The finitely generated Hausdorff spectra of a family of pro-\(p\) groups (Q2144378)

Let \(\Gamma\) be a countably based, infinite profinite group. Let \(\mathscr{S}\) be a filtration series of \(\Gamma\), that is, a descending sequence \(\Gamma = \Gamma_{0} \ge \Gamma_{1} \ge \Gamma_{i} \ge \cdots\) of open normal subgroups with trivial intersection. Let \(d^{\mathscr{S}}(x, y) = \inf \{ \vert \Gamma : \Gamma_{i} \vert^{-1} : x \equiv y \pmod{\Gamma_{i}} \}\) be the induced translation-invariant metric. \textit{Y. Barnea} and \textit{A. Shalev} [Trans. Am. Math. Soc. 349, No. 12, 5073--5091 (1997; Zbl 0892.20020)] have given a formula for the Hausdorff dimension of a closed subgroup \(H\) of \(\Gamma\) as \(\displaystyle \operatorname{hdim}^{\mathscr{S}}_{\Gamma}(H) = \liminf_{i \to \infty} \frac{\log \lvert H \Gamma_{i} : \Gamma_{i} \rvert}{\log \lvert \Gamma : \Gamma_{i} \rvert}\). \par The paper under review is concerned with the computation of the Hausdorff spectrum \(\operatorname{hspec}^{\mathscr{S}}_{\trianglelefteq}(\Gamma) = \{ \operatorname{hdim}^{\mathscr{S}}_{\Gamma}(H) : H \le \Gamma \text{ closed and normal} \}\) of \(\Gamma\), and of its finitely generated version \(\operatorname{hspec}^ {\mathscr{S}}_{\text{fg}} (\Gamma) = \{ \operatorname{hdim}^{\mathscr{S}}_{\Gamma}(H) : H \le \Gamma \text{ closed and finitely generated} \}\), for \(\mathscr{S}\) one of five standard filtration series, such as the lower \(p\)-central series. \par The Authors show among others that the finitely generated Hausdorff spectra of these groups consist of infinitely many \(p\)-adic rational numbers. This result supports the conjecture that there are no finitely generated pro-\(p\) groups with an uncountable, finitely generated Hausdorff spectrum.