathbb{N}$, and let $\mathcal{S}$ be a filtration series of $G$ such that $\operatorname{hdim}^{\mathcal{S}}_{G}(H)=1$. Then the normal Hausdorff spectrum of $G^{(m)}$ has the form

[TABLE]

where $\xi=\operatorname{hdim}^{\mathcal{S}}_{G}(Z)$.

Proof.

First let $K\trianglelefteq_{\mathrm{c}}G^{(m)}$, and define $\overline{K}(j),\underline{K}(j)$ for $1\leq j\leq m$ as in Corollary 4.3. From $\operatorname{hdim}^{\mathcal{S}}_{G}(H)=1$ we deduce that

[TABLE]

where $l=\#\{j\mid 1\leq j\leq m\text{ and }\overline{K}(j)=G\}$.

Conversely, for every $l\in\{0,1,\ldots,m\}$ and \beta\in\big{[}\nicefrac{{l}}{{m}},\nicefrac{{l+(m-l)\xi}}{{m}}\big{]} there is a normal subgroup

[TABLE]

where $U\leq_{\mathrm{c}}Z$ for $l<m$ has $\operatorname{hdim}^{\mathcal{S}}_{G}(U)=\nicefrac{{m}}{{m-l}}\,(\beta-\nicefrac{{l}}{{m}})\in[0,\xi]$; compare Corollary 4.2. This yields $\beta=\operatorname{hdim}^{\mathcal{S}^{(m)}}_{G^{(m)}}(K_{\beta})\in\operatorname{hspec}^{\mathcal{S}^{(m)}}_{\trianglelefteq}(G^{(m)}).$ ∎

Corollary 4.4 shows that, once $\operatorname{hdim}^{\mathcal{S}}_{G}(H)=1$, the general shape (e.g. the number of connected components) of the normal Hausdorff spectrum $\operatorname{hspec}^{\mathcal{S}^{(m)}}_{\trianglelefteq}(G^{(m)})$ depends only on the parameters $\xi=\operatorname{hdim}^{\mathcal{S}}_{G}(Z)$ and $m\in\mathbb{N}$. For instance, if $\xi<\nicefrac{{1}}{{m}}$, then $\operatorname{hspec}^{\mathcal{S}^{(m)}}_{\trianglelefteq}(G^{(m)})$ is the union of $m+1$ disjoint intervals, whereas for $\xi\geq\nicefrac{{1}}{{2}}$ we obtain $\operatorname{hspec}^{\mathcal{S}^{(m)}}_{\trianglelefteq}(G^{(m)})=[0,1-\nicefrac{{(1-\xi)}}{{m}}]\cup\{1\}$.

The proof of Theorem 1.1 in Sections 5 and 6 will give $\operatorname{hdim}^{\mathcal{S}}_{G}(H)=1$ for the standard filtrations $\mathcal{S}\in\{\mathcal{P},\mathcal{D},\mathcal{F}\}$ and $\xi=\operatorname{hdim}^{\mathcal{P}}_{G}(Z)=\operatorname{hdim}^{\mathcal{D}}_{G}(Z)=\nicefrac{{1}}{{3}}$ respectively $\xi=\operatorname{hdim}^{\mathcal{F}}_{G}(Z)=\nicefrac{{1}}{{p+1}}$; the assertion for $H$ is already a consequence of [4, Prop. 4.2]. We formulate a taylor-made corollary for these situations.

Corollary 4.5**.**

Let $m,n\in\mathbb{N}$ with $m\geq\max\{2,n-1\}$ and $n\geq 2$. Let $\mathcal{S}$ be a filtration series of $G$ such that $\operatorname{hdim}^{\mathcal{S}}_{G}(H)=1$ and $\operatorname{hdim}^{\mathcal{S}}_{G}(Z)=\nicefrac{{1}}{{n}}$. Then

[TABLE]

consists of $n$ disjoint intervals.

Proof.

From Corollary 4.4, we have

[TABLE]

For $m-n+1\leq l\leq m-1$ it is easy to verify that

[TABLE]

Hence it suffices to show that

[TABLE]

For $m=n-1$ this reduces to \big{[}0,\nicefrac{{1}}{{n}}\big{]}=\big{[}0,\tfrac{mn-(n-1)^{2}}{mn}\big{]}. Now suppose that $m\geq n$. Then the claim follows from

[TABLE]

5. The normal Hausdorff spectrum of $G$ with respect to

the $p$-power series

We continue to use the notation set up in Section 3 and establish that $\xi=\operatorname{hdim}^{\mathcal{P}}_{G}(Z)=\nicefrac{{1}}{{3}}$ and $\eta=\operatorname{hdim}^{\mathcal{P}}_{G}(H)=1$, with respect to the $p$-power series $\mathcal{P}$. In view of Corollary 4.2 this proves Theorem 1.1 for the $p$-power series. Indeed, $\operatorname{hdim}^{\mathcal{P}}_{G}(H)=1$ is already a consequence of [4, Prop. 4.2]. It remains to show that

[TABLE]

It is convenient to work with the finite quotients $G_{k}$, $k\in\mathbb{N}$, introduced in Section 3. Let $k\in\mathbb{N}$. From (3.3) and (3.4) we observe that

[TABLE]

Heuristically, $G_{k}^{\,p^{k}}$ is almost trivial (see Proposition 5.2) and the elementary abelian $p$-group $Z_{k}$ requires roughly half the number of generators compared to the elementary abelian $p$-group $H_{k}/Z_{k}$. This suggests that (3.4) should be true. We now work out the details.

