Dimensions of Zassenhaus filtration subquotients of some pro-$p$-groups

TL;DR

This paper calculates the dimensions of graded pieces in the Zassenhaus filtration for various finitely generated pro-$p$-groups, including free and Demushkin groups, providing a unifying framework for these computations.

Contribution

It introduces a unifying principle to compute the dimensions of Zassenhaus filtration subquotients across different classes of pro-$p$-groups.

Findings

Dimensions are explicitly computed for free pro-$p$-groups.

Dimensions are explicitly computed for Demushkin pro-$p$-groups.

A general method applies to free pro-$p$-products.

Abstract

We compute the ${\mathbb F}_p$-dimension of an $n$-th graded piece $G_{(n)}/G_{(n+1)}$ of the Zassenhaus filtration for various finitely generated pro-$p$-groups $G$. These groups include finitely generated free pro-$p$-groups, Demushkin pro-$p$-groups and their free pro-$p$ products. We provide a unifying principle for deriving these dimensions.