Pro-$p$ groups of positive rank gradient and Hausdorff dimension

TL;DR

This paper investigates the structure and Hausdorff dimension spectra of finitely generated pro-$p$ groups with positive rank gradient, establishing new results on subgroups, spectra, and their relation to series and filtrations.

Contribution

It proves that finitely generated pro-$p$ groups of positive rank gradient have full Hausdorff spectrum and extends this to direct products and various filtrations, resolving longstanding problems.

Findings

Finitely generated pro-$p$ groups of positive rank gradient have full Hausdorff spectrum [0,1].

Non-abelian free pro-$p$ groups and Demushkin groups also have full spectrum with respect to certain series.

The results generalize to direct products and arbitrary iterated verbal filtrations.

Abstract

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient. Motivated by the study of Hausdorff dimension, we show that finitely generated closed subgroups $H$ of infinite index in $G$ never contain any infinite subgroups $K$ that are subnormal in~$G$ via finitely generated successive quotients. This pro-$p$ version of a well-known theorem of Greenberg generalises similar assertions that were known to hold for non-abelian free pro-$p$ groups, non-soluble Demushkin pro-$p$ groups and other related pro-$p$ groups. The result we prove is reminiscent of Gaboriau's theorem for countable groups with positive first $\ell^2$-Betti number, but not quite a direct analogue. The approach via the notion of Hausdorff dimension in pro-$p$ groups also leads to our main results. We show that every finitely generated pro-$p$ group $G$ of positive rank gradient has full Hausdorff spectrum $\text{hspec}^\mathcal{F}(G) = [0,1]$ with respect to the Frattini series~$\mathcal{F}$. Using different, Lie-theoretic techniques we also prove that finitely generated non-abelian free pro-$p$ groups and non-soluble Demushkin groups $G$ have full Hausdorff spectrum $\text{hspec}^\mathcal{Z}(G) = [0,1]$ with respect to the Zassenhaus series~$\mathcal{Z}$. This resolves a long-standing problem in the subject of Hausdorff dimensions in pro-$p$ groups. The results about full Hausdorff spectra hold more generally for finite direct products of finitely generated pro-$p$ groups of positive rank gradient and for mixed finite direct products of finitely generated non-abelian free pro-$p$ groups and non-soluble Demushkin groups, respectively. Analogously, the aforementioned results with respect to the Frattini series generalise further to Hausdorff dimension functions with respect to arbitrary iterated verbal filtrations. Finally, we determine the normal Hausdorff spectra of such direct products.

Equations

$$\mathrm{RG}(G)=\underset{H{\le }_{\mathrm{o}}G}{\inf }\frac{d(H)-1}{|G:H|},$$

$$\mathrm{RG}(G,\mathcal{S})=\underset{i\to \infty }{\lim }\frac{d(G_i)-1}{|G:G_i|}$$

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

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

$$\mathcal{F}:$$

$$\mathcal{Z}:$$

$$\mathcal{P}:$$

$${\mathcal{P}}^{\ast }:{\Pi }_0^{\ast }(G)=G,\text{and}{\Pi }_i^{\ast }(G)={\Pi }_{i-1}^{\ast }(G{)}^p=⟨x^p\mid x\in {\Pi }_{i-1}^{\ast }(G)⟩\text{for i≥1.}$$

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

$$d=\max \{d(F_j)\mid 1\le j\le r\}\text{and}t=|\{j\mid 1\le j\le r,d(F_j)=d\}|.$$

$${\mathrm{hspec}}_{⊴}^{\mathcal{S}}(G)={\mathrm{hspec}}_{⊴}^{\mathcal{Z}}(G)=\{0,\text{nicefrac}1t,\dots ,\text{nicefrac}(t-1)t,1\}.$$

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

$$\frac{d(H\cap G_i)}{d(G_i)}\le \frac{(d(H)-1)|H:H\cap G_i|}{R|G:G_i|}\le \frac{d(H)|H{G}_i:G_i|}{R|G:H{G}_i||H{G}_i:G_i|}=\frac{d(H)}{R|G:H{G}_i|}\underset{i\to \infty }{\to }0.$$

$$\frac{d(H{G}_i)}{d(G_i)}\le \frac{(d(G)-1)|G:H{G}_i|}{R|G:G_i|}\le \frac{d(G)}{R|H{G}_i:G_i|}\underset{i\to \infty }{\to }0.\text{qed}$$

$${\log }_p|H\cap G_i:H\cap G_{i+1}|\le {\log }_p|H\cap G_i:\Phi (H\cap G_i)|=d(H\cap G_i)\text{for i≥0.}$$

$$\frac{{\log }_p|H\cap G_{i_0}:H\cap G_i|}{{\log }_p|G_{i_0}:G_i|}\le \frac{{\sum }_{j=i_0}^{i-1}d(H\cap G_j)}{{\sum }_{j=i_0}^{i-1}d(G_j)}\le \frac{{\sum }_{j=i_0}^{i-1}d(H\cap G_j)}{{\sum }_{j=i_0}^{i-1}nd(H\cap G_j)}=\frac{1}{n},$$

$${\mathrm{hdim}}_G^{\mathcal{F}}(H)=\underset{\begin{array}{c}i\to \infty \\ i\ge i_0\end{array}}{\underline{\lim }}\frac{\stackrel{\text{constant}}{\overbrace{{\log }_p|H:H\cap G_{i_0}|}}+{\log }_p|H\cap G_{i_0}:H\cap G_i|}{\underset{\text{constant}}{\underbrace{{\log }_p|G:G_{i_0}|}}+{\log }_p|G_{i_0}:G_i|}\le \frac{1}{n}.\text{qed}$$

$${\mathrm{hdim}}_G^{\mathcal{F}}(N)=\underset{i\to \infty }{\underline{\lim }}\frac{{\log }_p|N{G}_i:G_i|}{{\log }_p|G:G_i|}=1-\underset{i\to \infty }{\overline{\lim }}\frac{{\log }_p|G:N{G}_i|}{{\log }_p|G:G_i|}.$$

$${\log }_p|N{G}_i:N{G}_{i+1}|={\log }_p|G_{i}N/N:\Phi (G_{i}N/N)|=d(G_{i}N/N)\le d(N{G}_i)\text{for i≥0.}$$

$$\frac{{\log }_p|N{G}_{i_0}:N{G}_i|}{{\log }_p|G_{i_0}:G_i|}\le \frac{{\sum }_{j=i_0}^{i-1}d(N{G}_j)}{{\sum }_{j=i_0}^{i-1}d(G_j)}\le \frac{{\sum }_{j=i_0}^{i-1}d(N{G}_j)}{{\sum }_{j=i_0}^{i-1}nd(N{G}_j)}=\frac{1}{n},$$

$$\underset{i\to \infty }{\overline{\lim }}\frac{{\log }_p|G:N{G}_i|}{{\log }_p|G:G_i|}=\underset{\begin{array}{c}i\to \infty \\ i\ge i_0\end{array}}{\overline{\lim }}\frac{\stackrel{\text{constant}}{\overbrace{{\log }_p|G:N{G}_{i_0}|}}+{\log }_p|N{G}_{i_0}:N{G}_i|}{\underset{\text{constant}}{\underbrace{{\log }_p|G:G_{i_0}|}}+{\log }_p|G_{i_0}:G_i|}\le \frac{1}{n}.\text{qed}$$

$${\mathrm{hdim}}_G^{\mathcal{F}}(AB)$$

$$\le \underset{i\to \infty }{\underline{\lim }}\frac{{\log }_p|A{G}_i:G_i|+{\log }_p|B{G}_i:G_i|}{{\log }_p|G:G_i|}$$

$$=\underset{i\to \infty }{\lim }\frac{{\log }_p|A{G}_i:G_i|}{{\log }_p|G:G_i|}+\underset{i\to \infty }{\underline{\lim }}\frac{{\log }_p|B{G}_i:G_i|}{{\log }_p|G:G_i|}$$

$$={\mathrm{hdim}}_G^{\mathcal{F}}(B).$$

$$\frac{d(U)-1}{|G:U|}\to 0\text{as ∣G:HU∣→∞ and ∣KU:U∣→∞ simultaneously;}$$

$$d(U)\le d(U\cap H)+{\sum }_{i=1}^{s}d((U\cap K_{i-1})/(U\cap K_i)).$$

$$\frac{d(U\cap H)-1}{|G:U|}\le \frac{d(H)}{|G:HU|}\to 0\text{as ∣G:HU∣→∞.}$$

$$d((U\cap K_{i-1})/(U\cap K_i)\le (d(K_{i-1}/K_i)-1)|K_{i-1}U:K_{i}U|\le d(K_{i-1}/K_i)|G:KU|.$$

$$\frac{{\sum }_{i=1}^{s}d((U\cap K_{i-1})/(U\cap K_i))}{|G:U|}\le \frac{{\sum }_{i=1}^{s}d(K_{i-1}/K_i)}{|KU:U|}\to 0\text{as ∣KU:U∣→∞.}$$

$$\mathrm{RG}(\Gamma ,\mathcal{S})=\underset{i\to \infty }{\lim }\frac{d({\Gamma }_i)-1}{|G:{\Gamma }_i|}\ge b_1^{(2)}(\Gamma ),$$

$${\mathrm{RG}}_p(\Gamma ,\mathcal{S})=\underset{i\to \infty }{\lim }\frac{{\dim }_{{\mathbb{F}}_p}{\mathrm{H}}_1({\Gamma }_i,{\mathbb{F}}_p)}{|\Gamma :{\Gamma }_i|}.$$

$$\mathrm{RG}(G/N)=\underset{i\to \infty }{\lim }\frac{d(G_{i}N/N)-1}{|G/N:G_{i}N/N|}=\underset{i\to \infty }{\lim }\frac{d(G_i)-1}{|G:G_{i}N|}=|N|\mathrm{RG}(G).\text{qed}$$

Full text

The approach via the notion of Hausdorff dimension in pro-$p$ groups also leads to our main results. We show that every finitely generated pro-$p$ group $G$ of positive rank gradient has full Hausdorff spectrum $\operatorname{hspec}^{\mathcal{F}}(G)=[0,1]$ with respect to the Frattini series $\mathcal{F}$. Using different, Lie-theoretic techniques we also prove that finitely generated non-abelian free pro-$p$ groups and non-soluble Demushkin groups $G$ have full Hausdorff spectrum $\operatorname{hspec}^{\mathcal{Z}}(G)=[0,1]$ with respect to the Zassenhaus series $\mathcal{Z}$. This resolves a long-standing problem in the subject of Hausdorff dimensions in pro-$p$ groups.

In fact, the results about full Hausdorff spectra hold more generally for finite direct products of finitely generated pro-$p$ groups of positive rank gradient and for mixed finite direct products of finitely generated non-abelian free pro-$p$ groups and non-soluble Demushkin groups, respectively. The results with respect to the Frattini series generalise further to Hausdorff dimension functions with respect to arbitrary iterated verbal filtrations, for mixed finite direct products of finitely generated non-abelian free pro-$p$ groups and non-soluble Demushkin pro-$p$ groups.

Finally, we determine the normal Hausdorff spectra of such direct products.

The first author was supported by the Basque Government postdoctoral fellowship POS–2017–2–0031. The second author gratefully acknowledges the support of an Alexander von Humboldt Fellowship. The research was partially conducted in the framework of the DFG-funded research training group “GRK 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology”.

1. Introduction

We study the subgroup structure of finitely generated pro-$p$ groups that resemble free pro-$p$ groups. Here and throughout, $p$ denotes a prime number and, as usual in the context of profinite groups, subgroups are tacitly assumed to be closed. To make this explicit, we write $H\leq_{\mathrm{c}}G$, respectively $H\leq_{\mathrm{o}}G$, to indicate that a subgroup $H$ is closed, respectively open, in $G$. Furthermore, group-theoretic notions, such as generating properties, are considered topologically.

The classical Schreier index formula motivates the following notion, first introduced in [37]. The (absolute) rank gradient of a finitely generated pro-$p$ group $G$ is

[TABLE]

where $\operatorname{d}(H)$ denotes the minimal number of generators of a pro-$p$ group $H$ and the infimum is taken over all open subgroups $H\leq_{\mathrm{o}}G$. We recall that in the present context $\operatorname{RG}(G)$ coincides with the (relative) rank gradient

[TABLE]

with respect to any descending chain $\mathcal{S}\colon G\supseteq G_{1}\supseteq G_{2}\supseteq\ldots$ of open subgroups that provide a base of neighbourhoods of the identity element; the limit exists, because the sequence of rational numbers is monotone decreasing and bounded. In practice, it is convenient to work with descending chains of open normal subgroups $G_{i}$, $i\in\mathbb{N}_{0}$, that intersect in the identity; we call such chains filtration series, or filtrations for short.

The class of pro-$p$ groups of positive rank gradient includes, for instance:

• •

finitely generated non-abelian free pro-$p$ groups;

• •

non-soluble Demushkin pro-$p$ groups, which occur as analogues of surface groups and as Galois groups of maximal $p$-extensions of local fields containing $p$th roots of unity (compare [49]);

• •

pro-$p$ analogues of limit groups, defined and studied in [30];

• •

free pro-$p$ products of finitely many non-trivial finitely generated pro-$p$ groups, with at least two factors and excluding $C_{2}\amalg C_{2}$ (compare [46, Thm. 2.3]);

• •

pro-$p$ completions of groups of positive $p$-deficiency, as constructed in [48] or arising from similar constructions in [45, Cor. 1.2];

• •

pro-$p$ completions of certain Golod–Shafarevich groups, as seen in [11].

Recall that $K\leq_{\mathrm{c}}G$ is subnormal in $G$ if there exists a finite descending sequence of subgroups $G=K_{0}\trianglerighteq K_{1}\trianglerighteq\ldots\trianglerighteq K_{s}=K$, with $K_{i}$ normal in $K_{i-1}$ for $i\in\{1,\ldots,s\}$. A well-known theorem of Greenberg [21] implies that a finitely generated subgroup $\Delta$ of a finitely generated, non-abelian free discrete group $\Gamma$ has finite index in $\Gamma$ if and only if $\Delta$ contains a non-trivial subnormal subgroup of $\Gamma$. Our first result is a version of Greenberg’s theorem in the context of pro-$p$ groups.

Definition**.**

We say that a subgroup $K\leq_{\mathrm{c}}G$ is subnormal via finitely generated successive quotients if there is a subnormal chain $G=K_{0}\trianglerighteq K_{1}\trianglerighteq\ldots\trianglerighteq K_{s}=K$ such that each factor $K_{i-1}/K_{i}$, for $i\in\{1,\ldots,s\}$, is finitely generated.

Theorem 1.1**.**

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient, and let $H\leq_{\mathrm{c}}G$ be a finitely generated subgroup. Then the following are equivalent:

1. (1)

$H$* contains an infinite subgroup of $G$ that is subnormal via finitely generated successive quotients;* 2. (2)

$H$* contains an infinite normal subgroup of $G$;* 3. (3)

$H$* is open in $G$.*

In particular, every subgroup $K\leq_{\mathrm{c}}G$ that is subnormal via finitely generated successive quotients satisfies: $K$ is finitely generated if and only if $K$ is either finite or open.

Theorem 1.1 generalises, for instance, results of Lubotzky [41, Prop. 3.3], Kochloukova [29, Prop. 3.13], and Snopce and Zalesskii [53, Thm. 3.1]. We thank Mark Shusterman for alerting us to his work on pro-$p$ groups of hereditarily linearly increasing rank, where he already obtains closely related pro-$p$ versions of Greenberg’s theorem; compare [52, Thm. 2.7 and Cor. 2.8]. While we discovered Theorem 1.1 in a different way, it can also easily be proved by adapting Shusterman’s approach, the key ingredient being a simple, but effective argument of Abért and Nikolov [2, Prop. 13] (where the condition $\lvert N\rvert=\infty$ should be added). The point of view that an assumption on the increase in the number of generators upon passing to finite index subgroups (e.g. positive rank gradient) forms a good framework for proving generalisations of Greenberg’s theorem is already expressed in [52]. It remains an interesting problem to find out which consequences of the more special results mentioned above hold, in fact, for all pro-$p$ groups of positive rank gradient. Furthermore, Theorem 1.1 is reminiscent of a theorem of Gaboriau [19, Thm. 6.8] for countable groups with positive first $\ell^{2}$-Betti number; possible connections between these notions and results are discussed in Section 2.2. We are grateful to Steffen Kionke for drawing our attention to Gaboriau’s theorem.

Our proof of Theorem 1.1, indeed our motivation for proving it, stems from the study of Hausdorff dimensions of subgroups in pro-$p$ groups of positive rank gradient. The concept of Hausdorff dimension is commonly employed to measure the sizes of fractals and subsets of general metric spaces; see [18, 47]. In [9], Barnea and Shalev initiated the use of Hausdorff dimension to study the distribution of closed subgroups in finitely generated pro-$p$ groups; see also Shalev’s survey article [50, §4]. This idea has led to many interesting applications; compare the references given in [28].