First we compute the order of $G_{k}$, using the notation from Section 3.

Lemma 5.1**.**

The logarithmic order of $G_{k}$ is

[TABLE]

In particular,

[TABLE]

Proof.

Observe from $F/R_{k}\cong W_{k}\cong C_{p}\mathrel{\wr}C_{p^{k}}$ that

[TABLE]

By construction, $R_{k}/N_{k}$ is elementary abelian of exponent $p$. Moreover, (3.1) shows that $\{x^{p^{k}},y^{p}\}\cup\{[y_{0},y_{i}]\mid 1\leq i\leq(p^{k}-1)/2\}$ generates $R_{k}$ modulo $N_{k}$. In order to prove that the generators are independent, we construct a factor group $\widetilde{G}_{k}$ of $G_{k}$ that has the maximal possible logarithmic order $\log_{p}\lvert\widetilde{G}_{k}\rvert=p^{k}+k+2+(p^{k}-1)/2$.

Consider the finite $p$-group

[TABLE]

where $E$ is a free pro-$p$ group on $p^{k}$ generators with Frattini subgroup $\Phi(E)=[E,E]E^{p}$. Then the images of $\widetilde{y}_{0},\ldots,\widetilde{y}_{p^{k}-1}$ generate independently the elementary abelian quotient $M/\Phi(M)$ and the commutators $[\widetilde{y}_{i},\widetilde{y}_{j}]$, for $0\leq i<j\leq p^{k}-1$, together with the $p$th powers $\widetilde{y}_{0}^{\;p},\ldots,\widetilde{y}_{p^{k}-1}^{\;p}$ generate independently the elementary abelian group $\Phi(M)$. The latter can be checked by considering homomorphisms from $M$ onto groups of the form $C_{p}^{\,p^{k}-1}\times C_{p^{2}}$ and $C_{p}^{\,p^{k}-2}\times\mathrm{Heis}(\mathbb{F}_{p})$, where $\mathrm{Heis}(\mathbb{F}_{p})$ denotes the group of upper unitriangular $3\times 3$ matrices over $\mathbb{F}_{p}$. Next consider the action of the cyclic group $X=\langle\widetilde{x}\rangle\cong C_{p^{k+1}}$, with kernel $\langle\widetilde{x}^{p^{k}}\rangle\cong C_{p}$, on $M$ that is induced by

[TABLE]

This induces a permutation action on our chosen basis for the elementary abelian group $\Phi(M)$; the orbits are given by

[TABLE]

We define $\widetilde{M}=M/[\Phi(M),X]$ and, for simplicity, continue to write $\widetilde{y}_{0},\ldots,\widetilde{y}_{p^{k}-1}$ for the images of these elements in $\widetilde{M}$. Then

1. $\circ$

the images of $\widetilde{y}_{0},\ldots,\widetilde{y}_{p^{k}-1}$ generate independently the elementary abelian quotient $\widetilde{M}/\Phi(\widetilde{M})$ and 2. $\circ$

the elements $[\widetilde{y}_{0},\widetilde{y}_{i}]$, for $1\leq i\leq(p^{k}-1)/2$, together with $\widetilde{y}_{0}^{\;p}$ generate independently the elementary abelian group $\Phi(\widetilde{M})$.

In particular, this yields $\log_{p}\lvert\widetilde{M}\rvert=p^{k}+(p^{k}-1)/2+1$.

Finally, we put $\widetilde{y}=\widetilde{y}_{0}$ and form the semidirect product

[TABLE]

with the induced action. Upon replacing $x,y$ by $\widetilde{x},\widetilde{y}$, we see that all the defining relations of $G_{k}$ in (3.3) are valid in $\widetilde{G}_{k}$. Since $\log_{p}\lvert G_{k}\rvert\leq p^{k}+k+2+(p^{k}-1)/2=\log_{p}\lvert\widetilde{G}_{k}\rvert$, we conclude that $G_{k}\cong\widetilde{G}_{k}$. ∎

Our next aim is to prove the following structural result.

Proposition 5.2**.**

In the set-up from Section 3, for $k\geq 2$, the subgroup $G_{k}^{\,p^{k}}\leq G_{k}$ is elementary abelian and central in $G_{k}$; it is generated independently by $x^{p^{k}}$, $w=y_{p^{k}-1}\cdots y_{1}y_{0}$ and $v=w\cdot y_{p^{k}-1}^{-1}\cdots y_{1}^{-1}y_{0}^{-1}$.

Consequently

[TABLE]

and

[TABLE]

The proof requires a series of lemmata.

Lemma 5.3**.**

The elements

[TABLE]

are of order $p$ in $G_{k}$ and lie in $G_{k}^{\,p^{k}}\cap Z(G_{k})$.

Proof.

Recall that $H_{k}=\langle y_{0},y_{1},\ldots,y_{p^{k}-1}\rangle Z_{k}\leq G_{k}$ and observe that $[H_{k},H_{k}]$ is a central subgroup of exponent $p$ in $G_{k}$. Furthermore, $[y^{p},x]=1$ implies $y_{p^{k}-1}^{\,p}=\ldots=y_{0}^{\,p}$ in $G_{k}$. Thus (2.1) yields

[TABLE]

As $w\neq 1$ we deduce that $w$ has order $p$. Likewise one shows that $w^{\prime}$ has order $p$.

