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

Benjamin Klopsch and Anitha Thillaisundaram

Vol. 303 (2019), No. 2, 569–603

DOI: 10.2140/pjm.2019.303.569

Abstract

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

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

here ${hdim}_G^{\mathcal{P}}:\left\{X\mid X\subseteq G\right\}\to \left[0,1\right]$ denotes the Hausdorff dimension function associated to the $p$-power series $\mathcal{P}:G^{p^i}$, $i\in {\mathbb{N}}_0$. More precisely, we show that ${hspec}_{⊴}^{\mathcal{P}}\left(G\right)=\left[0,\frac{1}{3}\right]\cup \left\{1\right\}$ contains an infinite interval; this settles a question of Shalev. Furthermore, we prove that the normal Hausdorff spectra ${hspec}_{⊴}^{\mathcal{S}}\left(G\right)$ with respect to other filtration series $\mathcal{S}$ have a similar shape. In particular, our analysis applies to standard filtration series such as the Frattini series, the lower $p$-series and the modular dimension subgroup series.

Lastly, we pin down the ordinary Hausdorff spectra

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

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

Keywords

pro-$p$ groups, Hausdorff dimension, Hausdorff spectrum, normal Hausdorff spectrum

Mathematical Subject Classification 2010

Primary: 20E18

Secondary: 28A78

Milestones

Received: 4 December 2018

Revised: 5 July 2019

Accepted: 6 July 2019

Published: 4 January 2020