Let $G$ be a finitely generated infinite pro-$p$ group with a filtration $\mathcal{S}\colon G_{i}$, $i\in\mathbb{N}_{0}$. Then, $\mathcal{S}$ induces a translation-invariant metric on $G$; the distance between $x,y\in G$ is $d_{\mathcal{S}}(x,y)=\inf\{\lvert G:G_{i}\rvert^{-1}\mid x^{-1}y\in G_{i}\}$. It is known that the Hausdorff dimension of a closed subgroup $H\leq_{\mathrm{c}}G$ with respect to $d_{\mathcal{S}}$ can be obtained as a lower limit of logarithmic densities:

[TABLE]

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

[TABLE]

The Hausdorff dimension function and the resulting Hausdorff spectrum depend significantly on the choice of the filtration $\mathcal{S}$. With respect to general filtration series rather unexpected features can be observed, for a wide range of groups, including the groups considered in this paper; compare [28, Thm. 1.5]. Consequently, one typically concentrates on Hausdorff dimension with respect to standard group-theoretic filtration series of the pro-$p$ group $G$, such as: the Frattini series $\mathcal{F}$, the Zassenhaus series $\mathcal{Z}$ (also known as the dimension subgroup or Jennings–Lazard–Zassenhaus series), or the $p$-power series $\mathcal{P}$; these are defined as follows:

[TABLE]

Another natural choice of filtration, the lower central $p$-series, leads to somewhat unexpected effects in terms of Hausdorff dimension; compare [28]. One of the key features of Hausdorff dimension is that it agrees with the usual dimension of smooth submanifolds of euclidean space. Pleasingly, something similar holds in the $p$-adic setting: closed subgroups of $p$-adic analytic pro-$p$ groups, being $p$-adic analytic themselves, yield only finitely many values of Hausdorff dimensions with respect to $\mathcal{F}$, $\mathcal{Z}$ and $\mathcal{P}$; see [9, Thm. 1.1] and [28, Prop. 1.5]. Whether this property characterises $p$-adic analytic groups among finitely generated pro-$p$ groups is a long-standing open problem; see [28] for a partial positive answer, valid for soluble groups. While the Hausdorff spectra of finitely generated non-abelian free pro-$p$ groups were perhaps suspected to be quite large, no proof was found to show that they are merely infinite, with respect to $\mathcal{F}$, $\mathcal{Z}$ or $\mathcal{P}$, say.

In this paper we confine ourselves – similar to Shalev [50, §4.6] – to the Frattini series, the Zassenhaus series and related filtration series. For the groups we study, the $p$-power series is beyond current knowledge; even estimating the indices $\lvert F:F^{p^{i}}\rvert$ of the terms of the series, for finitely generated non-abelian free pro-$p$ groups $F$, amounts to estimating the corresponding Burnside numbers which are largely unknown.

Our main results solve problems highlighted in [9] and [50, §4.6]; in particular, they finally remove non-abelian free pro-$p$ groups from the list of potential counter-examples to the desired characterisation of $p$-adic analytic pro-$p$ groups $G$ by the finiteness of their Hausdorff spectra, with respect to $\mathcal{F}$ or $\mathcal{Z}$.

For the Frattini series $\mathcal{F}$, we can, in fact, remove a much larger class of potential counter-examples, by virtue of the techniques used to prove Theorem 1.1.

Theorem 1.2**.**

Let $G=H_{1}\times\ldots\times H_{r}$ be a non-trivial finite direct product of finitely generated pro-$p$ groups $H_{j}$ of positive rank gradient. Then $G$ has full Hausdorff spectrum $\operatorname{hspec}^{\mathcal{F}}(G)=[0,1]$ with respect to the Frattini series $\mathcal{F}$.

For the Zassenhaus series $\mathcal{Z}$, we rely on more intricate Lie-theoretic methods, which do not readily extend from free groups to general groups of positive rank gradient. In order to translate between certain subgroups of a given free pro-$p$ group $F$ and Lie subalgebras of the associated free restricted Lie algebra, we make use of power-commutator factorisations in $F$, which result from Hall’s commutator collection process.

Theorem 1.3**.**

Let $F$ be a finitely generated non-abelian free pro-$p$ group. Then $F$ has full Hausdorff spectrum $\operatorname{hspec}^{\mathcal{Z}}(F)=[0,1]$ with respect to the Zassenhaus series $\mathcal{Z}$.

It is worth comparing Theorem 1.3 to what was previously known about the Hausdorff spectrum $\operatorname{hspec}^{\mathcal{Z}}(F)$ of a non-abelian free pro-$p$ group $F$; see [50, §4.6]. Lie-theoretic arguments can be used to deduce with relative ease that non-trivial normal subgroups of $F$ have Hausdorff dimension $1$ with respect to $\mathcal{Z}$. In addition, Shalev [50, Thm. 4.10] sketched a high-level argument to locate finitely many rational values of the form $e^{-1}$, for certain $e\in\mathbb{N}$, in the Hausdorff spectrum $\operatorname{hspec}^{\mathcal{Z}}(F)$, but stressed at the time that it was “unclear how to construct subgroups of $F$ whose Hausdorff dimension is irrational.”

In addition to free groups, we also consider Demushkin pro-$p$ groups. They are, by definition, Poincaré duality pro-$p$ groups of dimension $2$ and thus can be regarded as pro-$p$ analogues of surface groups; indeed, the pro-$p$ completions of orientable surface groups are Demushkin groups. In this way they are rather close to finitely generated free pro-$p$ groups. Demushkin groups also occur naturally as Galois groups in algebraic number theory, as mentioned above. We refer to [55, Sec. 12.3] for the classification of Demushkin groups of depth $q\neq 2$ and further details; Demushkin groups of depth $q=2$ are treated in [33, 49].

The next result applies to mixed finite direct products of free and Demushkin pro-$p$ groups, equipped with an iterated verbal filtration, and can be established by adapting our proof of Theorem 1.2.

Definition**.**

A filtration $\mathcal{S}\colon G_{i}$, $i\in\mathbb{N}_{0}$, of $G$ such that $G_{i}$ is a verbal subgroup of $G_{i-1}$ for each $i\in\mathbb{N}$ is called an iterated verbal filtration.

Typical examples of iterated verbal filtrations are the Frattini filtration and the iterated $p$-power series defined by

[TABLE]

Theorem 1.4**.**

Let $G=E_{1}\times\ldots\times E_{r}$ be a finite direct product of finitely generated pro-$p$ groups $E_{j}$, each of which is either free or Demushkin. Suppose that at least one factor is non-soluble. Let $\mathcal{S}\colon G_{i}$, $i\in\mathbb{N}_{0}$, be an iterated verbal filtration of $G$. Then $G$ has full Hausdorff spectrum $\operatorname{hspec}^{\mathcal{S}}(G)=[0,1]$.

In a similar direction, we extend Theorem 1.3 to non-soluble Demushkin pro-$p$ groups and derive as a common generalisation a corresponding result for mixed finite direct products of free and Demushkin pro-$p$ groups.

Theorem 1.5**.**

Let $D$ be a non-soluble Demushkin pro-$p$ group. Then $D$ has full Hausdorff spectrum $\operatorname{hspec}^{\mathcal{Z}}(D)=[0,1]$ with respect to the Zassenhaus series $\mathcal{Z}$.

Corollary 1.6**.**

Let $G=E_{1}\times\ldots\times E_{r}$ be a finite direct product of finitely generated pro-$p$ groups $E_{j}$, each of which is either free or Demushkin. Suppose that at least one factor is non-soluble. Then $G$ has full Hausdorff spectrum $\operatorname{hspec}^{\mathcal{Z}}(G)=[0,1]$ with respect to the Zassenhaus series $\mathcal{Z}$.

The study of Hausdorff spectra of pro-$p$ right-angled Artin groups – defined via presentations analogously to the discrete case – provides a natural avenue for future research. Above we already formulated our results to cover the basic case of direct products of free pro-$p$ groups and, indeed, more general direct products. As a supplement, we can also describe the Hausdorff dimensions of normal subgroups in such direct products.

The normal Hausdorff spectrum of a finitely generated pro-$p$ group $G$, with respect to a filtration $\mathcal{S}$, is

[TABLE]

compare [50, §4.7] and [27].

Theorem 1.7**.**

Let $G=H_{1}\times\ldots\times H_{r}$ be a non-trivial finite direct product of finitely generated pro-$p$ groups $H_{j}$ of positive rank gradient. For $1\leq j\leq r$, set $\alpha_{j}=\operatorname{hdim}_{G}^{\mathcal{F}}(H_{j})$, with respect to the Frattini series $\mathcal{F}$. Then $G$ has normal Hausdorff spectrum $\operatorname{hspec}_{\trianglelefteq}^{\mathcal{F}}(G)=\{\sum_{j\in J}\alpha_{j}\mid J\subseteq\{1,\ldots,r\}\}$ with respect to $\mathcal{F}$.

Theorem 1.8**.**

Let $G=E_{1}\times\ldots\times E_{r}$ be a finite direct product of finitely generated pro-$p$ groups $E_{j}$, each of which is either free or Demushkin. Suppose that at least one factor is non-soluble. For $1\leq j\leq r$, set $\alpha_{j}=\operatorname{hdim}_{G}^{\mathcal{S}}(E_{j})$, with respect to an iterated verbal filtration $\mathcal{S}$ of $G$. Then $G$ has normal Hausdorff spectrum $\operatorname{hspec}_{\trianglelefteq}^{\mathcal{S}}(G)=\{\sum_{j\in J}\alpha_{j}\mid J\subseteq\{1,\ldots,r\}\}$ with respect to $\mathcal{S}$.

In addition to these general theorems we have the following concrete results for products of free pro-$p$ groups.

Theorem 1.9**.**

Let $G=F_{1}\times\ldots\times F_{r}$ be a non-trivial finite direct product of finitely generated free pro-$p$ groups $F_{j}$. Put

[TABLE]

Let $\mathcal{S}$ be an iterated verbal filtration of $G$ and let $\mathcal{Z}$ be the Zassenhaus series of $G$. Then $G$ has the normal Hausdorff spectra

[TABLE]

We conclude the introduction with a brief outline of the general strategy and the organisation of the paper, following a suggestion of the referee.

General Strategy and Organisation. Theorem 1.1, which is proved in Section 2, arises from the study of Hausdorff dimensions of subgroups of pro-$p$ groups $G$ of positive rank gradient. Its proof is based on the observation that finitely generated, infinite-index subgroups $H\leq_{\mathrm{c}}G$ have Hausdorff dimension [math], whereas infinite subgroups $K\leq_{\mathrm{c}}G$ that are subnormal via finitely generated successive quotients have Hausdorff dimension $1$.

The main idea underlying the proofs of Theorems 1.2, 1.3 and 1.4 is to exploit suitable variations of the following result.

Interval Theorem** ([28, Thm. 5.4]).**

Let $G$ be a countably based pro-$p$ group with a filtration series $\mathcal{S}$. Let $H\leq_{\mathrm{c}}G$ be a closed subgroup such that $\xi=\operatorname{hdim}_{G}^{\mathcal{S}}(H)>0$ is given by a proper limit. Suppose further that $\eta\in[0,\xi)$ is such that every finitely generated closed subgroup $K\leq_{\mathrm{c}}H$ satisfies $\operatorname{hdim}_{G}^{\mathcal{S}}(K)\leq\eta$. Then $(\eta,\xi]\subseteq\operatorname{hspec}^{\mathcal{S}}(G)$.

The theorem can be applied directly to pro-$p$ groups $G$ of positive rank gradient, equipped with the Frattini series $\mathcal{F}$. For this, one takes $H\trianglelefteq_{\mathrm{c}}G$ such that $\lvert H\rvert$ and $\lvert G:H\rvert$ are both infinite; then $\xi=1$ and $\eta=0$ yield $\operatorname{hspec}^{\mathcal{F}}(G)=[0,1]$. Proposition 2.10 provides a technical modification of the theorem, tailor-made for dealing with direct products; this approach leads to a proof of Theorem 1.2 in Section 2.

For non-abelian free pro-$p$ groups and for non-soluble Demushkin pro-$p$ groups, we generalise in Section 3 the relevant arguments. This allows us to deal in a similar way with Hausdorff dimension with respect to arbitrary iterated verbal filtrations, and we obtain Theorem 1.4.

In Section 4, we focus on non-abelian free pro-$p$ groups $F$, equipped with the Zassenhaus filtration $\mathcal{Z}$. A standard construction yields a (graded) free restricted ${\mathbb{F}}_{p}$-Lie algebra $\mathsf{R}=\mathsf{R}(F)$, associated to $F$ via $\mathcal{Z}$. Moreover, the Hausdorff dimension function with respect to $\mathcal{Z}$ has a natural Lie-theoretic counterpart, the so-called density function. *A priori * it is not clear that relevant density results for restricted Lie subalgebras of $\mathsf{R}$ can be ‘pulled back’ to Hausdorff dimension results for subgroups of $F$, but, since $F$ is free, the fact that finitely generated subalgebras of infinite codimension in $\mathsf{R}$ have density [math] (see Proposition 4.14) implies that finitely generated infinite-index subgroups of $F$ have Hausdorff dimension [math] with respect to $\mathcal{Z}$ (see Corollary 4.19). Using the Interval Theorem stated above, we obtain Theorem 1.3. In addition we derive Theorem 4.10 and Corollary 4.11 which show, in particular, that free ${\mathbb{F}}_{p}$-Lie algebras and free restricted ${\mathbb{F}}_{p}$-Lie algebras have full density spectra.

Much less seems to be known about the restricted Lie algebras associated to pro-$p$ groups that resemble free pro-$p$ groups. For instance, Lie-theoretic analogues of rank gradient seem to have received little attention. Even extending Theorem 1.3 to Demushkin pro-$p$ groups turns out to be somewhat subtle. By analysing relevant Hilbert–Poincaré series, we obtain Theorem 1.5 in Section 5. With some extra care we subsequently derive Corollary 1.6.

Theorems 1.7, 1.8 and 1.9 which deal with Hausdorff dimensions of normal subgroups are proved in Section 6.

Acknowledgement. We thank Andrei Jaikin-Zapirain for general comments and particularly for alerting us to results in [16] that led to a simplification of our proof of Theorem 1.3. Furthermore, his comments encouraged us to extend our result to Demushkin groups in Theorem 1.5. The third author thanks Ilir Snopche for several discussions about Demushkin groups.

2. Groups with positive rank gradient and the Frattini

series

In this section we establish Theorems 1.1 and 1.2, working with Hausdorff dimensions with respect to the Frattini series. We also present an alternative and more direct proof of Theorem 1.1, which was later suggested to us by Mark Shusterman. It is based on an idea that is used in the proofs of [2, Prop. 13] and [52, Thm. 2.7].

Since the Frattini filtration encodes subgroup generation, it is naturally linked to rank gradient. Other filtrations encode different properties of subgroups, linked to as yet unexplored group invariants, analogous to rank gradient. Our proof of Theorem 1.1 in the language of Hausdorff dimension allows for generalisations in this direction.

2.1. Finitely generated subgroups

We say that a closed subgroup $H$ of a finitely generated pro-$p$ group $G$ has strong Hausdorff dimension with respect to a filtration $\mathcal{S}\colon G_{i}$, $i\in\mathbb{N}_{0}$, if

[TABLE]

is given by a proper limit. We begin with an elementary, but central lemma.

Lemma 2.1**.**

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient, with a filtration $\mathcal{S}\colon G_{i}$, $i\in\mathbb{N}_{0}$.

1. (1)

If $H\leq_{\mathrm{c}}G$ is finitely generated and $\lvert G:H\rvert=\infty$, then $\lim\limits_{i\to\infty}\frac{\operatorname{d}(H\cap G_{i})}{\operatorname{d}(G_{i})}=0$. 2. (2)

If $H\leq_{\mathrm{c}}G$ is infinite, then $\lim\limits_{i\to\infty}\frac{\operatorname{d}(HG_{i})}{\operatorname{d}(G_{i})}=0$.

Proof.

Put $R=\operatorname{RG}(G)=\operatorname{RG}(G,\mathcal{S})>0$, and note that $\operatorname{d}(G_{i})\geq R\,\lvert G:G_{i}\rvert$ for all $i\in\mathbb{N}_{0}$.

First suppose that $H\leq_{\mathrm{c}}G$ is finitely generated and $\lvert G:H\rvert=\infty$. Then each $\operatorname{d}(H\cap G_{i})$ can be bounded by means of the Schreier index formula applied to $H\cap G_{i}\leq_{\mathrm{o}}H$, and $\lvert G:HG_{i}\rvert$ tends to infinity as $i$ does. This yields

[TABLE]

Now suppose that $H\leq_{\mathrm{c}}G$ is infinite. Then, similarly, each $\operatorname{d}(HG_{i})$ can be bounded by means of the Schreier index formula applied to $HG_{i}\leq_{\mathrm{o}}G$, and $\lvert HG_{i}:G_{i}\rvert$ tends to infinity as $i$ does. This yields

[TABLE]

We require two further auxiliary results, which are closely related to one another.