Clearly, $w=x^{-p^{k}}(xy)^{p^{k}}$ and $w^{\prime}=x^{-p^{k}}(xy^{-1})^{p^{k}}$ lie in $G_{k}^{\,p^{k}}$. In order to prove that $w$ is central, it suffices to check that $w$ commutes with the generators $x$ and $y$ of $G_{k}$. First we observe that, for $1\leq i\leq p^{k}-1$, the relation $[y_{0},y_{i},x]=1$ implies

[TABLE]

Since $[H_{k},H_{k}]$ is central in $G_{k}$, we deduce inductively that

[TABLE]

Likewise, using the relation $[y_{0},y_{i},y]=1$ and (5.2), we obtain

[TABLE]

A similar computation can be carried out for $w^{\prime}$. ∎

Lemma 5.4**.**

Putting

[TABLE]

the subgroup $\langle x^{p^{k}},w,v\rangle\leq G_{k}$ is isomorphic to $C_{p}\times C_{p}\times C_{p}$ and lies in $G_{k}^{\,p^{k}}\cap Z(G_{k})$.

Proof.

From the presentation (3.3) and from Lemma 5.3 it is clear that the subgroup $\langle x^{p^{k}},w,v\rangle\leq G_{k}$ is elementary abelian and lies in $G_{k}^{\,p^{k}}\cap Z(G_{k})$. Furthermore, in order to prove that $\langle x^{p^{k}},w,v\rangle\cong C_{p}\times C_{p}\times C_{p}$, it suffices to establish that $v\neq 1$.

Upon a similar rearrangement and cancellation as in the proof of Lemma 5.3, we obtain

[TABLE]

Recall that all commutators appearing in the above product are central in $G_{k}$. In particular, we have $[y_{0},y_{p^{k}-j}]=[y_{0},y_{p^{k}-j}]^{x^{i}}=[y_{i},y_{p^{k}-j+i}]$, for $1\leq j\leq p^{k}-1$ and $1\leq i\leq j-1$. This gives

[TABLE]

Taking note of the second statement in Lemma 5.1, it follows that $v\neq 1$. ∎

Lemma 5.5**.**

The group $\gamma_{2}(G_{k})\leq G_{k}$ has exponent $p$.

Proof.

Recall that $H_{k}=\langle y_{0},y_{1},\ldots,y_{p^{k}-1}\rangle Z_{k}\leq G_{k}$ satisfies: $[H_{k},H_{k}]$ is a central subgroup of exponent $p$ in $G_{k}$. Since $p$ is odd, (2.1) shows that it suffices to prove that $[y,x]$ has order $p$. But $[y,x]=y_{0}^{\,-1}y_{1}$; thus (2.1) and $y_{0}^{\,p}=x^{-1}y_{0}^{\,p}x=y_{1}^{\,p}$ imply $[y,x]^{p}=y_{0}^{\,-p}y_{1}^{\,p}=1$. ∎

Lemma 5.6**.**

The group $G_{k}$ has nilpotency class $p^{k}$, and $\gamma_{m}(G_{k})/\gamma_{m+1}(G_{k})$ is elementary abelian of rank at most $2$ for $2\leq m\leq p^{k}$.

Proof.

Let $2\leq m\leq p^{k}$. Since $G_{k}$ is a central extension of $Z_{k}$ by $W_{k}$, we deduce from Proposition 2.6 that

[TABLE]

and Lemma 5.5 shows that $\gamma_{m}(G_{k})/\gamma_{m+1}(G_{k})$ is elementary abelian of rank at most $2$. Again by Proposition 2.6, the nilpotency class of $G_{k}$ is at least $p^{k}$. Moreover, $\gamma_{p^{k}}(G_{k})Z_{k}=\langle w\rangle Z_{k}$, where $w\in Z(G_{k})$ by Lemma 5.3. We conclude that $G_{k}$ has nilpotency class precisely $p^{k}$. ∎

Lemma 5.7**.**

The group $G_{k}$ satisfies

[TABLE]

Proof.

Recall that $H_{k}=\langle y_{0},y_{1},\ldots,y_{p^{k}-1}\rangle Z_{k}\leq G_{k}$ has exponent $p^{2}$, and observe that Proposition 2.5 together with Lemma 5.5 yields $H_{k}^{\,p}=\langle y^{p}\rangle$. Every element $g\in G$ is of the form $g=x^{m}h$, with $0\leq m<p^{k+1}$ and $h\in H_{k}$. Using (2.3), based on Proposition 2.5 and Lemma 5.5, we conclude that

[TABLE]

and for $j\geq 2$,

[TABLE]

Proof of Proposition 5.2.

Apply Lemmata 5.4, 5.6 and 5.7. ∎

From Lemma 5.1 and Proposition 5.2 we deduce that

[TABLE]

On the other hand, we observe from Proposition 2.6 that

[TABLE]

hence

[TABLE]

Thus (5.1) follows from

[TABLE]

Remark 5.8**.**

In the literature, one sometimes encounters a variant of the $p$-power series, the iterated $p$-power series of $G$ which is recursively given by

[TABLE]

By a small modification of the proof of Lemma 5.7 we obtain inductively

[TABLE]

based on the commutator identities (2.3) for $r=1$. With Proposition 5.2 and Lemma 5.6 this yields $G_{k}^{\,p^{k}}\subseteq I_{k}(G_{k})\subseteq\langle x^{p^{k}}\rangle\gamma_{p^{k}}(G_{k})=G_{k}^{\,p^{k}}$. We conclude that the $p$-power series $\mathcal{P}$ and the iterated $p$-power series $\mathcal{I}$ of $G$ coincide.

