#### Vol. 303, No. 2, 2019

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A pro-$p$ group with infinite normal Hausdorff spectra

### Benjamin Klopsch and Anitha Thillaisundaram

Vol. 303 (2019), No. 2, 569–603
##### Abstract

Using wreath products, we construct a finitely generated pro-$p$ group $G$ with infinite normal Hausdorff spectrum

${hspec}_{⊴}^{\mathsc{P}}\left(G\right)=\left\{{hdim}_{G}^{\mathsc{P}}\left(H\right)\mid H{⊴}_{c}G\right\};$

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

Lastly, we pin down the ordinary Hausdorff spectra

${hspec}^{\mathsc{S}}\left(G\right)=\left\{{hdim}_{G}^{\mathsc{S}}\left(H\right)\mid H{\le }_{c}G\right\}$

with respect to the standard filtration series $\mathsc{S}$. The spectrum ${hspec}^{\mathsc{ℒ}}\left(G\right)$ for the lower $p$-series $\mathsc{ℒ}$ displays surprising new features.

##### Keywords
pro-$p$ groups, Hausdorff dimension, Hausdorff spectrum, normal Hausdorff spectrum
Primary: 20E18
Secondary: 28A78