Proposition 2.2**.**

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient, and let $H\leq_{\mathrm{c}}G$ be finitely generated with $\lvert G:H\rvert=\infty$. Then $H$ has strong Hausdorff dimension $\operatorname{hdim}_{G}^{\mathcal{F}}(H)=0$ with respect to the Frattini series $\mathcal{F}$ of $G$.

Proof.

We show that $\operatorname{hdim}_{G}^{\mathcal{F}}(H)\leq\nicefrac{{1}}{{n}}$ for any given $n\in\mathbb{N}$, and from the argument it will be clear that $H$ has strong Hausdorff dimension.

We write $\mathcal{F}\colon G_{i}=\Phi_{i}(G)$, $i\in\mathbb{N}_{0}$. Clearly, $\log_{p}\lvert G_{i}:G_{i+1}\rvert=\operatorname{d}(G_{i})$ for $i\geq 0$, and from $\Phi(H\cap G_{i})\subseteq H\cap\Phi(G_{i})=H\cap G_{i+1}$ we deduce that

[TABLE]

By Lemma 2.1, we find $i_{0}\in\mathbb{N}$ such that $\operatorname{d}(H\cap G_{i})/\operatorname{d}(G_{i})\leq\nicefrac{{1}}{{n}}$ for $i\geq i_{0}$. These observations yield, for $i\geq i_{0}$,

[TABLE]

and hence

[TABLE]

Proposition 2.3**.**

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient, and let $N\trianglelefteq_{\mathrm{c}}G$ be an infinite normal subgroup. Then $N$ has strong Hausdorff dimension $\operatorname{hdim}_{G}^{\mathcal{F}}(N)=1$ with respect to the Frattini series $\mathcal{F}$ of $G$.

Proof.

We write $\mathcal{F}\colon G_{i}=\Phi_{i}(G)$, $i\in\mathbb{N}_{0}$, and observe that

[TABLE]

Hence it suffices to show that the upper limit on the right-hand side equals [math] and the Hausdorff dimension will automatically be strong. Proceeding in a similar way to the proof of Proposition 2.2, we show that the upper limit is at most $\nicefrac{{1}}{{n}}$ for any given $n\in\mathbb{N}$.

Again, $\log_{p}\lvert G_{i}:G_{i+1}\rvert=\operatorname{d}(G_{i})$ for $i\geq 0$, and we observe that

[TABLE]

By Lemma 2.1, we find $i_{0}\in\mathbb{N}$ such that $\operatorname{d}(NG_{i})/\operatorname{d}(G_{i})\leq\nicefrac{{1}}{{n}}$ for $i\geq i_{0}$. These observations yield, for $i\geq i_{0}$,

[TABLE]

and hence

[TABLE]

From Proposition 2.2 we deduce the following corollary.

Corollary 2.4**.**

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient, and let $\mathcal{F}$ denote the Frattini series of $G$. Then the following hold.

1. (1)

If $A,B\leq_{\mathrm{c}}G$ are such that $AB=\{ab\mid a\in A,b\in B\}\leq_{\mathrm{c}}G$ is a subgroup and $A$ is finitely generated with $\lvert G:A\rvert=\infty$, then $\operatorname{hdim}_{G}^{\mathcal{F}}(AB)=\operatorname{hdim}^{\mathcal{F}}_{G}(B)$. 2. (2)

If $K\trianglelefteq_{\mathrm{c}}H\leq_{\mathrm{c}}G$ such that $H/K$ is finitely generated and $\lvert G:H\rvert=\infty$, then $\operatorname{hdim}^{\mathcal{F}}_{G}(H)=\operatorname{hdim}^{\mathcal{F}}_{G}(K)$.

Proof.

(1) Let $A,B\leq_{\mathrm{c}}G$ be such that $AB=\{ab\mid a\in A,b\in B\}\leq G$ and $A$ is finitely generated. Using Proposition 2.2, we obtain

[TABLE]

(2) The claim follows from (1), upon taking $B=K$ and $A=\langle a_{1},\ldots,a_{d}\rangle$ such that $AB=H$. ∎

We are ready to prove our first result, Theorem 1.1.

Proof of Theorem 1.1.

The implications $(3)\Rightarrow(2)\Rightarrow(1)$ are straightforward, and it remains to show $(1)\Rightarrow(3)$. Suppose that $H$ contains an infinite subgroup $K\leq_{\mathrm{c}}G$ that is subnormal via finitely generated successive quotients. This means: $K\leq_{\mathrm{c}}H$ with $\lvert K\rvert=\infty$ and there is a subnormal chain $G=K_{0}\trianglerighteq K_{1}\trianglerighteq\ldots\trianglerighteq K_{s}=K$ such that each factor $K_{i-1}/K_{i}$, for $i\in\{1,\ldots,s\}$, is finitely generated. By Proposition 2.2 it suffices to show that $\operatorname{hdim}_{G}^{\mathcal{F}}(K)=1$, where $\mathcal{F}$ denotes the Frattini series of $G$.

Choose $j\in\{0,1,\ldots,s\}$ maximal with respect to $\lvert G:K_{j}\rvert<\infty$. Clearly, if $j=s$ then $K\leq_{\mathrm{o}}G$ and hence $\operatorname{hdim}_{G}^{\mathcal{F}}(K)=1$. Now suppose that $j<s$. As $K_{j}\leq_{\mathrm{o}}G$, we find $i\in\mathbb{N}$ such that $\Phi_{i}(G)$, the $i$th term of the Frattini series, is contained in $K_{j}$. Replacing $G$ by $\Phi_{i}(G)$ and all other subgroups by their intersections with $\Phi_{i}(G)$, we may assume without loss of generality that $G=K_{0}=\ldots=K_{j}$ and $K_{j+1}\trianglelefteq_{\mathrm{c}}G$. Proposition 2.3 implies that $\operatorname{hdim}_{G}^{\mathcal{F}}(K_{j+1})=1$. Now, several applications of Corollary 2.4, part (2), yield that $1=\operatorname{hdim}_{G}^{\mathcal{F}}(K_{j+1})=\operatorname{hdim}_{G}^{\mathcal{F}}(K_{j+2})=\ldots=\operatorname{hdim}_{G}^{\mathcal{F}}(K_{s})$. ∎

As indicated above, we briefly record an alternative argument that is similar to the one used in the proofs of  [2, Prop. 13] and [52, Thm. 2.7].

Second proof of Theorem 1.1.

As in the first proof, it suffices to show $(1)\Rightarrow(3)$. Suppose that $H,K\leq_{\mathrm{c}}G$ and $G=K_{0}\trianglerighteq K_{1}\trianglerighteq\ldots\trianglerighteq K_{s}=K$ are given as before. For a contradiction, assume that $\lvert G:H\rvert=\infty$.

We show that, for $U\trianglelefteq_{\mathrm{o}}G$,

[TABLE]

this yields the required contradiction, because $\operatorname{RG}(G)>0$ and $\lvert G:H\rvert,\lvert K\rvert=\infty$.

Observe that

[TABLE]

From $\operatorname{d}(U\cap H)\leq(\operatorname{d}(H)-1)\lvert H:U\cap H\rvert+1\leq\operatorname{d}(H)\lvert HU:U\rvert+1$ we deduce that

[TABLE]

For $1\leq i\leq s$, we observe that $(U\cap K_{i-1})/(U\cap K_{i})\cong(UK_{i}\cap K_{i-1})/K_{i}$ and that $\lvert K_{i-1}:UK_{i}\cap K_{i-1}\rvert=\lvert K_{i-1}U:K_{i}U\rvert$; this gives

[TABLE]

We conclude that

[TABLE]

The claim (2.1) follows from (2.2), (2.3) and (2.4). ∎

2.2. Rank gradient and $\ell^{2}$-Betti

numbers

Theorem 1.1 can be regarded as a pro-$p$ analogue of Gaboriau’s theorem [19, Thm. 6.8] for countably infinite groups with positive first $\ell^{2}$-Betti number.

Several results in the literature point to a relationship between the rank gradient and the first $\ell^{2}$-Betti number of finitely generated groups. Let $\Gamma$ be a finitely generated residually finite group and denote its first $\ell^{2}$-Betti number by $b_{1}^{(2)}(\Gamma)$. By combining [2, Thm. 1] and [19, Cor. 3.23], it is established that

[TABLE]

for any descending chain $\mathcal{S}\colon\Gamma_{i}$, $i\in\mathbb{N}_{0}$, of finite-index normal subgroups $\Gamma_{i}\trianglelefteq_{\mathrm{f}}\Gamma$ with trivial intersection $\bigcap_{i}\Gamma_{i}=1$. In fact, in all known examples, the two quantities coincide, but it is not known whether this is true in general.

In order to provide a link between $\Gamma$ and the rank gradient of its pro-$p$ completion, it is useful to introduce the (relative) mod-$p$ homology gradient (or $p$-rank gradient) of $\Gamma$ with respect to a descending chain $\mathcal{S}\colon\Gamma_{i}$, $i\in\mathbb{N}_{0}$, consisting of finite-$p$-power-index normal subgroups; it is

[TABLE]

Notice that, if $\bigcap_{i}\Gamma_{i}=1$ and the completion $\widehat{\Gamma}_{\mathcal{S}}$ with respect to $\mathcal{S}$ and the pro-$p$ completion $\widehat{\Gamma}_{p}$ of $\Gamma$ coincide, then the (relative) mod-$p$ homology gradient coincides with the rank gradient of $\widehat{\Gamma}_{p}$. In general, it is not known under what conditions the equality $\operatorname{RG}_{p}(\Gamma,\mathcal{S})=\operatorname{RG}(\widehat{\Gamma}_{p})$ holds; compare [2]. We thank Andrei Jaikin-Zapirain for pointing this out to us.

Every group $\Gamma$ with positive mod-$p$ homology gradient has exponential subgroup growth and, if $\Gamma$ is finitely presented, it virtually maps onto a non-abelian free group; see [38] and [39], respectively. Inspired by Lück’s approximation theorem for finitely presented residually finite groups, Ershov and Lück [17] showed that $\operatorname{RG}_{p}(\Gamma,\mathcal{S})\geq b_{1}^{(2)}(\Gamma)$ for every finitely generated infinite group $\Gamma$, equipped with a descending chain $\mathcal{S}$ of finite-$p$-power-index normal subgroups that have trivial intersection. In particular, for every residually finite-$p$ group $\Gamma$ with $b_{1}^{(2)}(\Gamma)>0$, the pro-$p$ completion $\widehat{\Gamma}_{p}$ satisfies the hypotheses of Theorem 1.1.

It would be of considerable interest to prove a version of Theorem 1.1, where $G$ is taken to be finitely generated (discrete), residually finite and of positive rank gradient. As this would be an analogue of Gaboriau’s theorem for discrete groups of positive rank gradient, it might explain, or at least strengthen, the connection between rank gradient and first $\ell^{2}$-Betti number. A result like this is beyond the scope of the present paper. Nevertheless, without any extra work, we can state at least a partial result in this direction.

Corollary 2.5**.**

Let $\Gamma$ be a finitely generated, residually finite-$p$ group of positive mod-$p$ homology gradient, with pro-$p$ completion $\widehat{\Gamma}_{p}$. Suppose further:

• $(\dagger)$

For every finitely generated subgroup $\Delta\leq\Gamma$ with $\lvert\Gamma:\Delta\rvert=\infty$ the closure $\overline{\Delta}$ in $\widehat{\Gamma}_{p}$ satisfies $\lvert\widehat{\Gamma}_{p}:\overline{\Delta}\rvert=\infty$.

Then every finitely generated subgroup of $\Gamma$ that is subnormal via finitely generated successive quotients is either finite or of finite index in $\Gamma$.

Proof.

Let $\Delta\leq\Gamma$ be subnormal via finitely generated successive quotients and assume for a contradiction that $\Delta$ is finitely generated, infinite and of infinite index in $\Gamma$. Then, in accordance with our hypothesis, $\overline{\Delta}\leq_{\mathrm{c}}\widehat{\Gamma}_{p}$ has the corresponding properties, in contradiction to Theorem 1.1. ∎

Remark 2.6**.**

The condition $(\dagger)$ in Corollary 2.5 is weaker than being $p$-WLERF in [16], but imposes a noticeable restriction. For instance, non-abelian-free groups have to be excluded: it was shown in [7] that, for any $m\in\mathbb{N}_{0}$, the subgroup $\Delta=\langle x[y,x],y,z_{1},\ldots,z_{m}\rangle$ of the free group $\Psi$ on $m+2$ elements $x,y,z_{1},\ldots,z_{m}$ is dense in the pro-nilpotent topology of $\Psi$ (and thus, in the pro-$p$ topology for any $p$) but of infinite index in $\Psi$ (and, in fact, isolated). We thank Michal Ferov for alerting us to this example.

A slight modification shows that any group $\Gamma$ that is a free product of a virtually non-abelian-free group and any other finitely generated group does not satisfy $(\dagger)$.

Question 2.7**.**

Identify examples of finitely generated, residually finite-$p$ groups of positive mod-$p$ homology gradient that satisfy condition $(\dagger)$ in Corollary 2.5.

Some potential candidates include the examples constructed in [45] and [48].

Remark 2.8**.**

The arguments in the second proof of Theorem 1.1 can be used verbatim to prove an analogue of Corollary 2.5 for finitely generated, residually finite groups $\Gamma$ of positive rank gradient, where the pro-$p$ completion $\widehat{\Gamma}_{p}$ is replaced by the profinite completion $\widehat{\Gamma}$. We do not pursue this line of investigation here, but merely point out that the analogue of $(\dagger)$ in this setting is a weaker assumption, since free discrete groups do satisfy it; compare [22, Thm. 5.1].

We end this section with another natural question. It appears that all currently known examples of finitely generated pro-$p$ groups of positive rank gradient have positive $p$-deficiency (which gives a lower bound for the rank gradient), or satisfy a similar condition.

Question 2.9**.**

Is there a finitely generated pro-$p$ group of positive rank gradient whose $p$-deficiency in the sense of [48] (and in any similar sense) is [math]?

2.3. Hausdorff spectrum with respect to the Frattini series

Consider a direct product $G=A\times B$ of finitely generated pro-$p$ groups $A$ and $B$, equipped with a filtration series $\mathcal{S}\colon G_{i}$, $i\in\mathbb{N}_{0}$, and let $\mathcal{S}|_{A}\colon A_{i}=A\cap G_{i}$, $i\in\mathbb{N}_{0}$, and $\mathcal{S}|_{B}\colon B_{i}=B\cap G_{i}$, $i\in\mathbb{N}_{0}$, denote the induced filtration series. In general, there is no straightforward reduction of the Hausdorff dimension function $\operatorname{hdim}_{G}^{\mathcal{S}}$ to the related Hausdorff dimension functions $\operatorname{hdim}_{A}^{\mathcal{S|_{A}}}$ and $\operatorname{hdim}_{B}^{\mathcal{S|_{B}}}$. Even if $G_{i}=A_{i}\times B_{i}$ for all $i\in\mathbb{N}_{0}$ and if $H\leq_{\mathrm{c}}G$ decomposes as $H=C\times D$ with $C\leq_{\mathrm{c}}A$ and $D\leq_{\mathrm{c}}B$, it is not generally true that $\operatorname{hdim}_{G}^{\mathcal{S}}(H)=\operatorname{hdim}_{A}^{\mathcal{S}|_{A}}(C)\operatorname{hdim}_{G}^{\mathcal{S}}(A)+\operatorname{hdim}_{B}^{\mathcal{S}|_{B}}(D)\operatorname{hdim}_{G}^{\mathcal{S}}(B)$; the right-hand side merely gives a lower bound for the left-hand side.

Despite these difficulties the proof of the Interval Theorem [28, Thm. 5.4] can be slightly modified to yield the following convenient tool for investigating the Hausdorff spectra of finite direct products of pro-$p$ groups.

Proposition 2.10**.**

Let $G=H_{1}\times\ldots\times H_{r}$ be a finite direct product of countably-based pro-$p$ groups $H_{j}$, equipped with a filration $\mathcal{S}\colon G_{i}=H_{1,i}\times\ldots\times H_{r,i}$, $i\in\mathbb{N}_{0}$, that arises as the product of filtrations $\mathcal{S}_{j}\colon H_{j,i}$, $i\in\mathbb{N}_{0}$, of $H_{j}$ for $1\leq j\leq r$.

Let $K\leq_{\mathrm{c}}G$ be such that $K=K_{1}\times\ldots\times K_{r}$ with $K_{j}\leq_{\mathrm{c}}H_{j}$ for $1\leq j\leq r$. Suppose that $0\leq\varepsilon<\delta\leq 1$ are such that, for each $j\in\{1,\ldots,r\}$, the group $K_{j}$ has strong Hausdorff dimension $\operatorname{hdim}_{H_{j}}^{\mathcal{S}_{j}}(K_{j})=\delta$ and that all finitely generated subgroups $L\leq_{\mathrm{c}}K_{j}$ have Hausdorff dimension $\operatorname{hdim}_{H_{j}}^{\mathcal{S}_{j}}(L)\leq\varepsilon$.

