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

TL;DR

This paper constructs a specific pro-$p$ group for odd primes where the normal Hausdorff spectra with respect to five standard filtrations cover the entire range from 0 to 1, illustrating maximal diversity in subgroup dimensions.

Contribution

It introduces a 2-generated pro-$p$ group with full normal Hausdorff spectra across multiple standard filtration series, a novel example in the study of pro-$p$ groups.

Findings

Normal Hausdorff spectra equal to [0,1] for five filtration series

Constructs a 2-generated pro-$p$ group with maximal spectral diversity

Spectra cover the entire unit interval for all considered series

Abstract

For each odd prime $p$, we produce a $2$-generated pro-$p$ group $G$ whose normal Hausdorff spectra \[ \mathrm{hspec}_{\trianglelefteq}^{\mathcal{S}}(G) = \{ \mathrm{hdim}_{G}^{\mathcal{S}}(H)\mid H\trianglelefteq_\mathrm{c} G \} \] with respect to five standard filtration series $\mathcal{S}$ - namely the lower $p$-series, the dimension subgroup series, the $p$-power series, the iterated $p$-power series and the Frattini series - are all equal to the full unit interval $[0,1]$. Here $\mathrm{hdim}_G^{\mathcal{S}} \colon \{ X\mid X \subseteq G \} \to[0,1]$ denotes the Hausdorff dimension function associated to the natural translation-invariant metric induced by the filtration series $\mathcal{S}$.