One may further note another natural filtration series $\mathcal{N}\colon N_{i}$, $i\in\mathbb{N}_{0}$, of $G$, consisting of the open normal subgroups defined in Section 3, where we set $N_{0}=G$. As $N_{i}\leq G^{p^{i}}$ with $\log_{p}\lvert G^{p^{i}}:N_{i}\rvert\leq 4$ for all $i\in\mathbb{N}_{0}$, we see that the filtration series $\mathcal{P}$ and $\mathcal{N}$ induce the same Hausdorff dimension function on $G$.

6. The normal Hausdorff spectra of $G$ with respect to the

lower $p$-series, the dimension subgroup series and the Frattini series

We continue to use the notation set up in Section 3 and work with the finite quotients $G_{k}$, $k\in\mathbb{N}$, of the pro-$p$ group $G$. Our aim is to pin down the lower central series, the lower $p$-series, the dimension subgroup series and the Frattini series of $G_{k}$. Subsequently, it will be easy to complete the proof of Theorem 1.1.

Proposition 6.1**.**

The group $G_{k}$ is nilpotent of class $p^{k}$; its lower central series satisfies

[TABLE]

and, for $1\leq i\leq(p^{k}-1)/2$,

[TABLE]

with

[TABLE]

Proof.

By Lemma 5.6 the nilpotency class of $G_{k}$ is $p^{k}$. From $G_{k}=\langle x,y\rangle$ it is clear that $\gamma_{2}(G_{k})=\langle[x,y]\rangle\gamma_{3}(G_{k})$, and (3.3) gives $G_{k}/\gamma_{2}(G_{k})\cong C_{p^{k+1}}\times C_{p^{2}}$. From Lemma 5.1 we know that

[TABLE]

and the proof of Lemma 5.6 shows that

[TABLE]

Consequently, it suffices to prove that $[y,x,\overset{m-2}{\ldots},x,y]\in\gamma_{m+1}(G_{k})$ whenever $m$ is even. More generally, we consider the elements

[TABLE]

Writing $e_{i}=[y_{0},y_{i}]\in Z_{k}\subseteq Z(G_{k})$ for $i\in\mathbb{Z}$, we recall from Lemma 5.1 that

[TABLE]

Induction on $m$ shows that

[TABLE]

and we deduce that

[TABLE]

The identities

[TABLE]

imply that

[TABLE]

Now suppose that $m$ is even, and recall that $p\neq 2$. From (6.2) we obtain inductively $[y,x,\overset{m-2}{\ldots},x,y]=b_{0,m}\equiv b_{j_{0},m}$ modulo $\gamma_{m+1}(G_{k})$ for

[TABLE]

Consequently, it suffices to prove that $b_{j_{0},m}\in\gamma_{m+1}(G_{k})$. First suppose that $p^{k}+1\equiv_{4}m$ and hence $j_{0}=\frac{p^{k}+1}{2}-\frac{m}{2}$. From (6.1) and (5.2) we see that

[TABLE]

and similarly

[TABLE]

Hence $b_{j_{0},m}^{\,2}=b_{j_{0},m+1}^{\,-1}\in\gamma_{m+1}(G_{k})$, and $p\neq 2$ implies $b_{j_{0},m}\in\gamma_{m+1}(G_{k})$.

In the remaining case $p^{k}+1\equiv_{4}m+2$ we have $j_{0}=\frac{p^{k}+3}{2}-\frac{m}{2}$, and a slight variation of the argument above shows that $b_{j_{0},m}^{\,2}=b_{j_{0}-1,m+1}$, hence $b_{j_{0},m}\in\gamma_{m+1}(G_{k})$. ∎

Corollary 6.2**.**

For $2\leq m\leq p^{k}$ and $\nu(m)=\lfloor\frac{1}{2}(p^{k}-m+2)\rfloor$, we have

[TABLE]

and $\gamma_{m}(G_{k})\cap Z(G_{k})=\langle[y,x,\overset{p^{k}-1}{\ldots},x]\rangle\times(\gamma_{m}(G_{k})\cap Z_{k})\cong C_{p}^{\,\nu(m)+1}$. In particular,

[TABLE]

Proof.

Clearly, all non-trivial elements of the form $[y,x,\ldots,x,y]$ are central and of order $p$. By Proposition 6.1 and Lemma 5.5, also $[y,x,\overset{p^{k}-1}{\ldots},x]$ is central and of order $p$. Moreover, Proposition 6.1 shows that every $g\in\gamma_{2}(G_{k})$ can be written as

[TABLE]

where $\alpha(i),\beta(j)\in\{0,1,\ldots,p-1\}$ are uniquely determined by $g$. Furthermore, $g$ is central if and only if $\alpha(i)=0$ for $1\leq i\leq p^{k}-2$, and $g\in Z_{k}$ if and only if $\alpha(i)=0$ for $1\leq i\leq p^{k}-1$. ∎

Corollary 6.3**.**

The lower $p$-series of $G_{k}$ has length $p^{k}$ and satisfies:

[TABLE]

and, for $3\leq i\leq p^{k}$, the $i$th term is $P_{i}(G_{k})=\langle x^{p^{i-1}}\rangle\gamma_{i}(G_{k})$ so that

[TABLE]

with

[TABLE]

Proof.

