Hausdorff dimensions in $p$-adic analytic groups

TL;DR

This paper investigates the Hausdorff spectrum of finitely generated pro-$p$ groups, establishing that finiteness of the spectrum characterizes $p$-adic analyticity for soluble groups and exploring related filtration series.

Contribution

It proves that for soluble pro-$p$ groups, a finite Hausdorff spectrum implies $p$-adic analyticity, and extends the analysis to various filtration series and broader classes of groups.

Findings

Finiteness of the Hausdorff spectrum characterizes $p$-adic analyticity in soluble groups.

Existence of filtration series with infinite spectra in certain pro-$p$ groups.

Analysis of Hausdorff spectra for multiple filtration series.

Abstract

Let $G$ be a finitely generated pro-$p$ group, equipped with the $p$-power series. The associated metric and Hausdorff dimension function give rise to the Hausdorff spectrum, which consists of the Hausdorff dimensions of closed subgroups of $G$. In the case where $G$ is $p$-adic analytic, the Hausdorff dimension function is well understood; in particular, the Hausdorff spectrum consists of finitely many rational numbers closely linked to the analytic dimensions of subgroups of $G$. Conversely, it is a long-standing open question whether the finiteness of the Hausdorff spectrum implies that $G$ is $p$-adic analytic. We prove that the answer is yes, in a strong sense, under the extra condition that $G$ is soluble. Furthermore, we explore the problem and related questions also for other filtration series, such as the lower $p$-series, the Frattini series, the modular dimension subgroup series and quite general filtration series. For instance, we prove, for odd primes $p$, that every countably based pro-$p$ group $G$ with an open subgroup mapping onto 2 copies of the $p$-adic integers admits a filtration series such that the corresponding Hausdorff spectrum contains an infinite real interval.