A pro-2 group with full normal Hausdorff spectra (Q2170270)

Let \(\Gamma\) be a countably based, infinite profinite group and \({\mathcal S}\) a filtration series of \(\Gamma\), that is, a descending chain of open normal subgroups, \(\{\Gamma_i\}\), with trivial intersection. With respect to this filtration \(\Gamma\) has a naturally defined metric which yields the Hausdorff dimension of a closed subgroup \(H\) of \(\Gamma\), denoted \(\mathrm{hdim}_{\Gamma}^{\mathcal S}(H) \in [0,1]\). The Hasudorff spectrum of \(\Gamma\), with respect to \({\mathcal S}\), is a subset of \([0,1]\) given by the set of all \(\mathrm{hdim}_{\Gamma}^{\mathcal S}(H)\) where \(H\) runs through all closed subgroups of \(\Gamma\). The normal Hausdorff spectrum is when just normal closed subgroups of \(\Gamma\) are considered. Barnea and Shalev have given an algebraic formula for the Hausdorff dimension: \[ \mathrm{hdim}_{\Gamma}^{\mathcal S}(H) = \underline{\lim}_{i \rightarrow \infty} \frac{\log |H \Gamma_i : \Gamma_i |}{\log |\Gamma: \Gamma_i|} \] where \(\underline{\lim}_{i \rightarrow \infty}\) denotes the lower limit. This formulation makes calculating the (normal) Hausdorff spectrum more feasible. In this clearly written and technical paper the authors give an example of a 2-generated pro-2 group with full normal Hausdorff spectrum \([0,1]\), with respect to each of the four standard filtration series: the 2-power series, the lower 2-series, the Frattini series and the dimension subgroup series. This answers a question posed by Klopsch and the second author, about whether such a pro-2 group exists. An example for odd \(p\) was previously given by Klopsch and the first author.