The descriptions of $G_{k}/P_{2}(G_{k})$ and $P_{2}(G_{k})/P_{3}(G_{k})$ are straightforward. Let $i\geq 3$. Clearly, $P_{i}(G_{k})\supseteq\langle x^{p^{i-1}}\rangle\gamma_{i}(G_{k})$. In view of Proposition 6.1, it suffices to prove that $x^{p^{i-1}}$ is central modulo $\gamma_{i+1}(G_{k})$. Indeed, from Lemma 5.5 and Proposition 2.5 (recall that $p>2$) we obtain

[TABLE]

Corollary 6.4**.**

The dimension subgroup series of $G_{k}$ has length $p^{k}$. For $1\leq i\leq p^{k}$, the $i$th term is $D_{i}(G_{k})=G_{k}^{\,p^{l(i)}}\gamma_{i}(G_{k})$, where $l(i)=\lceil\log_{p}i\rceil$.

Furthermore, if $i$ is not a power of $p$, equivalently if $l(i+1)=l(i)$, then $D_{i}(G_{k})/D_{i+1}(G_{k})\cong\gamma_{i}(G_{k})/\gamma_{i+1}(G_{k})$ so that

[TABLE]

with

[TABLE]

whereas if $i=p^{l}$ is a power of $p$, equivalently if $l(i+1)=l+1$ for $l=l(i)$, then $D_{i}(G_{k})/D_{i+1}(G_{k})\cong\langle x^{p^{l}}\rangle/\langle x^{p^{l+1}}\rangle\times\langle y^{p^{l}}\rangle/\langle y^{p^{l+1}}\rangle\times\gamma_{i}(G_{k})/\gamma_{i+1}(G_{k})$ so that

[TABLE]

with

[TABLE]

In particular, for $p^{k-1}+1\leq i\leq p^{k}$ and thus $l(i)=k$,

[TABLE]

so that

[TABLE]

Proof.

For $i\in\mathbb{N}$ write $l(i)=\lceil\log_{p}i\rceil$. From [2, Thm. 11.2] and Lemma 5.5 we obtain $D_{i}(G_{k})=G_{k}^{\,p^{l(i)}}\gamma_{i}(G_{k})$. In particular, $D_{i}(G_{k})=1$ for $i>p^{k}$, by Proposition 6.1 and Corollary 6.3.

Now suppose that $1\leq i\leq p^{k}$ and put $l=l(i)$. From Lemma 5.7 we observe that $G_{k}^{\,p^{l}}\cap\gamma_{i}(G_{k})\subseteq\gamma_{p^{l}}(G_{k})$. If $l(i+1)=l$ then $\gamma_{p^{l}}(G_{k})\subseteq\gamma_{i+1}(G_{k})$, and hence

[TABLE]

Now suppose that $l(i+1)=l+1$, equivalently $i=p^{l}$. We observe that, modulo $H_{k}$, the $i$th factor of the dimension subgroup series is

[TABLE]

Comparing with the overall order of $G_{k}$, conveniently implicit in Corollary 6.3, we deduce that

[TABLE]

All remaining assertions follow readily from Proposition 6.1. ∎

Proposition 6.5**.**

The Frattini series of $G_{k}$ has length $k+2$ and satisfies:

[TABLE]

and, for $2\leq i\leq k$, the $i$th term is

[TABLE]

where

[TABLE]

lastly,

[TABLE]

Proof.

For ease of notation we set $c_{1}=y$ and, for $i\geq 2$,

[TABLE]

From Lemma 5.5 we observe that $c_{i}^{\,p}=z_{i}^{\,p}=1$ for $i\geq 2$; furthermore, the elements $z_{i}\in[H_{k},H_{k}]\subseteq Z_{k}$ are central in $G_{k}$. We claim that

[TABLE]

Indeed, $[c_{i},c_{1}]=[c_{i},y]=z_{i+1}$, and, modulo $\gamma_{i+j+1}(G_{k})$, the Hall–Witt identity gives

[TABLE]

hence $[c_{i},c_{j}]\equiv[c_{i+1},c_{j-1}]^{-1}$ from which the result follows by induction.

We use the generators specified in the statement of the proposition to define an ascending chain $1=L_{k+2}\leq L_{k+1}\leq\ldots\leq L_{1}\leq L_{0}=G_{k}$ so that each $L_{i}$ is the desired candidate for $\Phi_{i}(G_{k})$. For $1\leq i\leq k+1$ we deduce from Proposition 6.1 and Corollary 6.2 that

[TABLE]

where $C_{i}=\langle y^{p^{i}}\rangle\times\langle z_{j}\mid 2\nu(i-1)+3\leq j\leq p^{k}\text{ and }j\equiv_{2}1\rangle$ is central in $G_{k}$. (Note that the factor $\langle y^{p^{i}}\rangle$ vanishes if $i\geq 2$.) Applying (2.3), based on Proposition 2.5 and Lemma 5.5, we see that $[x^{p^{i}},G_{k}]=[x^{p^{i}},H_{k}]\subseteq\gamma_{p^{i}+1}(G_{k})$, hence $L_{i}\trianglelefteq G_{k}$ for $1\leq i\leq k+1$. Using also (6.3), we see that the factor groups $L_{i}/L_{i+1}$ are elementary abelian for $0\leq i\leq k+1$. In particular, this shows that $\Phi_{i}(G_{k})\subseteq L_{i}$ for $1\leq i\leq k+2$.