Then the Hausdorff spectrum of $G$ satisfies $(\varepsilon,\delta]\subseteq\operatorname{hspec}^{\mathcal{S}}(G)$.

Our strategy for proving Theorem 1.2 is to apply this result in a situation, where the $K_{j}\trianglelefteq_{\mathrm{c}}H_{j}$ are infinite normal subgroups of infinite index, with Hausdorff dimension $\delta=1$, and such that $\varepsilon$ can be taken to be [math].

Lemma 2.11**.**

Let $G$ be a finitely generated infinite pro-$p$ group, and let $N\trianglelefteq_{\mathrm{c}}G$ be a finite normal subgroup. Then $\operatorname{RG}(G/N)=\lvert N\rvert\,\operatorname{RG}(G)$.

Proof.

A filtration $\mathcal{S}\colon G_{i}$, $i\in\mathbb{N}$, of $G$ induces a filtration $\mathcal{S}|_{G/N}\colon G_{i}N/N$, $i\in\mathbb{N}$, of $G/N$. Since $N$ is finite, there exists $i_{0}\in\mathbb{N}$ such that $G_{i}\cap N=1$ for $i\geq i_{0}$. This implies

[TABLE]

We thank Mikhail Ershov for pointing out to us that the next result is essentially proved in [8, Thm. 3.1].

Proposition 2.12**.**

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient. Then $G$ admits an infinite normal subgroup $N\trianglelefteq_{\mathrm{c}}G$ such that $G/N$ has positive rank gradient. Consequently, $G$ is not commensurable to any just infinite pro-$p$ group.

Proof.

Following [8], for each normal subgroup $N\trianglelefteq_{\mathrm{c}}G$, we define $s(N,G)$ to be the infimum, over all (countable) sets $X$ of normal generators of $N$ in $G$, of the quantity $\sum_{x\in X}p^{-\nu(x)}$, where $\nu(x)=\sup\{k\in\mathbb{N}_{0}\mid x\in G^{p^{k}}\}$ for each $x$. We observe that Theorem 3.1 in [8], which is stated and proved for finitely generated discrete groups, translates almost verbatim to the pro-$p$ context:

[TABLE]

As $\operatorname{RG}(G)>0$, we find $k\in\mathbb{N}$ such that $\operatorname{RG}(G)-p^{-k}>0$. As $G$ is infinite, so is the open normal subgroup $G^{p^{k}}$, and we find among a finite generating set at least one element $x\in G^{p^{k}}\smallsetminus G^{p^{k+1}}$ whose normal closure $N=\langle x\rangle^{G}$ is infinite. We observe that $s(N,G)=p^{-k}$ and conclude that

[TABLE]

Repeating the above procedure, we find an ascending chain $N=N_{1}\lneqq N_{2}\lneqq\ldots$ of normal subgroups of $G$ such that the factor groups $N_{i+1}/N_{i}$, $i\in\mathbb{N}$, are infinite. In particular, $G$ has infinitely many commensurability classes of normal subgroups. This implies that $G$ is not (abstractly) commensurable to a just infinite pro-$p$ group. ∎

Corollary 2.13**.**

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient. Then $G$ has an infinite normal subgroup $N\trianglelefteq_{\mathrm{c}}G$ with $\lvert G:N\rvert=\infty.$

Proof.

Assume, for a contradiction, that $G$ has no infinite normal subgroup $N\trianglelefteq_{\mathrm{c}}G$ with $\lvert G:N\rvert=\infty$. Using Zorn’s Lemma we see that $G$ has a just infinite quotient $G/N$ whose kernel $N$ must be finite. Lemma 2.11 implies that $G/N$ has positive rank gradient, in contradiction to Proposition 2.12. ∎

Proof of Theorem 1.2.

Let $G=H_{1}\times\ldots\times H_{r}$ be a non-trivial finite direct product of finitely generated pro-$p$ groups $H_{i}$ of positive rank gradient. Observe that the Frattini series $\mathcal{F}$ of $G$ decomposes as the product of the Frattini series $\mathcal{F}_{j}$ of the direct factors $H_{j}$ for $1\leq j\leq r$.

By Corollary 2.13, for each $j\in\{1,\ldots,r\}$, the group $H_{j}$ has an infinite normal subgroup $N_{j}\trianglelefteq_{\mathrm{c}}H_{j}$ with $\lvert G_{j}:N_{j}\rvert=\infty$. From Proposition 2.3 we deduce that $N_{j}$ has strong Hausdorff dimension $\operatorname{hdim}_{H_{j}}^{\mathcal{F}_{j}}(N_{j})=1$; from Proposition 2.2 we conclude that, finitely generated subgroups $L\leq_{\mathrm{c}}N_{j}$ have Hausdorff dimension [math] with respect to $\mathcal{F}_{j}$. Thus, Proposition 2.10 shows that $G$ has full Hausdorff spectrum. ∎

3. Hausdorff spectrum with respect to iterated filtration

series

In this section we prove Theorem 1.4, which generalises Theorem 1.2 for certain groups to iterated verbal filtration series, defined in Section 1; the key task is to extend Propositions 2.2 and 2.3.

Let $G$ be a finitely generated pro-$p$ group equipped with a filtration $\mathcal{S}\colon G_{i}$, $i\in\mathbb{N}_{0}$. Let $H\leq_{\mathrm{c}}G$ be finitely generated with $\lvert G:H\rvert=\infty$. In order to prove that $H$ has strong Hausdorff dimension $\operatorname{hdim}_{G}^{\mathcal{S}}(H)=0$, in analogy to Proposition 2.2, it suffices to find $i_{0}\in\mathbb{N}$ and an unbounded monotone increasing sequence $(m_{i})_{i\geq i_{0}}$ such that

[TABLE]

For, if (3.1) holds, then we obtain for $i\geq i_{0}$,

[TABLE]

and hence

[TABLE]

Moreover, the argument shows that $H$ has strong Hausdorff dimension $\operatorname{hdim}_{G}^{G_{i}}(H)=0$.

Similarly, if $N\trianglelefteq_{\mathrm{c}}G$ is an infinite normal subgroup, we can prove that $N$ has strong Hausdorff dimension $\operatorname{hdim}_{G}^{\mathcal{S}}(N)=1$ in analogy to Proposition 2.3, once we have identified $i_{0}\in\mathbb{N}$ and an unbounded monotone increasing sequence $(m_{i})_{i\geq i_{0}}$ such that

[TABLE]

As above, let $H\leq_{\mathrm{c}}G$ be finitely generated with $\lvert G:H\rvert=\infty$ and let $N\trianglelefteq_{\mathrm{c}}G$ be infinite normal. A natural way to guarantee that conditions (3.1) and (3.3) hold is to identify an unbounded increasing sequence $(m_{i})_{i\in\mathbb{N}}$ such that

1. (a)

$G_{i}$ maps homomorphically onto $(H\cap G_{i})^{m_{i}}$, respectively $(NG_{i})^{m_{i}}$, where 2. (b)

the image of $G_{i+1}$ is contained in $(H\cap G_{i+1})^{m_{i}}$, respectively $(NG_{i+1})^{m_{i}}$.

One way of ensuring that (b) holds is by requiring that $G_{i+1}$ be a verbal subgroup of $G_{i}$, for each $i$. Recall that a verbal subgroup of a group $G$ is one generated by all values $w(g_{1},\ldots,g_{r})$, for $g_{1},\ldots,g_{r}\in G$, of group words $w=w(X_{1},\ldots,X_{r})$ in a given set $\mathcal{W}$. Examples of filtrations satisfying this include the Frattini series, which we already covered, and the iterated $p$-power series defined in Section 1. One could even take different sets $\mathcal{W}_{i}$ of words at each stage to define $G_{i}$.

One way of ensuring that (a) holds is, for instance, by arranging that $G_{i}$ maps onto the free pro-$p$ group on $m_{i}\operatorname{d}(H\cap G_{i})$, respectively $m_{i}\operatorname{d}(NG_{i})$, generators.

Proof of Theorem 1.4.

The main task is to extend Propositions 2.2 and 2.3 to the situation, where (i) $G$ is either a finitely generated non-abelian free pro-$p$ group or a non-soluble Demushkin pro-$p$ group and (ii) the Hausdorff dimension function is associated to an iterated verbal filtration $\mathcal{S}\colon G_{i}$, $i\in\mathbb{N}_{0}$, of $G$. In particular, $G$ has positive rank gradient.

First suppose that $H\leq_{\mathrm{c}}G$ is finitely generated and that $\lvert G:H\rvert=\infty$. Lemma 2.1 ensures that for each $n\in\mathbb{N}$, there exists $i_{0}\in\mathbb{N}$ such that $n\operatorname{d}(H\cap G_{i})\leq\operatorname{d}(G_{i})$ for $i\geq i_{0}$. Now, if $G$ is free, then so is each $G_{i}$; hence $G_{i}$ maps homomorphically onto the direct product of $n$ copies of $H\cap G_{i}$. It is known that, if $G$ is a non-soluble Demushkin group, then so is $G_{i}$; compare [14, 3, 49]. In this case $G_{i}$ maps homomorphically onto a free pro-$p$ group of rank $\lfloor\operatorname{d}(G_{i})/2\rfloor$ and consequently onto the direct product of $\lfloor n/2\rfloor-1$ copies of $H\cap G_{i}$; compare [55, Thm. 12.3.1] and [33, 49]. Still considering $i\geq i_{0}$, write $m$ for the number of copies of $H\cap G_{i}$ that $G_{i}$ maps onto ($n$ in the first case, $\lfloor n/2\rfloor-1$ in the second case). Since $G_{i+1}$ is a verbal subgroup of $G_{i}$, there is an induced epimorphism $G_{i}/G_{i+1}\to((H\cap G_{i})/(H\cap G_{i+1}))^{m}$. Repeating this argument for $n\to\infty$, we construct an unbounded monotone increasing sequence $(m_{i})_{i\geq i_{0}}$ such that (3.1) holds. Proceeding as in (3.2), we obtain $\operatorname{hdim}_{G}^{\mathcal{S}}(H)=0$.

Next suppose that $N\trianglelefteq_{\mathrm{c}}G$ is infinite. Arguing very similarly, we obtain $i_{0}\in\mathbb{N}$ and an unbounded monotone increasing sequence $(m_{i})_{i\geq i_{0}}$ such that (3.3) holds. From this we derive that $\operatorname{hdim}_{G}^{\mathcal{S}}(N)=1$.

Now let $G=E_{1}\times\ldots\times E_{r}$ be a non-trivial finite direct product of finitely generated pro-$p$ groups $E_{j}$, each of which is either free or Demushkin, and let $G$ be equipped with an iterated verbal filtration $\mathcal{S}\colon G_{i}$, $i\in\mathbb{N}_{0}$. Suppose further that $E_{1}$ is non-soluble. In order to deduce that $\operatorname{hspec}^{\mathcal{S}}(G)=[0,1]$ from Proposition 2.10, as in the proof of Theorem 1.2, we only need to explain how to deal with possible soluble direct factors $E_{j}$ in the decomposition of $G$. Indeed, let $E=E_{j}$ be one of these soluble factors. Then each of the groups $E\cap G_{i}$, $i\in\mathbb{N}_{0}$, is of the form $1$, $C_{2}$, ${\mathbb{Z}}_{p}$ or ${\mathbb{Z}}_{p}\ltimes{\mathbb{Z}}_{p}$ and can be generated by at most $2$ elements. From this we deduce that

[TABLE]

because $(E_{1}\cap G_{i})/(E_{1}\cap G_{i+1})$ maps onto $((E\cap G_{i})/(E\cap G_{i+1}))^{m_{i}}$ for an unbounded monotone increasing sequence $(m_{i})_{i\geq i_{0}}$, by an argument very similar to the one given above for $H\leq_{\mathrm{c}}G$. ∎

Remark 3.1**.**

(1) Perhaps it is not too surprising that our approach works for filtrations whose terms are recursively defined as verbal subgroups, because such filtrations grow at least as fast as the Frattini series. To wit, it is not hard to show that every proper verbal subgroup of a finitely generated pro-$p$ group $G$ is contained in the Frattini subgroup $\Phi(G)$ of $G$, because the elementary abelian $p$-group $G/\Phi(G)$ has no proper non-trivial verbal subgroups.

(2) It is an interesting open problem to identify further finitely generated pro-$p$ groups of positive rank gradient – apart from free and Demushkin groups – for which the conclusion of Theorem 1.4 holds, possibly even by a similar reasoning.

4. Non-abelian free pro-$p$ groups and the Zassenhaus

series

This section contains the proof of Theorem 1.3 which relies on Lie-theoretic techniques. In the context of graded (restricted) Lie algebras, the concept of density, which we recall in Definition 4.3, is a natural counterpart to the notion of Hausdorff dimension.

Results for free Lie algebras in [6, 43] imply that non-zero ideals and finitely generated graded subalgebras of free Lie algebras have density $1$ and [math], respectively. We establish analogous results for restricted Lie algebras; see Propositions 4.14 and 4.12. A Lie-theoretic counterpart to the Interval Theorem, stated at the end of Section 1, yields that free (restricted) Lie algebras have full density spectrum; see Theorem 4.10 and Corollary 4.11. To obtain the full Hausdorff spectrum for free pro-$p$ groups from that of free restricted Lie algebras, we need to translate between subgroups of a free pro-$p$ group $F$ and subalgebras of the associated free restricted ${\mathbb{F}}_{p}$-Lie algebra. This is possible due to consequences of a result of Ershov and Jaikin-Zapirain [16, Prop. 3.2], which in turn relies on power-commutator factorisations in $F$ resulting from Hall’s commutator collection process.

It would be very interesting to extend these methods so that they cover other pro-$p$ groups resembling free pro-$p$ groups and their corresponding restricted ${\mathbb{F}}_{p}$-Lie algebras. Already the case of one-relator groups poses challenges; we deal with Demushkin pro-$p$ groups in Section 5.

Notation*.*

Group commutators are written as $[x,y]=x^{-1}y^{-1}xy$. Lie commutators are denoted by $[\mathsf{x},\mathsf{y}]=\mathsf{x}.\mathrm{ad}(\mathsf{y})$. Group and Lie commutators of higher degree are left-normed; for instance, $[x,y,z]=[[x,y],z]$ and $[\mathsf{x},\mathsf{y},\mathsf{z}]=[[\mathsf{x},\mathsf{y}],\mathsf{z}]$.

Additionally, we write $\langle X\rangle_{\mathrm{Lie}}$ respectively, $\langle X\rangle_{\mathrm{res.\,Lie}}$ for the Lie subalgebra, respectively, restricted Lie subalgebra generated by a set $X$. We also write $\langle X\rangle_{L}$ for the (restricted) Lie ideal generated by $X$ in $L$.

4.1. Associated restricted Lie

algebras

Recall from Section 1 the definition of the Zassenhaus series $\mathcal{Z}\colon Z_{i}(G)$, $i\in\mathbb{N}$, of a finitely generated pro-$p$ group $G$. It is the fastest descending series with $[Z_{i}(G),Z_{j}(G)]\subseteq Z_{i+j}(G)$ and $Z_{i}(G)^{p}\subseteq Z_{ip}(G)$ for all $i,j\geq 1$. Lazard [40, Thm. 5.6] proved that

[TABLE]

where $\gamma_{i}(G)$ denotes the $i$th term of the lower central series of $G$. Each $Z_{n}(G)$ is open in $G$ and the Zassenhaus series provides a filtration of $G$; compare [15, Sec. 11.3]. Note that each quotient $\mathsf{R}_{n}(G)=Z_{n}(G)/Z_{n+1}(G)$ is an elementary abelian $p$-group, which can be regarded as an ${\mathbb{F}}_{p}$-vector space. The direct sum

[TABLE]

carries naturally the structure of a finitely generated graded restricted ${\mathbb{F}}_{p}$-Lie algebra, with the Lie bracket and the $p$-map induced from group commutators and the $p$-power map on (representatives of) homogeneous elements; compare [15, Sec. 12] or [50]. For convenience we recall what this means and introduce the usual notation; see [25, §V.7] or [5, §1.11].

