The lower $p$-central series of a free profinite group and the shuffle algebra

TL;DR

This paper provides a combinatorial description of the second cohomology group of certain quotients of free profinite groups using the shuffle algebra, revealing new algebraic structures related to the lower p-central series.

Contribution

It introduces a novel combinatorial approach to understanding the cohomology of free profinite groups via shuffle algebra, connecting group theory and algebraic combinatorics.

Findings

Describes $H^2(S/S^{(n,p)},\mathbb{Z}/p)$ in terms of shuffle algebra

Establishes a connection between lower p-central series and algebraic combinatorics

Provides explicit combinatorial descriptions for cohomology groups

Abstract

For a prime number $p$ and a free profinite group $S$ on the basis $X$, let $S^{(n,p)}$, $n=1,2,\ldots$ be the lower $p$-central filtration of $S$. For $p>n$, we give a combinatorial description of $H^2(S/S^{(n,p)},\mathbb{Z}/p)$ in terms of the Shuffle algebra on $X$.