Clearly, for each $i\in\{0,\ldots,k+1\}$, the value of $\log_{p}\lvert L_{i}/L_{i+1}\rvert=d(L_{i}/L_{i+1})$ is bounded by the number of explicit generators used to define $L_{i}$ modulo $L_{i+1}$; these numbers are specified in the statement of the proposition and a routine summation shows that they add up to the logarithmic order $\log_{p}\lvert G_{k}\rvert$, as given in Lemma 5.1. Therefore each $L_{i}/L_{i+1}$ has the expected rank and it suffices to show that $\Phi_{i}(G_{k})\supseteq L_{i}$ for $1\leq i\leq k+1$.

Let $i\in\{1,\ldots,k+1\}$. It is enough to show that the following elements which generate $L_{i}$ as a normal subgroup belong to $\Phi_{i}(G_{k})$:

[TABLE]

Clearly, $x^{p^{i}}\in\Phi_{i}(G_{k})$ and, applying (2.3), based on Proposition 2.5 and Lemma 5.5, we see by induction on $i$ that

[TABLE]

Now let $2\nu(i-1)+3\leq j\leq p^{k}$ with $j\equiv_{2}1$. By Corollary 6.2 and reverse induction on $j$ it suffices to show that $z_{j}$ is contained in $\Phi_{i}(G_{k})$ modulo $\gamma_{j+1}(G_{k})$. This follows from (6.3) and the fact that $c_{\nu(i-1)+1},c_{j-\nu(i-1)-1}\in\Phi_{i-1}(G_{k})$ by induction on $i$. ∎

Using Corollary 4.2, we can now complete the proof of Theorem 1.1: it suffices to compute $\operatorname{hdim}^{\mathcal{S}}_{G}(Z)$ and $\operatorname{hdim}^{\mathcal{S}}_{G}(H)$ for the standard filtration series $\mathcal{S}\in\{\mathcal{L},\mathcal{D},\mathcal{F}\}$.

Corollary 6.3 implies

[TABLE]

Corollary 6.4 implies

[TABLE]

Lastly, Proposition 6.5 implies

[TABLE]

Remark 6.6**.**

From (5.3), (6.4), (6.5), (6.6), (6.7) and the fact that subgroups of Hausdorff dimension $1$ automatically have strong Hausdorff dimension we conclude that $Z$ and $H$ have strong Hausdorff dimension in $G$ with respect to all standard filtration series $\mathcal{P}$, $\mathcal{D}$, $\mathcal{F}$ and $\mathcal{L}$.

7. The entire Hausdorff spectra of $G$ with respect to the

standard filtration series

We continue to use the notation set up in Section 3 to study and determine the entire Hausdorff spectra of the pro-$p$ group $G$, with respect to the standard filtration series $\mathcal{P},\mathcal{D},\mathcal{F},\mathcal{L}$.

Proof of Theorem 1.3.

As in Sections 2 and 3, we write $W=G/Z\cong C_{p}\mathrel{\hat{\wr}}\mathbb{Z}_{p}$, and we denote by $\pi\colon G\rightarrow W$ the canonical projection with $\ker\pi=Z$.

First suppose that $\mathcal{S}$ is one of the filtration series $\mathcal{P},\mathcal{D},\mathcal{F}$ on $G$. By Remark 6.6, the group $H$ has strong Hausdorff dimension $1$ in $G$ with respect to $\mathcal{S}$. As every finitely generated subgroup of $H$ is finite, it follows from [4, Thm. 5.4] that $\operatorname{hspec}^{\mathcal{S}}(G)=[0,1]$.

It remains to pin down the Hausdorff spectrum of $G$ with respect to the lower $p$-series $\mathcal{L}\colon P_{i}(G)$, $i\in\mathbb{N}$, on $G$. By Remark 6.6, the normal subgroups $Z,H\trianglelefteq_{\mathrm{c}}G$ have strong Hausdorff dimensions $\operatorname{hdim}^{\mathcal{L}}_{G}(Z)=\nicefrac{{1}}{{5}}$ and $\operatorname{hdim}^{\mathcal{L}}_{G}(H)=\nicefrac{{3}}{{5}}$. From Corollary 2.4, Lemma 2.2 and Corollary 2.11 we deduce that $\operatorname{hspec}^{\mathcal{L}}(G)$ contains

[TABLE]

Thus it suffices to show that

[TABLE]

First we prove the second inclusion. Let $K\leq_{\mathrm{c}}G$ be any closed subgroup with $\operatorname{hdim}^{\mathcal{L}}_{G}(K)>\nicefrac{{3}}{{5}}$. In particular, this implies $K\not\subseteq H$ and hence $KH\leq_{\mathrm{o}}G$.

