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

TL;DR

This paper computes the finitely generated Hausdorff spectra of a specific family of pro-$p$ groups across five standard filtration series, revealing infinitely many rational numbers and supporting the conjecture about the spectrum's cardinality.

Contribution

It provides the first detailed computation of finitely generated Hausdorff spectra for this family of pro-$p$ groups across multiple filtration series, using a technical approach.

Findings

Finitely generated Hausdorff spectra consist of infinitely many rational numbers.

Supports the conjecture that no finitely generated pro-$p$ group has uncountable spectrum.

Provides detailed spectra for five standard filtration series.

Abstract

Recently the first example of a family of pro-$p$ groups, for $p$ a prime, with full normal Hausdorff spectrum was constructed. In this paper we further investigate this family by computing their finitely generated Hausdorff spectrum with respect to each of the five standard filtration series: the $p$-power series, the iterated $p$-power series, the lower $p$-series, the Frattini series and the dimension subgroup series. Here the finitely generated Hausdorff spectra of these groups consist of infinitely many rational numbers, and their computation requires a rather technical approach. This result also gives further evidence to the non-existence of a finitely generated pro-$p$ group with uncountable finitely generated Hausdorff spectrum.