Abstract

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

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

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

Mathematical Subject Classification 2010

Milestones

Authors