具有完全正规Hausdorff谱的pro- $p$群
A pro-$p$ group with full normal Hausdorff spectra
摘要：
对于每一个奇素数$p$，我们产生一个$2$生成的pro-$p$群$G$，它的正规Hausdorff谱\[ \mathrm{hspec}_{\trianglelefteq}^{\mathcal{S}}(G) = \{ \mathrm{hdim}_{G}^{\mathcal{S}}(H)\mid H\trianglelefteq_\mathrm{c} G \} \]关于五个标准过滤级数$\mathcal{S}$ -即下$p$ -级数，维数子群级数，$p$-幂级数，迭代$p$ -幂级数和Frattini级数-都等于全单位区间$[0,1]$ .这里，$\mathrm{hdim}_G^{\mathcal{S}} \colon \{ X\mid X \subseteq G \} \to[0,1]$表示与由过滤级数$\mathcal{S}$引起的自然消隐不变度量相关联的豪斯多夫维数函数。
英文摘要：
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 $\mathcal{S}$.
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