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- Title
A pro‐p group with full normal Hausdorff spectra.
- Authors
Heras, Iker de las; Klopsch, Benjamin
- Abstract
For each odd prime p, we produce a 2‐generated pro‐p group G whose normal Hausdorff spectra hspec⊴S(G)=hdimGS(H)∣H⊴cG\begin{equation*}\hskip7pc \operatorname{hspec}_{\trianglelefteq }^{\mathcal {S}}(G) = {\left\lbrace \operatorname{hdim}_{G}^{\mathcal {S}}(H)\mid H\trianglelefteq _\mathrm{c} G \right\rbrace}\hskip-7pc \end{equation*}with respect to five standard filtration series S$\mathcal {S}$, 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 hdimGS:{X∣X⊆G}→[0,1]$\operatorname{hdim}_G^{\mathcal {S}} : \lbrace X\mid X \subseteq G \rbrace \rightarrow [0,1]$ denotes the Hausdorff dimension function associated to the natural translation‐invariant metric induced by the filtration series S$\mathcal {S}$.
- Subjects
FRACTAL dimensions; POWER series; POINCARE series
- Publication
Mathematische Nachrichten, 2022, Vol 295, Issue 1, p89
- ISSN
0025-584X
- Publication type
Article
- DOI
10.1002/mana.202000164