We denote by $\mathcal{L}|_{H}$ and $\mathcal{L}|_{H\pi}$ the filtration series induced by $\mathcal{L}$ on $H$, via intersection, and on $H\pi=HZ/Z$, via subsequent reduction modulo $Z$. We write $\mathcal{L}$ for the filtration series $\mathcal{L}|_{W}$ induced on $W=G/Z$, as it coincides with the lower $p$-series of the quotient group. Using Corollary 2.11 and Lemma 2.2, we see that $(K\cap H)\pi$ has strong Hausdorff dimension

[TABLE]

in $H\pi$ with respect to $\mathcal{L}|_{H\pi}$. Applying Lemma 2.2 twice, we deduce that

[TABLE]

For $\alpha<\nicefrac{{1}}{{2}}$ we obtain $\operatorname{hdim}^{\mathcal{L}}_{G}(K)<\nicefrac{{4}}{{5}}$ and there is nothing further to prove. Now suppose that $\alpha\geq\nicefrac{{1}}{{2}}$. It suffices to show that $K\cap Z\leq_{\mathrm{o}}Z$ and hence $\operatorname{hdim}^{\mathcal{L}}_{G}(K\cap Z)=\nicefrac{{1}}{{5}}$: with this extra information we can refine the analysis in (7.2) and use Corollary 2.11 once more to deduce that

[TABLE]

Let us prove that $K\cap Z\leq_{\mathrm{o}}Z$. As $KH\leq_{\mathrm{o}}G$, we have $KH=\langle x^{p^{n}}\rangle H$, where $n=\log_{p}\lvert G:KH\rvert\in\mathbb{N}_{0}$. Using Lemma 2.2, we deduce from $\alpha\geq\nicefrac{{1}}{{2}}$ that

[TABLE]

At this point it is useful to recall our analysis of $\operatorname{hspec}^{\mathcal{L}}(W)$ in the proof of Theorem 2.10 and also the computations carried out in the proof of Proposition 6.5, involving the elements $c_{i}=[y,x,\overset{i-1}{\ldots},x]$ and $z_{i}=[c_{i-1},y]$. In particular, for $i\in\mathbb{N}$ with $i\geq 3$ we have

[TABLE]

compare Corollary 6.3. From (7.3) and the proof of Theorem 2.10 we deduce that, subject to replacing $K$ by a suitable open subgroup $\widetilde{K}=K\cap\langle x^{p^{\widetilde{n}}}\rangle H$ with $\widetilde{n}\geq n$ if necessary, we find $m\geq(p^{n}+1)/2$ and $a_{1},\ldots,a_{m}\in K\cap H$ so that

[TABLE]

and the numbers

[TABLE]

form a strictly increasing sequence $1\leq d(1)<\ldots<d(m)<p^{n}$. Commuting $a_{1},\ldots,a_{m}$ repeatedly with $x^{p^{n}}$, we see as in the proof of Theorem 2.10 that

[TABLE]

For every $k\in\mathbb{N}$ with $k>p^{n}$ and $k\equiv_{2}1$, the pigeonhole principle (Dirichlet’s ‘Schubfachprinzip’) yields $i,j\in\mathbb{N}$ with $i>j\geq 1$ and $i+j=k$, and we find $g_{i},g_{j}\in K\cap H$ with $g_{i}\equiv_{P_{i+1}(G)Z}c_{i}$ and $g_{j}\equiv_{P_{j+1}(G)Z}c_{j}$ so that (6.3) gives

[TABLE]

But this implies $K\cap Z\supseteq\langle z_{j}\mid j>p^{n}\text{ and }j\equiv_{2}1\rangle=P_{p^{n}+1}(G)\cap Z$ and thus $K\cap Z\leq_{\mathrm{o}}Z$. This concludes the proof of the second inclusion in (7.1).

Finally we prove the first inclusion in (7.1). Let $\xi\in(\nicefrac{{2}}{{5}},\nicefrac{{4}}{{5}})$. Choose $m,n\in\mathbb{N}$ such that $1\leq m<\nicefrac{{p^{n}}}{{2}}$ and

[TABLE]

Consider the group $K=\langle x^{p^{n}},y_{0},y_{1},\ldots,y_{m-1}\rangle$. Using the proof of Theorem 2.10 and Lemma 2.2, we show below that $K$ has Hausdorff dimension

[TABLE]

In a similar, but much more straightforward way, we see that $ZK$ has strong Hausdorff dimension

[TABLE]

An application of [4, Thm. 5.4] yields $L\leq_{\mathrm{c}}G$ with $K\leq L\leq ZK$ such that $\operatorname{hdim}^{\mathcal{L}}_{G}(L)=\xi$.

The key to (7.4) consists in showing that

[TABLE]

First we examine the lower limit on the left-hand side, restricting to indices of the form $i=p^{k}+1$, $k\in\mathbb{N}$. Let $i=p^{k}+1$, where $k\geq n$. Recall that $G_{k}=G/\langle x^{p^{k+1}},[x^{p^{k}},y]\rangle^{G}$ and consider the canonical projection $\varrho_{k}\colon G\to G_{k}$, $g\mapsto\overline{g}$. As before, we write $H_{k}=H\varrho_{k}$. Furthermore, we observe that $Z_{k}=\langle\overline{x}^{p^{k}}\rangle Z\varrho_{k}$ with $\lvert Z_{k}:Z\varrho_{k}\rvert=p$. By Corollary 6.3, we have

[TABLE]

and hence

[TABLE]

Observe that

[TABLE]

From Lemma 5.1 we see that $Z\varrho_{k}\cong C_{p}^{\,(p^{k}+1)/2}$ and further we deduce that

[TABLE]

This yields

[TABLE]

In order to establish (7.5) it now suffices to prove that

[TABLE]

Our analysis above yields

[TABLE]

Setting

[TABLE]

and recalling the notation $c_{1}=y=y_{0}$, we conclude that

[TABLE]

Next we consider the set

[TABLE]

Each element $y_{j}$ can be written (modulo $Z$) as a product

[TABLE]

using the elements $c_{i}=[y,x,\overset{i-1}{\ldots},x]$ introduced in the proof of Proposition 6.5. In this product decomposition, the exponents should be read modulo $p$, and the elementary identity $(1+t)^{j+p^{n}}=(1+t)^{j}(1+t^{p^{n}})$ in $\mathbb{F}_{p}[\![t]\!]$ translates to

[TABLE]

compare (2.4). Inductively, we obtain

[TABLE]

Observe that $\lvert D_{0}\rvert=2m-1$ and that, for each $k\in\mathbb{N}_{0}$, the set $(2kp^{n}+D_{0})\cup((2k+1)p^{n}+D_{0})$ consists of $2m-1$ odd and $2m-1$ even numbers.

