A pro‐pgroup with full normal Hausdorff spectra

Abstract

For each odd prime p, we produce a 2‐generated pro‐p group G whose normal Hausdorff spectra 84.0pthspec⊴S(G)={}prefixhdimGscriptSfalse(Hfalse)∣H⊴normalcG\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$\mathcal {S}$.

The finitely generated Hausdorff spectra of a family of pro-p groups

“…As the terms of the filtration series L, D and M can be found in [5], it remains to settle the filtration series P, F, and I.…”

“…Next, in analogue to the even prime case, one can determine the Frattini series F precisely, which we include here for completeness. Following the notation in [5], we write…”

“…Over the past few years, the study of the normal Hausdorff spectra of finitely generated pro-p groups has received quite a bit of attention. Indeed, the first examples of finitely generated pro-p groups with infinite normal Hausdorff spectra, with respect to the series P, I, L, F, D were constructed in [15], and the first family of finitely generated pro-p groups G(p) with full normal Hausdorff spectra [0, 1] was constructed in [5] and [6].…”

“…Here M denotes a natural filtration series that arises from the construction of G(p); see Section 2 for details. We note that for p = 2 the construction of the pro-2 group G(2) (which was given in [6]) in the above family is different from the odd prime case (given in [5]); see Section 2 for precise details. Now to prove Theorem 1.1, we will focus solely on the pro-2 group G(2).…”

“…From [5] and [6], we have that for these pro-p groups G(p), for p a prime, the subgroup Z has strong Hausdorff dimension 1 with respect to the filtration series M, L, D, P, I, and F. Here we recall that M : M 0 ≥ M 1 ≥ • • • stands for the natural filtration series of G(p) where each M i is the subgroup of G(p) corresponding to N i /N , and here…”