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Zassenhaus filtrations as intersections

Jan Minac, Nguyen Duy Tan, Nguyen Thi Tra

TL;DR

The paper reframes the $p$-Zassenhaus filtration $G_{(n)}$ of profinite groups as intersections of kernels of representations into unipotent groups arising from rank-$n$ multiplicative systems, forging a link between filtration theory and representation/cohomology. It proves a main theorem: for a free pro-$p$ group $S$ and $R\le S_{(n)}$, with $G=S/R$, the subgroup $G_{(n+1)}$ is the intersection of kernels of all representations $G\to U(A)$ associated with rank-$n$ multiplicative systems having $A_{1,n+1}=\mathbb{F}_p$. The work also builds a non-degenerate cohomological pairing between certain Zassenhaus quotients and Massey-product obstructions, connecting these pairings to the kernel-unipotent framework and to Pontryagin duality in the free case. Together, these results bridge Zassenhaus filtrations with representation theory and cohomology, offering new tools for analyzing Galois and profinite groups via linear representations and higher-order cohomological operations.

Abstract

Zassenhaus filtrations of profinite groups are an important tool to study profinite groups.In this paper, we describe Zassenhaus filtrations of profinite groups as intersections of kernels of certain representations. In this way we introduce a link between studying profinite groups with methods of Zassenhaus filtrations and representation theory.

Zassenhaus filtrations as intersections

TL;DR

The paper reframes the -Zassenhaus filtration of profinite groups as intersections of kernels of representations into unipotent groups arising from rank- multiplicative systems, forging a link between filtration theory and representation/cohomology. It proves a main theorem: for a free pro- group and , with , the subgroup is the intersection of kernels of all representations associated with rank- multiplicative systems having . The work also builds a non-degenerate cohomological pairing between certain Zassenhaus quotients and Massey-product obstructions, connecting these pairings to the kernel-unipotent framework and to Pontryagin duality in the free case. Together, these results bridge Zassenhaus filtrations with representation theory and cohomology, offering new tools for analyzing Galois and profinite groups via linear representations and higher-order cohomological operations.

Abstract

Zassenhaus filtrations of profinite groups are an important tool to study profinite groups.In this paper, we describe Zassenhaus filtrations of profinite groups as intersections of kernels of certain representations. In this way we introduce a link between studying profinite groups with methods of Zassenhaus filtrations and representation theory.
Paper Structure (3 sections, 16 theorems, 55 equations)

This paper contains 3 sections, 16 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Multiplicative systems
  3. Pairings

Key Result

Theorem 1.1

Let $S$ be a free pro-$p$-group and $n\geq1$. Then $S_{(n)}$ is the intersection of all kernels of linear representations $\rho:G\to {\rm GL}_n({\mathbb F}_p)$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2: =Theorem \ref{['thm:main']}
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6: Dwy
  • ...and 22 more