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

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 $\mathcal{S}$.