Zassenhaus and lower central filtrations of pro-$p$ groups considered as modules

TL;DR

This paper investigates the structure of Zassenhaus and lower central filtrations in finitely generated pro-$p$ groups, especially in the semisimple case and for groups with low cohomological dimension, using an isotypical framework.

Contribution

It introduces an isotypical approach to analyze filtrations of pro-$p$ groups, focusing on the semisimple case and finitely presented groups with cohomological dimension ≤ 2.

Findings

Characterization of filtrations in the semisimple case

Results for finitely presented groups with low cohomological dimension

Insights into the module structure of these filtrations

Abstract

The goal of this paper is to study Zassenhaus and lower central filtrations of finitely generated pro-$p$ groups in an isotypical context. We shall focus on the semisimple case. Particular attention is given for finitely presented groups of cohomological dimension less than or equal two.