Definition 4.1**.**

A restricted Lie algebra (or Lie $p$-algebra) $\mathsf{R}=(\mathsf{R},[\cdot,\cdot],{\cdot}^{[p]})$ over the field ${\mathbb{F}}_{p}$ consists of a ${\mathbb{F}}_{p}$-Lie algebra $\mathsf{R}$, with Lie bracket $[\cdot,\cdot]$, and a $p$-map $\mathsf{R}\to\mathsf{R}$, $\mathsf{x}\mapsto\mathsf{x}^{[p]}$, satisfying the following properties:

• (i)

$[\mathsf{x},\mathsf{y}^{[p]}]=\mathsf{x}.\mathrm{ad}(\mathsf{y})^{p}$ for $\mathsf{x},\mathsf{y}\in\mathsf{R}$,

• (ii)

$(\alpha\mathsf{x})^{[p]}=\alpha\mathsf{x}^{[p]}$ for $\alpha\in{\mathbb{F}}_{p}$ and $\mathsf{x}\in\mathsf{R}$, (note that $\alpha^{p}=\alpha$ for $\alpha\in{\mathbb{F}}_{p}$)

• (iii)

$(\mathsf{x}+\mathsf{y})^{[p]}=\mathsf{x}^{[p]}+\mathsf{y}^{[p]}+\sum\limits_{i=1}^{p-1}s_{i}(\mathsf{x},\mathsf{y})$ for $\mathsf{x},\mathsf{y}\in R$, where $i\,s_{i}(\mathsf{x},\mathsf{y})$ is the coefficient of $\lambda^{i-1}$ in the generic expansion of $\mathsf{x}.(\mathrm{ad}(\lambda\mathsf{x}+\mathsf{y}))^{p-1}$.

The restricted Lie algebra $\mathsf{R}$ is ($\mathbb{N}$-)graded, if $\mathsf{R}=\bigoplus_{n=1}^{\infty}\mathsf{R}_{n}$ decomposes as a direct sum of subspaces, which we refer to as homogeneous components, such that $[\mathsf{R}_{m},\mathsf{R}_{n}]\subseteq\mathsf{R}_{m+n}$ and $\mathsf{R}_{n}^{\,[p]}\subseteq\mathsf{R}_{pn}$ for $m,n\in\mathbb{N}$.

Restricted Lie subalgebras and restricted Lie ideals of $\mathsf{R}$ are defined in the usual way. A restricted Lie subalgebra $\mathsf{S}$ of $\mathsf{R}$ forms a subideal if there exists a finite sequence

[TABLE]

of restricted Lie subalgebras of $\mathsf{R}$, where each term is a Lie ideal in the preceding one.

Remark 4.2**.**

For every prime $p$, the $p$-Zassenhaus series of a discrete group $\Gamma$ is defined in the same way as for pro-$p$ groups and gives rise to a corresponding graded restricted ${\mathbb{F}}_{p}$-Lie algebra. If $\Gamma$ is finitely generated, with pro-$p$ completion $G=\widehat{\Gamma}_{p}$, the quotients $\Gamma/Z_{n}(\Gamma)$ are canonically isomorphic to $G/Z_{n}(G)$ for $n\in\mathbb{N}$. Furthermore, the associated restricted ${\mathbb{F}}_{p}$-Lie algebra $\mathsf{R}(\Gamma)$ is canonically isomorphic to $\mathsf{R}(G)$.

If $\Gamma$ is free on $d\geq 2$ generators, then $G=\widehat{\Gamma}_{p}$ is a free pro-$p$ group on $d$ generators, and $\mathsf{R}(\Gamma)\cong\mathsf{R}(G)$ is the free restricted Lie algebra on $d$ generators, equipped with the natural grading; see [40, Thm. 6.5].

The (lower) density of subalgebras in graded restricted ${\mathbb{F}}_{p}$-Lie algebras and in graded ${\mathbb{F}}_{p}$-Lie algebras is defined as follows; see [43, 6] for related notions, indeed closely related for graded subalgebras.

Definition 4.3**.**

Let $\mathsf{R}=\bigoplus_{n=1}^{\infty}\mathsf{R}_{n}$ be a non-zero graded restricted ${\mathbb{F}}_{p}$-Lie algebra such that each homogeneous component $\mathsf{R}_{n}$ is finite dimensional. For each $n\in\mathbb{N}$, denote by $\mathsf{R}_{\geq n}=\bigoplus_{m=n}^{\infty}\mathsf{R}_{m}$ the ideal of $\mathsf{R}$ consisting of all elements of degree at least $n$, and set $\mathsf{R}_{>n}=\mathsf{R}_{\geq n+1}$.

Let $\mathsf{S}$ be a restricted Lie subalgebra of $\mathsf{R}$. For each $n\in\mathbb{N}$, put

[TABLE]

so that $\mathsf{S}_{\mathrm{grd}}=\bigoplus_{n=1}^{\infty}\mathsf{S}_{n}$ is a graded restricted Lie subalgebra of $\mathsf{R}$.

The density of $\mathsf{S}$ in $\mathsf{R}$ is defined as

[TABLE]

The density spectrum of the graded restricted Lie algebra $\mathsf{R}$ is defined as

[TABLE]

Analogously, we define the density of Lie subalgebras $\mathsf{S}$ in a graded ${\mathbb{F}}_{p}$-Lie algebra $\mathsf{L}=\bigoplus_{n=1}^{\infty}\mathsf{L}_{n}$, with finite dimensional homogeneous components $\mathsf{L}_{n}$, and we employ the same notation; in particular, $\mathsf{L}_{\geq n}=\bigoplus_{m=n}^{\infty}\mathsf{L}_{m}$ and $\mathsf{L}_{>n}=\mathsf{L}_{\geq n+1}$ for $n\in\mathbb{N}$.

In our setting, $G$ is a finitely generated infinite pro-$p$ group with associated graded restricted ${\mathbb{F}}_{p}$-Lie algebra $\mathsf{R}(G)$. The Hausdorff dimension of a subgroup $H\leq_{\mathrm{c}}G$ with respect to the Zassenhaus series $\mathcal{Z}$ of $G$ coincides with the density of the associated graded restricted Lie subalgebra

[TABLE]

of $\mathsf{R}(G)$, that is $\operatorname{hdim}_{G}^{\mathcal{Z}}(H)=\operatorname{den}_{\mathsf{R}(G)}(\mathsf{R}_{G}(H))$; compare [50, §4.1].

Note that, if $H$ is (sub)normal in $G$, then $\mathsf{R}_{G}(H)$ is a restricted Lie (sub)ideal in $\mathsf{R}(G)$; in the converse direction, if the restricted Lie subalgebra $\mathsf{R}_{G}(H)$ is finitely generated, so is the subgroup $H$. However, in general the mapping $H\mapsto\mathsf{R}_{G}(H)$ from closed subgroups of $G$ to restricted Lie subalgebras of $\mathsf{R}(G)$ is neither injective nor surjective. Furthermore, the respective minimal numbers of generators may change substantially; in particular, a finitely generated subgroup need not produce a finitely generated restricted Lie subalgebra. Nor does a free subgroup need to produce a free restricted Lie subalgebra; compare Section 4.3.

In Section 4.2 we compute the density spectrum of free (restricted) Lie algebras. In Section 4.3 we provide a bridge to the Hausdorff spectrum of the corresponding free pro-$p$ groups; as indicated, the transition requires some care.

4.2. The density spectrum of finitely generated free

(restricted) Lie algebras

Let $d\in\mathbb{N}$ with $d\geq 2$. In this section, $\mathsf{R}$ denotes a free restricted ${\mathbb{F}}_{p}$-Lie algebra on $d$ generators $\mathsf{x}_{1},\ldots,\mathsf{x}_{d}$, and $\mathsf{L}$ denotes a free ${\mathbb{F}}_{p}$-Lie algebra on $\mathsf{x}_{1},\ldots,\mathsf{x}_{d}$ (embedded as a Lie subalgebra into $\mathsf{R}$). The Zassenhaus series of $\mathsf{R}$ – or indeed, on a free pro-$p$ group $F$ on $d$ generators, if we think of $\mathsf{R}$ as $\mathsf{R}=\mathsf{R}(F)$ – produces the natural grading $\mathsf{R}=\bigoplus_{n=1}^{\infty}\mathsf{R}_{n}$ associated with Lie words. This induces a compatible grading $\mathsf{L}=\bigoplus_{n=1}^{\infty}\mathsf{L}_{n}$ on $\mathsf{L}$.

In fact, for all practical purposes, $\mathsf{R}$ and $\mathsf{L}$ can be constructed as subalgebras of the free associative algebra

$\mathsf{A}={\mathbb{F}}_{p}\langle\mathsf{x}_{1},\ldots,\mathsf{x}_{d}\rangle$, turned into a restricted ${\mathbb{F}}_{p}$-Lie algebra via $[\mathsf{u},\mathsf{v}]=\mathsf{u}\mathsf{v}-\mathsf{v}\mathsf{u}$ and $\mathsf{u}^{[p]}=\mathsf{u}^{p}$ for $\mathsf{u},\mathsf{v}\in\mathsf{A}$. From this perspective, the homogeneous components $\mathsf{R}_{n}$ and $\mathsf{L}_{n}$ arise as intersections of $\mathsf{R}$, respectively $\mathsf{L}$, with the ${\mathbb{F}}_{p}$-subspace $\mathsf{A}_{n}$ spanned by all products of length $n$ in $\mathsf{x}_{1},\ldots,\mathsf{x}_{d}$; compare [5, §2].

It is well-known that each homogeneous component $\mathsf{L}_{n}$ of $\mathsf{L}$ is spanned, as an ${\mathbb{F}}_{p}$-vector space, by linearly independent basic Lie commutators in $\mathsf{x}_{1},\ldots,\mathsf{x}_{d}$ of weight $n$; compare [23, §11] or [5, §2]. We recall the relevant notions and record the analogous result for the free restricted Lie algebra $\mathsf{R}$.

Definition 4.4**.**

The weight of a non-zero Lie commutator in a countable set $X$ of free generators is defined inductively: each generator $\mathsf{x}\in X$ has weight $\operatorname{wt}(\mathsf{x})=1$, and, if $\mathsf{c}_{1},\mathsf{c}_{2}$ are Lie commutators in $X$ such that $[\mathsf{c}_{1},\mathsf{c}_{2}]\neq 0$, then $[\mathsf{c}_{1},\mathsf{c}_{2}]$ is a Lie commutator of weight $\operatorname{wt}([\mathsf{c}_{1},\mathsf{c}_{2}])=\operatorname{wt}(\mathsf{c}_{1})+\operatorname{wt}(\mathsf{c}_{2})$.

The $X$-basic commutators are also defined inductively. The $X$-basic commutators of weight $1$ are the generators contained in $X$ in some well-order. For each $n\in\mathbb{N}$ with $n\geq 2$, after defining $X$-basic commutators of weight less than $n$, the $X$-basic commutators of weight $n$ are those Lie commutators $[\mathsf{u},\mathsf{v}]$ such that:

1. (1)

$\mathsf{u},\mathsf{v}$ are $X$-basic commutators with $\operatorname{wt}(\mathsf{u})+\operatorname{wt}(\mathsf{v})=n$, 2. (2)

$\mathsf{u}>\mathsf{v}$; and if $\mathsf{u}=[\mathsf{y},\mathsf{z}]$ with $X$-basic commutators $\mathsf{y},\mathsf{z}$, then $\mathsf{v}\geq\mathsf{z}$.

Lastly, the well-order is extended, subject to the condition that $\mathsf{u}<\mathsf{v}$ whenever $\operatorname{wt}(\mathsf{u})<\operatorname{wt}(\mathsf{v})$ and arbitrarily among the new $X$-basic commutators of weight $n$. When the generating set $X$ is clear from the context, we speak of basic commutators, and a similar terminology is used for group commutators.

For the following fundamental result we refer to [5, §2 Thm. 1 and §2 Prop. 14].

Proposition 4.5**.**

Let $\mathsf{R}=\bigoplus_{n=1}^{\infty}\mathsf{R}_{n}$ be the free restricted ${\mathbb{F}}_{p}$-Lie algebra on $\mathsf{x}_{1},\ldots,\mathsf{x}_{d}$, and let $\mathsf{L}\leq\mathsf{R}$ be the free Lie algebra on $\mathsf{x}_{1},\ldots,\mathsf{x}_{d}$, as above. For each $n\in\mathbb{N}$, with $n=p^{j}m$ and $p\nmid m$, the ${\mathbb{F}}_{p}$-space $\mathsf{R}_{n}$ is spanned by the linearly independent elements

[TABLE]

In particular, $\dim_{{\mathbb{F}}_{p}}(\mathsf{R}_{n})=\sum_{i=0}^{j}\dim_{{\mathbb{F}}_{p}}(\mathsf{L}_{n/p^{i}})$.

Witt’s formula gives $\dim_{{\mathbb{F}}_{p}}(\mathsf{L}_{n})=\tfrac{1}{n}\sum_{l\mid n}\mu(l)d^{n/l}$, where $\mu\colon\mathbb{N}\to\{0,1,-1\}$ denotes the classical Möbius function; see [25, Ex. 2 on p. 194] or [5, §3 Thm. 1]. This implies

[TABLE]

where $n=p^{j}m$ with $p\nmid m$ as in Proposition 4.5.

We obtain simple, but significant consequences for the dimensions of homogeneous components of $\mathsf{R}$ and $\mathsf{L}$, including their relation to one another; see [26, Lem. 4.3].

Lemma 4.6**.**

In the setup described above,

1. (1)

\dim_{{\mathbb{F}}_{p}}(\mathsf{R}_{n})=\big{(}1+o(1)\big{)}\frac{d^{n}}{n}* and \dim_{{\mathbb{F}}_{p}}(\mathsf{L}_{n})=\big{(}1+o(1)\big{)}\frac{d^{n}}{n} as $n\to\infty$;* 2. (2)

\dim_{{\mathbb{F}}_{p}}(\mathsf{R}/\mathsf{R}_{>n})=\big{(}1+o(1)\big{)}\frac{d^{n+1}}{n(d-1)}* and \dim_{{\mathbb{F}}_{p}}(\mathsf{L}/\mathsf{L}_{>n})=\big{(}1+o(1)\big{)}\frac{d^{n+1}}{n(d-1)} as $n\to\infty$.*

Consequently, for every graded Lie subalgebra $\mathsf{S}\leq\mathsf{R}$ we have $\operatorname{den}_{\mathsf{R}}(\mathsf{S})=\operatorname{den}_{\mathsf{L}}(\mathsf{S}\cap\mathsf{L})$.

Proof.

The estimates (1) and (2) are proved in [26, Lem. 4.3]. We justify the conclusion for graded Lie subalgebras $\mathsf{S}\leq\mathsf{R}$. For $m\in\mathbb{N}$ we write

[TABLE]

By (2), we have

[TABLE]

Clearly, for $n\in\mathbb{N}$ the inequalities

[TABLE]

hold. Dividing by $\sum_{m=1}^{n}\dim_{{\mathbb{F}}_{p}}(\mathsf{R}_{m})$ or, asymptotically equivalent, by $\sum_{m=1}^{n}\dim_{{\mathbb{F}}_{p}}(\mathsf{L}_{m})$, and passing to the lower limit as $n\to\infty$, we obtain

[TABLE]

The next lemma is elementary, but central for our constructions.

Lemma 4.7**.**

In the setup described above, let $\mathsf{M}\leq\mathsf{L}$ be a Lie subalgebra. Suppose that $\mathsf{M}=\langle\mathsf{y}\mid\mathsf{y}\in Y\rangle_{{\mathbb{F}}_{p}}$, as an ${\mathbb{F}}_{p}$-vector space, and let $\mathsf{S}=\langle\mathsf{M}\rangle_{\mathrm{res.\,Lie}}\leq\mathsf{R}$ be the restricted Lie subalgebra generated by $\mathsf{M}$. Then we have

[TABLE]

Moreover, if $\mathsf{M}\trianglelefteq\mathsf{L}$ is a Lie ideal, then $[\mathsf{S},\mathsf{R}]\subseteq\mathsf{M}$ and consequently $\mathsf{S}\trianglelefteq\mathsf{R}$ is a restricted Lie ideal.

Proof.

The first equation is easily verified, using the defining properties of the $p$-map. For the second equation, we may choose $Y=\{\mathsf{y}_{1},\mathsf{y}_{2},\ldots\}$ to be an ${\mathbb{F}}_{p}$-basis for $\mathsf{M}$ in the following way. Each basis element $\mathsf{y}_{i}$ can be written as a linear combination of basic commutators for $\mathsf{L}$; let $\mathsf{c}_{i}$ denote the smallest basic commutator appearing in this expression, without loss of generality with coefficient $1$. Arrange that these smallest terms form an ascending chain $\mathsf{c}_{1}<\mathsf{c}_{2}<\ldots$ of basic commutators. Using Proposition 4.5 and working modulo $\mathsf{L}$, we see that the elements $\mathsf{y}^{[p]^{j}}$, for $\mathsf{y}\in Y$ and $j\in\mathbb{N}$, are linearly independent. Indeed, it is enough to monitor the appearance of the linearly independent elements $\mathsf{c}_{i}^{\,[p]^{j}}$ in the decompositions of the elements in question, modulo $\mathsf{L}$. In particular, we obtain

[TABLE]

Finally, suppose that $\mathsf{M}\trianglelefteq\mathsf{L}$ is a Lie ideal. The description of $\mathsf{S}$ in (4.1) and the defining properties of the $p$-map imply that $[\mathsf{S},\mathsf{R}]\subseteq\mathsf{M}$. ∎

From [43, Thm. 4.1] we obtain the following useful consequence.

