A pro-p group with full normal Hausdorff spectra

Abstract

[EN] For each odd prime p, we produce a 2-generated pro-p group G whose normal Hausdorff spectra

hspec((sic))(S)(G) = {hdim(G)(S) (H) vertical bar (sic)(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) : {X vertical bar X subset of G} -> [0,1] denotes the Hausdorff dimension function associated to the natural translation-invariant metric induced by the filtration series S.