Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cohomology and the Combinatorics of Words for Magnus Formations

Ido Efrat

Abstract

For a prime number $p$ and a free pro-$p$ group $G$ on a totally ordered basis $X$, we consider closed normal subgroups $G^Φ$ of $G$ which are generated by $p$-powers of iterated commutators associated with Lyndon words in the alphabet $X$. We express the profinite cohomology group $H^2(G/G^Φ)$ combinatorically, in terms of the shuffle algebra on $X$. This partly extends existing results for the lower $p$-central and $p$-Zassenhaus filtrations of $G$.

Cohomology and the Combinatorics of Words for Magnus Formations

Abstract

For a prime number and a free pro- group on a totally ordered basis , we consider closed normal subgroups of which are generated by -powers of iterated commutators associated with Lyndon words in the alphabet . We express the profinite cohomology group combinatorically, in terms of the shuffle algebra on . This partly extends existing results for the lower -central and -Zassenhaus filtrations of .
Paper Structure (8 sections, 14 theorems, 53 equations)

This paper contains 8 sections, 14 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Binomial maps
  3. Words
  4. Magnus formations
  5. The fundamental matrix
  6. Unitriangular representations
  7. The isomorphism theorem
  8. The shuffle algebra

Key Result

Lemma 2.2

The map $e$ is binomial if and only if the following two conditions hold:

Theorems & Definitions (36)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Lemma 3.1
  • Proposition 4.1
  • proof
  • ...and 26 more