Proposition 4.8**.**

Let $\mathsf{L}$ be a free non-abelian ${\mathbb{F}}_{p}$-Lie algebra on finitely many generators. If $\mathsf{M}\lneqq\mathsf{L}$ is a finitely generated proper graded Lie subalgebra, then $\operatorname{den}_{\mathsf{L}}(\mathsf{M})=0$.

Example 4.9*.*

We provide an explicit example of a finitely generated proper Lie subalgebra $\mathsf{M}\lneqq\mathsf{L}$ that is not graded and has density $\operatorname{den}_{\mathsf{L}}(\mathsf{M})=1$; this should be contrasted with [6, Thm. 1].

Suppose that $\mathsf{L}$ is free on $\{\mathsf{x},\mathsf{y}\}$. Then the Lie subalgebra $\mathsf{M}=\langle\mathsf{x}+[\mathsf{x},\mathsf{y}],\mathsf{y}\rangle_{\mathrm{Lie}}$ does not contain $\mathsf{x}$ and is thus properly contained in $\mathsf{L}$. Indeed, $\mathsf{L}$ maps, via $\mathsf{x}\mapsto 1$ and $\mathsf{y}\mapsto D$, onto the split extension ${\mathbb{F}}_{p}D\rightthreetimes{\mathbb{F}}_{p}[t]$ of the abelian Lie algebra ${\mathbb{F}}_{p}[t]$ by the $1$-dimensional Lie algebra ${\mathbb{F}}_{p}D$ spanned by the derivation $D\colon{\mathbb{F}}_{p}[t]\to{\mathbb{F}}_{p}[t]$, $f(t)\mapsto tf(t)$. The image of $\mathsf{M}$ under this epimorphism is the proper subalgebra ${\mathbb{F}}_{p}D\rightthreetimes(1+t){\mathbb{F}}_{p}[t]$.

On the other hand, the graded Lie algebra $\mathsf{M}_{\mathrm{grd}}$, constructed in Definition 4.3, is equal to $\mathsf{L}$; hence $\operatorname{den}_{\mathsf{L}}(\mathsf{M})=1$.

Proposition 4.8 implies, in particular, that proper graded Lie subalgebras of finite codimension in $\mathsf{L}$ are never finitely generated. It is now easy to manufacture, for any given $\alpha\in(0,1)$, a graded Lie subalgebra $\mathsf{M}\leq\mathsf{L}$ such that $\operatorname{den}_{\mathsf{L}}(\mathsf{M})=\alpha$: one builds $\mathsf{M}=\bigcup_{i=1}^{\infty}\mathsf{M}(i)$ as the union of an ascending chain $\mathsf{M}(i)$, $i\in\mathbb{N}$, of suitably chosen finitely generated graded Lie subalgebras, in analogy to the construction underlying [28, Thm. 5.4]. Below we provide a detailed argument, following closely an analogous argument for pro-$p$ groups, given in [28, Proof of Prop. 5.2].

The purpose is to prepare the transition to a corresponding result for the free restricted Lie algebra $\mathsf{R}$. This transition is not straightforward, because an analogue of the Schreier formula implies that every restricted Lie subalgebra $\mathsf{S}$ of finite codimension in $\mathsf{R}$ is finitely generated and thus constitutes a potential obstacle to an analogous construction; see [31, Thm. 2] and also [10] for the analogue of Schreier’s formula.

Theorem 4.10**.**

Let $\mathsf{L}$ be a non-abelian free ${\mathbb{F}}_{p}$-Lie algebra on finitely many generators. Then there exists, for each $\alpha\in[0,1]$, a Lie subalgebra $\mathsf{M}\leq\mathsf{L}$ that can be generated freely by a suitable collection of basic commutators for $\mathsf{L}$ and satisfies $\operatorname{den}_{\mathsf{L}}(\mathsf{M})=\alpha$. In particular, $\operatorname{dspec}(\mathsf{L})=[0,1]$.

Proof.

Recall that $\mathsf{L}$ is a free Lie algebra on generators $\mathsf{x}_{1},\ldots,\mathsf{x}_{d}$, for $d\in\mathbb{N}$ with $d\geq 2$. Clearly, the trivial subalgebra $\{0\}$ and $\mathsf{L}$ have densities [math] and $1$, respectively.

Suppose now that $\alpha\in(0,1)$. Observe that the sequence

[TABLE]

is strictly increasing. It suffices to produce a Lie subalgebra $\mathsf{M}=\bigoplus_{i}\mathsf{M}_{i}\leq\mathsf{L}$ which is generated by basic commutators such that

1. (i)

$\alpha-\nicefrac{{1}}{{l(n)}}\leq\tfrac{1}{l(n)}\sum_{i=1}^{n}\dim_{{\mathbb{F}}_{p}}(\mathsf{M}_{i})$ for all $n\in\mathbb{N}$; and 2. (ii)

$\frac{1}{l(n)}\sum_{i=1}^{n}\dim_{{\mathbb{F}}_{p}}(\mathsf{M}_{i})\leq\alpha$ for infinitely many $n\in\mathbb{N}$.

Indeed, we can replace any generating set for $\mathsf{M}$ consisting of basic commutators by an irredundant subset that is a free generating set for $\mathsf{M}$; see [51, Proof of Thm. 2] and compare the introductory comments in [10].

We construct a Lie subalgebra $\mathsf{M}$ satisfying (i) and (ii) inductively as the union $\mathsf{M}=\bigcup_{k=1}^{\infty}\mathsf{M}(k)$ of an ascending chain of Lie subalgebras $\mathsf{M}(1)\subseteq\mathsf{M}(2)\subseteq\ldots$, where each term $\mathsf{M}(k)=\langle Y_{k}\rangle_{\mathrm{Lie}}$ is generated by a finite set $Y_{k}$ of basic commutators of weight at most $k$ and $Y_{1}\subseteq Y_{2}\subseteq\ldots$.

Let $k\in\mathbb{N}$. For $k=1$, choose $a\in\{0,1,\ldots,d-1\}$ such that $\alpha-\nicefrac{{1}}{{d}}\leq\nicefrac{{a}}{{d}}\leq\alpha$ and let $\mathsf{M}(1)$ denote the Lie subalgebra generated by $Y_{1}=\{\mathsf{x}_{1},\ldots,\mathsf{x}_{a}\}$. Observe that $\mathsf{M}(1)$ is a proper finitely generated graded Lie subalgebra of $\mathsf{L}$.

Now suppose that $k\geq 2$. Suppose further that we have already constructed a proper finitely generated Lie subalgebra $\mathsf{M}(k-1)=\langle Y_{k-1}\rangle_{\mathrm{Lie}}\leq\mathsf{L}$ which is generated by a finite set of basic commutators of weight at most $k-1$, and that the inequalities in (i) hold for $\mathsf{M}(k-1)$ in place of $\mathsf{M}$ and $1\leq n\leq k-1$. For $n\geq k$, consider

[TABLE]

First suppose that $\beta_{k}\leq\alpha$. Since

[TABLE]

we find a finite set $Z_{k}\subseteq\mathsf{L}_{k}\smallsetminus\mathsf{M}(k-1)_{k}$, consisting of basic commutators of weight $k$, such that

[TABLE]

Let $\mathsf{M}(k)$ be the Lie subalgebra generated by $Y_{k}=Y_{k-1}\cup Z_{k}$. Observe that $\mathsf{M}(k)$ is a proper finitely generated graded Lie subalgebra of $\mathsf{L}$ and that the inequality in (i), respectively (ii), holds for $\mathsf{M}(k)$ in place of $\mathsf{M}$ and $1\leq n\leq k$, respectively $n=k$.

Now suppose that $\beta_{k}>\alpha$. By Proposition 4.8, we find a minimal $k_{0}\geq k+1$ such that $\beta_{k_{0}}\leq\alpha$. Putting $\mathsf{M}(k)=\ldots=\mathsf{M}(k_{0}-1)$ and $Y_{k}=\ldots=Y_{k_{0}-1}$, we return to the previous case for $k_{0}>k$ in place of $k$. ∎

The proof of Theorem 4.10 does not simply carry over, mutatis mutandis, to free restricted Lie algebras, because Proposition 4.8 is lacking a direct analogue, as explained before. Nevertheless, we can deduce the analogous result as follows.

Corollary 4.11**.**

Let $\mathsf{R}$ be a non-abelian free restricted ${\mathbb{F}}_{p}$-Lie algebra on finitely many generators. Then there exists, for each $\alpha\in[0,1]$, a graded restricted Lie subalgebra $\mathsf{S}\leq\mathsf{R}$ that can be generated by basic commutators and satisfies $\operatorname{den}_{\mathsf{R}}(\mathsf{S})=\alpha$. In particular, $\operatorname{dspec}(\mathsf{R})=[0,1]$.

Proof.

For $\alpha\in[0,1]$, Theorem 4.10 yields a Lie subalgebra $\mathsf{M}\leq\mathsf{L}$, generated by basic Lie commutators and such that $\operatorname{den}_{\mathsf{L}}(\mathsf{M})=\alpha$. Let $\mathsf{S}=\langle\mathsf{M}\rangle_{\mathrm{res.\,Lie}}$ be the restricted Lie subalgebra of $\mathsf{R}$ that is generated by $\mathsf{M}$. By Lemma 4.7, we have $\mathsf{S}\cap\mathsf{L}=\mathsf{M}$. Hence Lemma 4.6 implies that $\operatorname{den}_{\mathsf{R}}(\mathsf{S})=\operatorname{den}_{\mathsf{L}}(\mathsf{M})=\alpha$. ∎

It is interesting to give an alternative, more direct proof of Corollary 4.11, which we proceed to do now.

Proposition 4.12**.**

If $\mathsf{S}$ is a non-zero restricted Lie subideal of the free restricted $\mathbb{F}_{p}$-Lie algebra $\mathsf{R}$ then $\operatorname{den}_{\mathsf{R}}(\mathsf{S})=1$.

Proof.

As $\operatorname{den}_{\mathsf{R}}(\mathsf{S})=\operatorname{den}_{\mathsf{R}}(\mathsf{S}_{\mathrm{grd}})$, we may assume that $\mathsf{S}$ is graded. Then $\mathsf{M}=\mathsf{S}\cap\mathsf{L}$ is a non-zero graded subideal of the free Lie algebra $\mathsf{L}$. By Lemma 4.6 and [6, Thm. 2], we have $\operatorname{den}_{\mathsf{R}}(\mathsf{S})=\operatorname{den}_{\mathsf{L}}(\mathsf{M})=1$. ∎

Example 4.13*.*

We provide an explicit example of a finitely generated restricted Lie subalgebra $\mathsf{S}\leq\mathsf{R}$ such that the Lie algebra $\mathsf{S}\cap\mathsf{L}$ is not finitely generated.

Indeed, suppose that $\mathsf{R}$ is free on $\{\mathsf{x},\mathsf{y}\}$ and $\mathsf{L}\leq\mathsf{R}$, as before; i.e., $d=2$ and $\mathsf{x}=\mathsf{x}_{1}$, $\mathsf{y}=\mathsf{x}_{2}$. Then the restricted Lie subalgebra $\mathsf{S}=\langle\mathsf{x},\mathsf{y}^{[p]}\rangle_{\mathrm{res.\,Lie}}\leq\mathsf{R}$ induces the Lie subalgebra

[TABLE]

which is not finitely generated. Observe that $\mathsf{L}=\langle\mathsf{x},\mathsf{y}\rangle_{\mathrm{Lie}}$ maps onto the split extension ${\mathbb{F}}_{p}D\rightthreetimes{\mathbb{F}}_{p}[t]$ as described in Example 4.9. The image of $\mathsf{S}\cap\mathsf{L}$ under this epimorphism is the infinite dimensional abelian Lie algebra ${\mathbb{F}}_{p}[t^{p}]$.

The following proposition can be seen as a strong analogue of Proposition 4.8.

Proposition 4.14**.**

Let $\mathsf{S}\leq\mathsf{R}$ be a finitely generated restricted Lie subalgebra of the free restricted $\mathbb{F}_{p}$-Lie algebra $\mathsf{R}$ such that $\dim_{{\mathbb{F}}_{p}}(\mathsf{R}/\mathsf{S})=\infty$. Then $\mathsf{S}$ has density $\operatorname{den}_{\mathsf{R}}(\mathsf{S})=0$.

Proof.

By [31, Thm. 3] (also refer to [10]), the restricted Lie subalgebra $\mathsf{S}$ is a free factor of a restricted Lie subalgebra $\mathsf{U}$ of finite codimension in $\mathsf{R}$. Then $\mathsf{U}=\mathsf{S}\ast\mathsf{T}$ is a split extension $\mathsf{U}=\mathsf{S}\rightthreetimes\mathsf{I}$ of $\mathsf{I}=\ker(\eta)$ by $\mathsf{S}$, where $\eta\colon\mathsf{U}\rightarrow\mathsf{S}$ denotes the retract that restricts to the identity on $\mathsf{S}$ and maps $\mathsf{T}$ to $\{0\}$. Since $\mathsf{S}$ has infinite codimension in $\mathsf{R}$, we deduce that $\mathsf{I}\neq\{0\}$. By [32, Lem. 10], the restricted Lie algebra $\mathsf{U}$ contains a restricted Lie ideal $\mathsf{J}\trianglelefteq\mathsf{R}$ of finite codimension. Hence $\mathsf{I}\cap\mathsf{J}\neq\{0\}$ is a restricted Lie subideal of $\mathsf{R}$ and $\operatorname{den}_{\mathsf{R}}(\mathsf{I}\cap\mathsf{}J)=1$, by Proposition 4.12. Thus $\operatorname{den}_{\mathsf{R}}(\mathsf{I})=1$ and the claim follows from

[TABLE]

Putting together Propositions 4.12 and 4.14 and [32, Lem. 10], we obtain the following.

Corollary 4.15**.**

Let $\mathsf{S}$ be a finitely generated restricted Lie subalgebra of the free restricted $\mathbb{F}_{p}$-Lie algebra $\mathsf{R}$. Then the following are equivalent:

1. (1)

$\mathsf{S}$* contains a non-trivial restricted Lie subideal of $\mathsf{R}$;* 2. (2)

$\mathsf{S}$* has finite codimension in $\mathsf{R}$;* 3. (3)

$\mathsf{S}$* has density $\operatorname{den}_{\mathsf{R}}(\mathsf{S})=1$.*

An alternative proof of Corollary 4.11 can now be obtained by modifying the proof of Theorem 4.10 as follows. The restricted Lie ideal $\mathsf{N}=\langle Y_{1}\rangle_{\mathsf{R}}\trianglelefteq\mathsf{R}$ generated by $Y_{1}$ is of infinite codimension and has density $1$ in $\mathsf{R}$. For each $k\in\mathbb{N}$, let $\mathsf{S}(k)\leq\mathsf{R}$ denote instead the restricted Lie subalgebra generated by $Y_{k}$ and replace occurrences of $\mathsf{L}_{k}$ by $\mathsf{N}_{k}=\mathsf{N}\cap\mathsf{R}_{k}$ so that $\mathsf{S}$ is constructed inside $\mathsf{N}$.

4.3. From free restricted Lie algebras to free pro-$p$

groups

Let $G$ be a finitely generated pro-$p$ group, with associated restricted ${\mathbb{F}}_{p}$-Lie algebra $\mathsf{R}(G)$; see Section 4.1. Let $\eta\colon G\to\mathsf{R}(G)$ denote the standard map, given by $1\mapsto 0$ and

[TABLE]

We observed before that, in general, the mapping $H\mapsto\mathsf{R}_{G}(H)$ from closed subgroups of $G$ to restricted Lie subalgebras of $\mathsf{R}(G)$ is not particularly well behaved. Let us illustrate some unpleasant phenomena by giving two concrete examples.

Example 4.16*.*

Consider the cyclotomic extension of the ring of $p$-adic integers, $\mathfrak{O}=\mathbb{Z}_{p}[\zeta]$, where $\zeta$ denotes a primitive $p^{2}$th root of unity. Recall that the cyclotomic field $\mathbb{Q}_{p}(\zeta)$ is a totally ramified extension of $\mathbb{Q}_{p}$ of degree $r=\varphi(p^{2})=(p-1)p$. Indeed, $\pi=\zeta-1$ is a uniformising element and

[TABLE]

Furthermore, we have $\pi^{r}\mathfrak{O}=p\mathfrak{O}$.

We consider the semidirect product $G=T\ltimes A$, where $T=\langle s\rangle\cong\mathbb{Z}_{p}$,

[TABLE]

and the action of $T$ on $A$ is given by $(b^{s})\psi=b\psi\cdot\zeta$ for $b\in A$. The group $G$ is a torsion-free, metabelian $p$-adic analytic pro-$p$ group of dimension $r+1$.

The first terms of the Zassenhaus series of $G$ can be computed with ease; a description of the entire series would be cumbersome. We obtain, for $1\leq n\leq p^{2}$,

[TABLE]

From this it is easy to check that in terms of the standard map $\eta\colon G\to\mathsf{R}(G)$ we have, for $1\leq n\leq r$,

[TABLE]

Furthermore, we obtain $(a_{i}\eta)^{[p]}=0$ for $0\leq i\leq p-2$. This implies, for instance, that the restricted Lie algebra $\mathsf{R}_{G}(H)$ corresponding to $H=\langle a_{0}\rangle\cong{\mathbb{Z}}_{p}$ is neither $1$-generated nor free.