For each $j\in D$ with $j\equiv_{2}0$ there exists $g_{j}\in L$ with $g_{j}\equiv_{P_{j+1}(G)Z}c_{j}$ and we deduce that

[TABLE]

For $i=2p^{n}q+r\in\mathbb{N}$, where $q,r\in\mathbb{N}_{0}$ with $0\leq r<2p^{n}$, the count

[TABLE]

yields

[TABLE]

From Corollary 6.3 we observe that, for $i\geq 3$,

[TABLE]

These estimates show that (7.6) holds. ∎

Appendix A The case $p=2$

When $p$ is even, Theorems 1.1 and 1.3, and all the results of Sections 2 and 4, hold with corresponding proofs. The structural results of Sections 5 and 6 however are slightly different and we now sketch these differences below; for complete details, we refer the reader to the supplement [8].

Firstly, for $p=2$,

[TABLE]

for $k\in\mathbb{N}$, and

[TABLE]

is a presentation of $G$ as a pro-$2$ group.

Next, we have $\log_{2}|G_{k}|=2^{k}+2^{k-1}+k+2$ and the exponent of $\gamma_{2}(G_{k})$ is 4. With regards to Lemma 5.3 , the elements

[TABLE]

are of order $2$ in $G_{k}$ and lie in $G_{k}^{\,2^{k}}$. In particular the subgroup $\langle x^{2^{k}},w,[w,x]\rangle$ is isomorphic to $C_{2}\times C_{2}\times C_{2}$ and lies in $G_{k}^{\,2^{k}}$. Hence, for $k\geq 2$,

[TABLE]

and

[TABLE]

Lemma 5.7 is slightly different; here the group $G_{k}$ satisfies $G_{k}^{\,2}\subseteq\langle x^{2},y^{2}\rangle\gamma_{2}(G_{k})$ and

[TABLE]

The proof is similar, but one needs the fact

[TABLE]

which is proved by induction, using

[TABLE]

Furthermore, $[y,x,\overset{i}{\ldots},x]^{2}\equiv[y,x,\overset{2i-1}{\ldots},x,y]$ modulo $\gamma_{2i+2}(G_{k})$.

The group $G_{k}$ is nilpotent of class $2^{k}+1$; its lower central series satisfies

[TABLE]

and, for $1\leq i\leq 2^{k-1}$,

[TABLE]

with

[TABLE]

The proof of the above is similar to that for the odd prime case, however here one takes

[TABLE]

For the $m\equiv_{4}0$ case, noting that $e_{2^{k-1}}=[w,x]\in\gamma_{2^{k}+1}(G_{k})$, we have $b_{j_{0},m}\equiv b_{j_{0},m+1}$ modulo $\gamma_{m+1}(G_{k})$. The $m\equiv_{4}2$ case is similar.

The lower $2$-series of $G_{k}$ has length $2^{k}+1$ and satisfies the corresponding form, based on the lower central series of $G_{k}$ above.

The dimension subgroup series of $G_{k}$ has length $2^{k}+2$. For $1\leq i\leq 2^{k}+2$, the $i$th term is $D_{i}(G_{k})=G_{k}^{\,2^{l(i)}}\gamma_{\lceil\nicefrac{{i}}{{2}}\rceil}(G_{k})^{2}\gamma_{i}(G_{k})$, where $l(i)=\lceil\log_{2}i\rceil$.

Furthermore, if $i$ is not a power of $2$, equivalently if $l(i+1)=l(i)$, then $D_{i}(G_{k})/D_{i+1}(G_{k})\cong\gamma_{\lceil\nicefrac{{i}}{{2}}\rceil}(G_{k})^{2}\gamma_{i}(G_{k})/\gamma_{\lceil\nicefrac{{i+1}}{{2}}\rceil}(G_{k})^{2}\gamma_{i+1}(G_{k})$ so that

[TABLE]

with

[TABLE]

whereas if $i=2^{l}$ is a power of $2$, equivalently if $l(i+1)=l+1$ for $l=l(i)$, then $D_{i}(G_{k})/D_{i+1}(G_{k})\cong\langle x^{2^{l}}\rangle/\langle x^{2^{l+1}}\rangle\times\langle y^{2^{l}}\rangle/\langle y^{2^{l+1}}\rangle\times\langle[y,x,\overset{i-3}{\ldots},x,y]\rangle\gamma_{i}(G_{k})/\gamma_{i+1}(G_{k})$ so that

[TABLE]

with

[TABLE]

In particular, for $2^{k-1}+1\leq i\leq 2^{k}$ and thus $l(i)=k$,

[TABLE]

so that

[TABLE]

Lastly, the Frattini series of $G_{k}$ has the corresponding form, though it has length $k+1$.