A pro-p group with full normal Hausdorff spectra

2020/04/06 by Iker de las Heras, Heras, Iker de las, Benjamin Klopsch +1
Mathematics · #20E18 #28A78 #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2004.02846

openalex publication_date 2020/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For each odd prime p, we produce a 2-generated pro-p group G whose normal Hausdorff spectra hspec_{\trianglelefteq}^S(G) = \ hdim_G^S(H)| H\trianglelefteq_c G \ with respect to five standard filtration series 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 hdim_G^S \colon \ X| X ⊆ G \ →[0,1] denotes the Hausdorff dimension function associated to the natural translation-invariant metric induced by the filtration series S.

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