Example 4.17*.*

The Sylow pro-$p$ subgroup $G=\mathrm{Aut}^{1}({\mathbb{F}}_{p}(\!(t)\!))$ of $\mathrm{Aut}({\mathbb{F}}_{p}(\!(t)\!))$, the automorphism group of the local field ${\mathbb{F}}_{p}(\!(t)\!)$, consists of all $g\in\mathrm{Aut}({\mathbb{F}}_{p}(\!(t)\!))$ such that $t.g\equiv_{t^{2}}t$. The bijection $G\to t+t^{2}{\mathbb{F}}_{p}[\![t]\!]$, $g\mapsto t.g$ provides a realisation of $G$ as a group of formal power series under substitution; in this form the group is known as the Nottingham group over ${\mathbb{F}}_{p}$; compare [12]. For simplicity we exclude the case $p=2$, which requires special attention.

The group $G$ is a $2$-generated, hereditarily just infinite pro-$p$ group with many interesting properties; a base for the neighbourhoods of the identity element is given by the descending chain of open normal subgroups

[TABLE]

The depth of $g\in G$ is defined as $d(g)=\inf\{m\in\mathbb{N}\mid g\not\in G_{m+1}\}$. Using [12, Rem. 1 and Lem. 1], it is easy to see that for all $n\in\mathbb{N}$,

[TABLE]

Conversely, this means that $G_{m}\subseteq Z_{n(m)}(G)$, but $G_{m}\not\subseteq Z_{n(m)+1}(G)$, where $n(m)=m-1-\lfloor(m-2)/p\rfloor$.

Consider the procyclic subgroup $H=\langle h\rangle\cong{\mathbb{Z}}_{p}$ whose generator $h\in G$ is given by $t.h=t+t^{3}$. Thus $d(g)=2$ and, for $k\in\mathbb{N}_{0}$, it is known that $d_{k}=d(h^{p^{k}})$ satisfies $d_{k}\equiv_{p}d(h)=2$ and $d_{k+1}\geq pd_{k}+2$; compare [12, Lem. 1 and Thm. 6]. (In fact, $d_{k}=2(p^{k+1}-1)/(p-1)$ for all $k\in\mathbb{N}_{0}$, in other words: $h$ is $2$-ramified; compare [44].)

For $k\in\mathbb{N}_{0}$, we observe that n(d_{k})=\big{(}(p-1)d_{k}-(p-2)\big{)}/p, hence

[TABLE]

and we conclude that $(h^{p^{k}}\eta)^{[p]}=0$, where $\eta\colon G\to\mathsf{R}(G)$ denotes the standard map. This shows that, restricted to $\mathsf{R}_{G}(H)$, the $p$-map is the null map. Consequently, the infinite dimensional, abelian restricted Lie algebra $\mathsf{R}_{G}(H)$ is not finitely generated.

For certain types of pro-$p$ groups $G$, the mapping $H\mapsto\mathsf{R}_{G}(H)$ may display considerably better properties, but little in this direction seems to be known. In the following we make use of a consequence that is easily derived from the proof of [16, Prop. 3.2]. (Note that in the statement of [16, Prop. 3.2] it is assumed from the start that $F$ is free and that $W$ is a valuation; we consider situations that are, a priori, more general.)

Proposition 4.18**.**

Let $G$ be a finitely generated pro-$p$ group, and let $X\subseteq G$ be such that the standard map $\eta\colon G\to\mathsf{R}(G)$ induces a bijection from $X$ onto $Y=X\eta$. Suppose that the restricted Lie subalgebra $\langle Y\rangle_{\mathrm{res.\,Lie}}\leq\mathsf{R}(G)$ is freely generated by $Y$.

Then the pro-$p$ group $H=\langle X\rangle\leq_{\mathrm{c}}G$ is freely generated by the $1$-convergent subset $X$, and $\mathsf{R}_{G}(H)=\langle Y\rangle_{\mathrm{res.\,Lie}}$.

Proof.

One easily verifies that $X$ is $1$-convergent. Next we show that $H$ is freely generated by the $1$-convergent set $X$; compare [55, §5.1]. As explained in [16, §3.1] every element of $h\in H$ admits (possibly not uniquely) a power-commutator factorisation

[TABLE]

where $\mathcal{C}$ is the family (‘multiset’ in the parlance of [16]) of $X$-basic group commutators and each $j_{c,k}$ lies in $\{0,1,\ldots,p-1\}$. Tacitly, we have fixed an ordering of the countable family $\mathcal{C}\times\mathbb{N}_{0}$, i.e., a bijection from $\mathbb{N}$ to $\mathcal{C}\times\mathbb{N}_{0}$, so that we know in which order the infinite, but in any case convergent product (4.2) is to be carried out.

Furthermore, $H$ is freely generated by $X$ if and only if the neutral element $1$ has only one power-commutator factorisation, namely the trivial one resulting from choosing $j_{c,k}=0$ for all $(c,k)\in\mathcal{C}\times\mathbb{N}_{0}$. (Implicitly, this criterion or a similar one is also used for establishing the basic part of “(ii) implies (i)” in [16, Prop. 3.2].) The mapping

[TABLE]

is a pseudo-valuation on $H$, in the sense of [16]. Adapting the argument that establishes “(iii) implies (ii)” in [16, Prop. 3.2] (to the present situation $(H,W,X,Y)$ in place of $(F,W,X,S)$ in [16]), we see that, indeed, any $h\in H$ that admits a non-trivial power-commutator factorisation cannot be equal to $1$. Thus $H$ is freely generated by $X$.

It remains to prove that $\mathsf{R}_{G}(H)=\langle Y\rangle_{\mathrm{res.\,Lie}}$. Clearly, $\mathsf{R}_{G}(H)\supseteq\langle Y\rangle_{\mathrm{res.\,Lie}}$ holds, and the reverse inclusion is derived again from power-commutator factorisations, as in “(iii) implies (iv)” in [16, Prop. 3.2]: briefly speaking, $h\in H\smallsetminus\{1\}$ with power-commutator factorisation (4.2) is mapped under $\eta$ to the finite linear combination

[TABLE]

of iterated commutators in $Y$, where $\mathcal{M}=\{(c,k)\in\mathcal{C}\times\mathbb{N}_{0}\mid p^{k}\operatorname{wt}(c)=d(h)\}$ and $c^{\mathrm{Lie}}$ is the basic Lie commutator in $Y$ that corresponds to the basic group commutator $c$ in $X$. ∎

Corollary 4.19**.**

Let $G$ be a finitely generated pro-$p$ group and let $H\leq_{\mathrm{c}}G$ be a subgroup. Suppose that the corresponding restricted Lie subalgebra $\mathsf{R}_{G}(H)\leq\mathsf{R}(G)$ is free. Then $\mathrm{d}(H)$, the minimal number of generators for $H$, is equal to the free restricted rank of $\mathsf{R}_{G}(H)$. In particular, $H$ is finitely generated if and only if $\mathsf{R}_{G}(H)$ is.

Proof.

Suppose that $\mathsf{R}_{G}(H)=\langle Y\rangle_{\mathrm{res.\,Lie}}$ is freely generated by a set $Y$ of homogeneous elements; compare [31, Lem. 2] and the comments in [10]. We choose $X\subseteq H$ such that the standard map $\eta\colon G\to\mathsf{R}(G)$ induces a bijection from $X$ onto $Y$. By Proposition 4.18, the group $\langle X\rangle$ is freely generated by $X$ and $\mathsf{R}_{G}(\langle X\rangle)=\mathsf{R}_{G}(H)$. Since $\langle X\rangle\leq H$, this implies $\langle X\rangle=H$. ∎

Proof of Theorem 1.3.

Let $F=\langle x_{1},\ldots,x_{d}\rangle$ be a finitely generated free pro-$p$ group of free rank $d\geq 2$. Then $\mathsf{R}(F)$ is a free restricted $\mathbb{F}_{p}$-Lie algebra and every restricted Lie subalgebra of $\mathsf{R}(F)$ is itself free. Set $N=\langle x_{2},\ldots,x_{d}\rangle^{F}\trianglelefteq_{\mathrm{c}}F$. Proposition 4.12 shows that $\operatorname{hdim}_{F}^{\mathcal{Z}}(N)=1$. By Corollary 4.19 and Proposition 4.14, every finitely generated subgroup $H\leq_{\mathrm{c}}F$ with $H\subseteq N$ has Hausdorff dimension $\operatorname{hdim}_{G}^{\mathcal{Z}}(H)=0$. Thus the Interval Theorem, stated at the end of Section 1, shows that $\operatorname{hspec}^{\mathcal{Z}}(F)=[0,1]$. ∎

5. Non-soluble Demushkin pro-$p$ groups and the Zassenhaus

series

In this section we extend the results obtained in Section 4.3 to cover non-soluble Demushkin pro-$p$ groups and, more generally, mixed finite direct products of finitely generated free and Demushkin pro-$p$ groups. In particular, we prove Theorem 1.5 and Corollary 1.6.

Let $D$ be a non-soluble Demushkin pro-$p$ group. According to the classification of Demushkin groups (see [55, Sec. 12.3] and [33, 49]) the pro-$p$ group $D$ admits a finite one-relator presentation of the form

[TABLE]

where $d\geq 3$ and one of the following holds (in this context $p^{\infty}$ is to be interpreted as [math]):

1. $\circ$

$d\equiv_{2}0$ and $r=x_{1}^{\,p^{f}}[x_{1},x_{2}]\cdots[x_{d-1},x_{d}]$ with $f\in\mathbb{N}\cup\{\infty\}$ such that $p^{f}\neq 2$; 2. $\circ$

$d\equiv_{2}1$, $p=2$ and $r=x_{1}^{\,2}x_{2}^{\,2^{f}}[x_{2},x_{3}][x_{4},x_{5}]\cdots[x_{d-1},x_{d}]$ with $f\in\mathbb{N}_{\geq 2}\cup\{\infty\}$; 3. $\circ$

$d\equiv_{2}0$, $p=2$, and $r=x_{1}^{\,2+2^{f}}[x_{1},x_{2}][x_{3},x_{4}]\cdots[x_{d-1},x_{d}]$ with $f\in\mathbb{N}_{\geq 2}\cup\{\infty\}$ 4. $d\equiv_{2}0$, $p=2$, and or $r=x_{1}^{\,2}[x_{1},x_{2}]x_{3}^{\,2^{f}}[x_{3},x_{4}]\cdots[x_{d-1},x_{d}]$ with $f\in\mathbb{N}_{\geq 2}$.

5.1. Associated restricted Lie algebras and free subalgebras

The graded restricted $\mathbb{F}_{p}$-Lie algebra associated to $D$ with respect to the Zassenhaus series has the form

[TABLE]

where $\mathsf{R}$ denotes the free restricted $\mathbb{F}_{p}$-Lie algebra on $\mathsf{x}_{1},\ldots,\mathsf{x}_{d}$ and $\langle\mathsf{r}\rangle_{\mathsf{R}}$ denotes the restricted Lie ideal generated by the homogeneous Lie element $\mathsf{r}$ given by

[TABLE]

or one of the following:

[TABLE]

Indeed, the presentation (5.1) is strongly free, as can be seen, for instance, from Anick’s criterion; compare [20, §3] and [35, 36]. Applying [20, Thm. 2.12], which is based on ideas put forward in [35], we obtain the described presentation for $\mathsf{D}$.

Let $\mathsf{L}$ denote the free $\mathbb{F}_{p}$-Lie algebra on $\mathsf{x}_{1},\ldots,\mathsf{x}_{d}$, embedded in $\mathsf{R}$, as in Section 4.2. For the next steps leading up to Proposition 5.4, we focus on the ‘generic’ case (5.2), where $\mathsf{r}\in\mathsf{L}$ (and the specific shape of $\mathsf{r}$ is of no further relevance). In the proof of Proposition 5.4, we explain how the remaining ‘exceptional’ cases in (5.3) can be reduced to the ‘generic’ case. Assuming (5.2), we put

[TABLE]

the Lie ideal generated by $\mathsf{r}$ in $\mathsf{L}$. From Lemma 4.7 we deduce the following result.

Lemma 5.1**.**

In the setup described above, $\langle\mathsf{I}\rangle_{\mathrm{res.\,Lie}}=\mathsf{J}$ and $\mathsf{I}=\mathsf{L}\cap\mathsf{J}$.

By [34, Thm. 3], we find a (free) graded Lie subalgebra $\mathsf{M}\leq\mathsf{L}$ with $\mathsf{M}\cap\mathsf{I}=0$ and such that $\mathsf{M}+\mathsf{I}$ has finite co-dimension in $\mathsf{L}$. Put

[TABLE]

Lemma 5.2**.**

In the setup described above, $\mathsf{S}\cap\mathsf{J}=0$. Consequently, $(\mathsf{S}+\mathsf{J})/\mathsf{J}\leq\mathsf{R}/\mathsf{J}\cong\mathsf{D}$ is a free restricted graded Lie subalgebra.

Proof.

Suppose that $\mathsf{h}\in\mathsf{S}\cap\mathsf{J}$. Recalling that $\mathsf{S}=\langle\mathsf{M}\rangle_{\mathrm{res.\,Lie}}$ and $\mathsf{J}=\langle\mathsf{I}\rangle_{\mathrm{res.\,Lie}}$, we apply Lemma 4.7 to write

[TABLE]

where $i$ and $j$ run through unspecified finite index sets and

[TABLE]

We observe that for every $\mathsf{m}\in\mathsf{M}$,

[TABLE]

In view of (4.1), this implies $\mathsf{h}\in\mathrm{Z}_{\mathsf{S}}(\mathsf{M})=\mathrm{Z}(\mathsf{S})=\{0\}$. ∎

Lemma 5.3**.**

Let $\mathsf{K}\leq\mathsf{L}$ be a Lie subalgebra of finite codimension $t$, say, in the free Lie algebra $\mathsf{L}$. Then, for $n\in\mathbb{N}$, we have

[TABLE]

Proof.

Write $\mathsf{L}=\mathsf{K}\oplus\langle\mathsf{z}_{1},\ldots,\mathsf{z}_{t}\rangle_{\mathbb{F}_{p}}$, as an $\mathbb{F}_{p}$-vector space. From Lemma 4.7 we conclude that

[TABLE]

Proposition 5.4**.**

The restricted $\mathbb{F}_{p}$-Lie algebra $\mathsf{D}=\mathsf{R}(D)$ contains a non-abelian free restricted graded Lie subalgebra $\mathsf{E}$ of density $\operatorname{den}_{\mathsf{D}}(\mathsf{E})=1$.

Proof.

First we continue to work in the setup that we built from (5.2). We consider the free restricted Lie subalgebra $\mathsf{E}$ corresponding to $(\mathsf{S}+\mathsf{J})/\mathsf{J}\leq\mathsf{R}/\mathsf{J}\cong\mathsf{D}$. Let $t=\dim_{\mathbb{F}_{p}}(\mathsf{L}/(\mathsf{M}+\mathsf{I}))$. Observe that $\mathsf{x}_{2}$ and $\mathsf{x}_{4}$ generate a free Lie subalgebra of rank $2$, modulo $\mathsf{J}$. Using Lemmata 4.6 and 5.3, we obtain

[TABLE]

It remains to deal with the ‘exceptional’ cases in (5.3), which we ignored so far. We show that these can be reduced to the ‘generic’ case (5.2).

Suppose that $\mathsf{D}\cong\mathsf{R}/\mathsf{J}$, where $\mathsf{J}=\langle\mathsf{r}\rangle_{\mathsf{R}}$ for $\mathsf{r}=\mathsf{x}_{1}^{\,[2]}+[\mathsf{x}_{2},\mathsf{x}_{3}]+\ldots+[\mathsf{x}_{d-1},\mathsf{x}_{d}]$, with $d\equiv_{2}1$ and $p=2$. Then the restricted Lie subalgebra

[TABLE]

has co-dimension $1$ in $\mathsf{R}$. Moreover, writing $\mathsf{a}=\mathsf{x}_{1}^{\,[2]}$ and $\mathsf{b}_{i}=\mathsf{x}_{i}$, $\mathsf{c}_{i}=[\mathsf{x}_{i},\mathsf{x}_{1}]$ for $2\leq i\leq d$, we see that $\widetilde{\mathsf{R}}/\mathsf{J}=\widetilde{\mathsf{R}}/\langle\mathsf{r},[\mathsf{r},\mathsf{x}_{1}]\rangle_{\widetilde{\mathsf{R}}}$ has the finite presentation

[TABLE]

Thus $\widetilde{\mathsf{R}}$ has a presentation of the form (5.2) and, with small modifications, we can argue as in the ‘generic’ case. (For instance, in the special case $d=3$, we adapt the argument in (5.4) as follows: the elements $\mathsf{x}_{1}$ and $\mathsf{x}_{3}$ generate, modulo $\mathsf{J}$, a Lie subalgebra that maps, via $\mathsf{x}_{1}+\mathsf{J}\mapsto 1$ and $\mathsf{x}_{3}+\mathsf{J}\mapsto D$, onto the Lie algebra ${\mathbb{F}}_{p}D\rightthreetimes{\mathbb{F}}_{p}[t]$ discussed in Example 4.9; this yields the considerably weaker, but still sufficient estimate \dim_{{\mathbb{F}}_{p}}\!\big{(}\mathsf{R}/(\mathsf{J}+\mathsf{R}_{>n})\big{)}\geq n+1 for $n\in\mathbb{N}$.)

