Standard Hausdorff spectrum of compact $\mathbb{F}_p[[t]]$-analytic groups

TL;DR

This paper investigates the Hausdorff spectrum of compact $F_p[[t]]$-analytic groups, revealing it contains a full interval for soluble groups and identifying isolated points for classical Chevalley groups.

Contribution

It establishes the structure of the Hausdorff spectrum for a broad class of $F_p[[t]]$-analytic groups, including solubility conditions and spectral gaps.

Findings

Hausdorff spectrum contains a real interval for soluble groups

Spectrum is not full for classical Chevalley groups, with 1 as an isolated point

Spectrum coincides with the full unit interval for soluble groups

Abstract

We prove that the $\mathbb{F}_p[[t]]$-standard Hausdorff spectrum of a compact $\mathbb{F}_p[[t]]$-analytic group contains a real interval and that it coincides with the full unit interval when the group is soluble. Moreover, we show that the $\mathbb{F}_p[[t]]$-standard Hausdorff spectrum of classical Chevalley groups over $\mathbb{F}_p[[t]]$ is not full, since 1 is an isolated point thereof.