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

TL;DR

The paper develops an equivariant Gocha framework for finitely generated pro-$p$ groups by incorporating a cyclic automorphism group $\Delta$ and decomposing invariants into irreducible components, focusing on the semisimple case and groups with ${\rm cd}(G)\le 2$. It proves an equivariant analogue of the Mináč–Rogelstad–Tân relations, introducing $gocha^*({\mathbb A},t)$, $gocha({\mathbb A},t)$ and $gocha_{\chi_0}({\mathbb A},t)$, and deriving how the equivariant coefficients $a_n^{\chi}$ relate to $w_n^{\chi}$ via Möbius-type formulas. The work establishes that these equivariant Gocha-series are independent of the base ring ${\mathbb A}$ in the cd$\le 2$ finitely presented setting, and provides criteria ensuring all eigenspaces of the filtration Lie algebra ${\mathcal L}({\mathbb A},G)$ are infinite dimensional, by analyzing the entropy $L_{\chi_0}(G)$ and the growth of $b_{\chi_0,n}$. Through explicit constructions—free pro-$p$ groups, mild quadratic quotients, and FAB examples from algebraic number theory—the paper demonstrates the ubiquity of infinite-dimensional eigenspaces and offers practical tools (Lyndon resolutions, Gocha-type series, and representation-theoretic decompositions) for probing lower central and Zassenhaus filtrations as ${\mathbb A}[\Delta]$-modules.

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.

Theorems & Definitions (28)

• Theorem : Alternative des Gocha
• Theorem A
• Theorem B
• Example 1: Cohomological dimension $2$
• Example 2: FAB example
• proof
• proof
• proof
• proof : Proof of Formula \ref{['JeLaRep']}
• proof