Finally, consider $\mathsf{D}\cong\mathsf{R}/\mathsf{J}$, where $\mathsf{J}=\langle\mathsf{r}\rangle_{\mathsf{R}}$ for $\mathsf{r}=\mathsf{x}_{1}^{\,[2]}+[\mathsf{x}_{1},\mathsf{x}_{2}]+\ldots+[\mathsf{x}_{d-1},\mathsf{x}_{d}]$, with $d\equiv_{2}0$ and $p=2$. Then the restricted Lie subalgebra

[TABLE]

has co-dimension $1$ in $\mathsf{R}$. Moreover, writing $\mathsf{a}=\mathsf{x}_{1}^{\,[2]}$, and $\mathsf{b}_{i}=\mathsf{x}_{i}$, $\mathsf{c}_{i}=[\mathsf{x}_{i},\mathsf{x}_{1}]$ for $2\leq i\leq d$, one sees that $\mathsf{S}/\mathsf{J}=\mathsf{S}/\langle\mathsf{r},[\mathsf{r},\mathsf{x}_{1}]\rangle_{\mathsf{S}}$ has the finite presentation

[TABLE]

Thus $\mathsf{S}$ has a presentation of the form (5.2) and, again with small modifications, we can argue as in the ‘generic’ case. ∎

5.2. The Hausdorff spectrum

We continue to consider the non-soluble Demushkin pro-$p$ group $D$ on $d$ generators, with presentation (5.1), and the associated restricted $\mathbb{F}_{p}$-Lie algebra $\mathsf{D}=\mathsf{R}(D)$. In the following we employ considerations that can be found in a very similar form, for instance, in [43], [54, §3] or [42, §2]. But we need more careful estimates, going somewhat further than computing the entropy of relevant Lie algebras. We adopt, with small changes, the notation employed in [42].

Lemma 5.5**.**

In the set-up described above, we have

[TABLE]

where $\varepsilon\in\mathbb{R}$ with $0<\varepsilon<1$ satisfies $(d-\varepsilon)\varepsilon=1$.

Proof.

Classical theorems of Jennings and Lazard (compare [15, §12]) show that a universal restricted enveloping algebra of $\mathsf{D}$ is provided by the graded group algebra

[TABLE]

where $\mathsf{I}^{n}$ denotes the $n$th power of the augmentation ideal $\mathsf{I}=\mathrm{ker}(\mathbb{F}_{p}[D]\to\mathbb{F}_{p})$.

We consider the Hilbert–Poincaré series associated to the restricted Lie algebra $\mathsf{D}$ and to the associative algebra $\mathsf{A}$; they are defined as

[TABLE]

where

[TABLE]

From Labute’s work on mild pro-$p$ groups [35] (see also [36, 20]) it is known that

[TABLE]

In accordance with [15, Thm. 12.16], we have the formal identity

[TABLE]

Let $\mu\colon\mathbb{N}\to\{0,1,-1\}$ denote the classical Möbius function, and write

[TABLE]

where $\log(1+X)=\sum_{k=1}^{\infty}(-1)^{k+1}k^{-1}X^{k}$ denotes the formal logarithm series. As shown in [42, Prop. 2.8] this implies that, for $n=p^{k}m$ with $p\nmid m$,

[TABLE]

where the integer summands are given by the formula

[TABLE]

We choose $\varepsilon\in\mathbb{R}_{>0}$ such that

[TABLE]

From (5.5) we conclude that, for $n\in\mathbb{N}$,

[TABLE]

and

[TABLE]

In particular,

[TABLE]

is positive for all sufficiently large $n$. Hence there is a constant $K\in\mathbb{N}$ such that

[TABLE]

while in the other direction we have, for similar reasons,

[TABLE]

These bounds yield the desired asymptotic estimate for $\dim_{{\mathbb{F}}_{p}}(\mathsf{D}_{n})$. The estimate for $\dim_{{\mathbb{F}}_{p}}(\mathsf{D}/\mathsf{D}_{>n})$ follows by standard manipulations; compare [26, Proof of Lem. 4.3]. ∎

Proposition 5.6**.**

Let $\mathsf{S}$ be a finitely generated free restricted graded Lie subalgebra of $\mathsf{D}$. Then $\operatorname{den}_{\mathsf{D}}(\mathsf{S})=0$.

Proof.

We continue to use the notation from the proof of Lemma 5.5. Now we consider the corresponding Hilbert–Poincaré series for the finitely generated free restricted graded Lie subalgebra $\mathsf{S}\leq\mathsf{D}$ and its universal enveloping algebra $\mathsf{B}$. Based on the ‘restricted Poincaré–Birkhoff–Witt Theorem’, we take $\mathsf{B}$ to be the graded subalgebra of $\mathsf{A}$ that is generated by the image of $\mathsf{S}$ under the canonical embedding $\mathsf{D}\hookrightarrow\mathsf{A}$. Clearly, we may assume that $\mathsf{S}$ is non-abelian and thus free on $r\geq 2$ generators.

We write

[TABLE]

where

[TABLE]

The universal restricted enveloping algebra $\mathsf{B}$ is a free associative $\mathbb{F}_{p}$-algebra in non-commuting homogeneous generators $\mathsf{y}_{1},\ldots,\mathsf{y}_{r}$, say. For each $i\in\mathbb{N}$, let

[TABLE]

denote the number of generators $\mathsf{y}_{s}$ of weight $i$, and set $e=\max\{i\in\mathbb{N}\mid r_{i}\neq 0\}$. Then [4, Lem. 1.2] shows that

[TABLE]

where $\lambda\in\mathbb{R}_{>1}$ and $\nu_{i}\in\mathbb{C}\smallsetminus\mathbb{R}$ with $\lvert\nu_{i}\rvert\leq\lambda$ for $1\leq i<e$. In analogy to $b_{n}$ and $w_{n}$, we consider

[TABLE]

Similar to the earlier situation we have, for $n=p^{k}m$ with $p\nmid m$,

[TABLE]

as a by-product we see that each $\widetilde{w}_{n}$ is an integer. We observe that

[TABLE]

due to the fact that $\mathrm{Re}(\nu_{1}^{\,n}),\ldots,\mathrm{Re}(\nu_{e-1}^{\,n})\geq 0$ for infinitely many $n\in\mathbb{N}$ (compare [13, Ch. I, §5, Thm. VI]), and that

[TABLE]

This implies that, for infinitely many $n\in\mathbb{N}$,

[TABLE]

and hence

[TABLE]

while, for all $n\in\mathbb{N}$,

[TABLE]

Progressing along an infinite increasing sequence of natural numbers $n$ for which the estimates above are valid, we conclude that

[TABLE]

This implies that $\lambda\leq d-\varepsilon$. Next we claim that

[TABLE]

Indeed, the characteristic polynomial

[TABLE]

associated to the series in (5.6) satisfies $r_{i}\geq 0$ for $1\leq i\leq e$. Hence $f$ has precisely one positive real root, namely $\lambda$, and thus cannot be divisible by the polynomial $t^{2}-dt+1$ which has two real roots, namely $d-\varepsilon$ and $\varepsilon$. But $t^{2}-dt+1$ is irreducible over $\mathbb{Z}$, as $d\geq 3$. Thus $\lambda\neq d-\varepsilon$.

From Lemma 5.5, (5.7) and (5.8) we deduce that

[TABLE]

Proof of

Theorem 1.5.

From Propositions 5.4 and 4.18 we conclude that there is a free subgroup $F\leq_{\mathrm{c}}D$ such that $\mathsf{E}=\mathsf{R}_{D}(F)$ is a free restricted Lie subalgebra of $\mathsf{D}=\mathsf{R}(D)$ and $\operatorname{hdim}_{D}^{\mathcal{Z}}(F)=\operatorname{den}_{\mathsf{D}}(\mathsf{E})=1$. By Corollary 4.19 and Proposition 5.6, every finitely generated subgroup $H\leq_{\mathrm{c}}F$ has Hausdorff dimension $\operatorname{hdim}_{D}^{\mathcal{Z}}(H)=0$. Thus the Interval Theorem, stated at the end of Section 1, shows that $\operatorname{hspec}^{\mathcal{Z}}(F)=[0,1]$. ∎

Proof of

Corollary 1.6.

From Lemmata 4.6 and 5.5 it is easy to see that soluble factors can be neglected. The assertion now follows directly from Proposition 2.10, Theorem 1.3 and Theorem 1.5. ∎

6. Normal and finitely generated Hausdorff

spectra

In this section we prove Theorems 1.7, 1.8 and 1.9. Recall from the introduction that the normal Hausdorff spectrum of a finitely generated pro-$p$ group $G$, with respect to a filtration $\mathcal{S}$, is $\operatorname{hspec}_{\trianglelefteq}^{\mathcal{S}}(G)=\{\operatorname{hdim}_{G}^{\mathcal{S}}(H)\mid H\trianglelefteq_{\mathrm{c}}G\}$.

Lemma 6.1**.**

Let $H$ be a finitely generated pro-$p$ group of positive rank gradient, and let $N\trianglelefteq_{\mathrm{c}}H$ be an infinite normal subgroup. Then, $[N,H]=\langle[x,y]\mid x\in N,\,y\in H\rangle\trianglelefteq_{\mathrm{c}}H$ is also infinite.

Proof.

Assume for a contradiction that $[N,H]$ is finite. By Lemma 2.11, $G=H/[N,H]$ has positive rank gradient, and $N/[N,H]\subseteq\mathrm{Z}(G)$ shows that $G$ has infinite centre $\mathrm{Z}(G)$.

On the other hand, Theorem 1.1 implies that $\mathrm{Z}(G)$ is locally finite. For any finite central subgroup $K\trianglelefteq_{\mathrm{c}}G$, the rank gradient of $G/K$ is bounded by $d(G)-1$; thus Lemma 2.11 implies that $\mathrm{Z}(G)$ is finite; compare [1, Thm. 4]. ∎

Proposition 6.2**.**

Let $G=H_{1}\times\ldots\times H_{r}$ be a non-trivial direct product of finitely generated pro-$p$ groups $H_{j}$ of positive rank gradient, and let $N\trianglelefteq_{\mathrm{c}}G$. Then there exist normal subgroups $K_{j},N_{j}\trianglelefteq_{\mathrm{c}}H_{j}$, for $1\leq j\leq r$, such that

[TABLE]

and $K_{j}$ is infinite if and only if $N_{j}$ is infinite, for each $j\in\{1,\ldots,r\}$.

Proof.

For $1\leq j\leq r$, let $N_{j}$ be the image of $N$ under the natural projection $G\to H_{j}$ and set $K_{j}=[N,H_{j}]=[N_{j},H_{j}]$. Now apply Lemma 6.1. ∎

Proof of Theorem 1.7.

Consider first normal subgroups $N\trianglelefteq_{\mathrm{c}}G=H_{1}\times\ldots\times H_{r}$ that decompose as $N=N_{1}\times\ldots\times N_{r}$, with $N_{j}\trianglelefteq_{\mathrm{c}}H_{j}$ for $1\leq j\leq r$. Set $J=\{j\mid 1\leq j\leq r,\,\lvert N_{j}\rvert=\infty\}$. A straightforward calculation, based on Proposition 2.3 and the fact that the the Frattini series $\mathcal{F}$ of $G$ admits a direct product decomposition, yields $\operatorname{hdim}_{G}^{\mathcal{F}}(N)=\sum_{j\in J}\alpha_{j}$. Finally, by Proposition 6.2, all normal subgroups of $G$ have Hausdorff dimension of the described form. ∎

Proof of Theorem 1.8.

We argue as in the proof of Theorem 1.7, using the extension of Proposition 2.3 that we provided in the proof of Theorem 1.4. ∎

Proof of Theorem 1.9.

Clearly, we may assume that

[TABLE]

If $G$ is abelian, then $d=1$ and $F_{1}\cong\ldots\cong F_{t}\cong{\mathbb{Z}}_{p}$, while the remaining factors are trivial. The filtrations $\mathcal{S}$ and $\mathcal{Z}$ consist of selected terms of the $p$-power series, possibly with repetitions. It is easy to verify that closed (normal) subgroups $H\leq_{\mathrm{c}}G$ have the expected Hausdorff dimension $\operatorname{hdim}_{G}^{\mathcal{S}}(H)=\operatorname{hdim}_{G}^{\mathcal{Z}}(H)=\dim(H)/\dim(G)$; compare [9, Thm. 1.1].

Now suppose that $d\geq 2$, equivalently that $F_{1}$ is non-abelian. First we consider the iterated verbal filtration $\mathcal{S}\colon G_{i}$, $i\in\mathbb{N}_{0}$. Suppose that, for $i\in\mathbb{N}$, the group $G_{i}$ is generated by all values of group words in a set $\mathcal{W}_{i}$, say, and write $G_{i}=\mathcal{W}_{i}(G_{i-1})$ for short. Let $1\leq j\leq r$. By Theorem 1.8, it suffices to prove that

[TABLE]

Suppose that $t<k\leq r$, thus $d=d(F_{1})>d(F_{k})=\tilde{d}$. We may regard $F_{k}=\langle x_{1},\ldots,x_{\tilde{d}}\rangle$ as a finitely generated subgroup of infinite index in $F_{1}=\langle x_{1},\ldots,x_{d}\rangle$. Observe that, for $i\in\mathbb{N}$,

[TABLE]

Hence the extension of Proposition 2.2 provided in the proof of Theorem 1.4 yields

[TABLE]

Thus (6.1) follows from

[TABLE]

Finally, we consider the Zassenhaus series $\mathcal{Z}$. Lemma 4.6 implies that, for $t<j\leq r$,

[TABLE]

Suppose that $N\trianglelefteq_{\mathrm{c}}G=F_{1}\times\ldots\times F_{r}$ decomposes as $N=N_{1}\times\ldots\times N_{r}$, and let $J=\{j\mid 1\leq j\leq t,\,\lvert N_{j}\rvert=\infty\}$. Propostion 4.12 shows that, for $j\in J$,

[TABLE]

Thus we obtain

[TABLE]

By Proposition 6.2, all normal subgroups of $G$ have Hausdorff dimension of this form. ∎

There is also a natural interest in the finitely generated Hausdorff spectrum of a finitely generated pro-$p$ group $G$, with respect to a filtration $\mathcal{S}$, defined as

[TABLE]

compare [50, §4.7] and [27].

Proposition 2.2, its extension (given in the proof of Theorem 1.4) and Corollary 4.19 show that

[TABLE]

provided that

1. $\circ$

$G$ is a finitely generated pro-$p$ group of positive rank gradient, equipped with the Frattini series $\mathcal{S}=\mathcal{F}$, or 2. $\circ$

$G$ is a finitely generated non-abelian free pro-$p$ group or a non-soluble Demushkin pro-$p$ group, equipped with an iterated verbal filtration $\mathcal{S}$, or 3. $\circ$

$G$ is a finitely generated free pro-$p$ group, equipped with the Zassenhaus series $\mathcal{S}=\mathcal{Z}$.

Surprisingly, the description of the finitely generated Hausdorff spectra of finite direct products $G=F_{1}\times\ldots\times F_{r}$ of non-abelian free pro-$p$ groups $F_{j}$, with respect to the Frattini series $\mathcal{F}$ (or more general iterated verbal filtrations), appears to be more difficult than expected. For simplicity, suppose that $G=F\times F$ is the direct product of two copies of a free pro-$p$ group $F$ with $d(F)=2$. It is easily seen that $\operatorname{hspec}_{\mathrm{fg}}^{\mathcal{F}}(G)\supseteq\{0,\nicefrac{{1}}{{2}},1\}$. However, the reverse inclusion does not seem to be clear, due to the existence of ‘diagonal subgroups’ of the form

[TABLE]

where $\varphi\colon U\to V$ is an isomorphism between open subgroups $U,V\leq_{\mathrm{o}}F$, necessarily of the same index $\lvert F:U\rvert=\lvert F:V\rvert$, but otherwise unrestricted.

Question 6.3**.**

Is it true that diagonal subgroups $H_{\varphi}\leq_{\mathrm{c}}F\times F=G$ have Hausdorff dimension $\operatorname{hdim}_{G}^{\mathcal{F}}(H_{\varphi})=\nicefrac{{1}}{{2}}$ with respect to the Frattini series $\mathcal{F}$?