Abstract

AbstractFor a prime number p and a free profinite group S on the basis X, let $S_{\left (n,p\right )}$ , $n=1,2,\dotsc ,$ be the p-Zassenhaus filtration of S. For $p>n$ , we give a word-combinatorial description of the cohomology group $H^2\left (S/S_{\left (n,p\right )},\mathbb {Z}/p\right )$ in terms of the shuffle algebra on X. We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.

Highlights

  • The purpose of this paper is to study the p-Zassenhaus filtration of a free profinite group S and its cohomology by means of the combinatorics of words

  • P is a fixed prime number, and we recall that the p-Zassenhaus filtration of a profinite group G is given by G(n,p) = ipj≥n G(i) pj, n = 1,2, . . . – that is, G(n,p) is generated as a profinite group by all pj-powers of elements of the i th term of the lower central filtration G(i) of G for ipj ≥ n

  • In the present paper we focus on the cohomology group H2 G/G(n,p) for a profinite group G and n ≥ 2

Read more

Summary

Introduction

The purpose of this paper is to study the p-Zassenhaus filtration of a free profinite group S and its cohomology by means of the combinatorics of words. We modify these methods in several aspects: mainly, whereas in the lower p-central case one should consider words w of arbitrary lengths, in the case of Zassenhaus filtration we need to restrict to words of lengths n/pk , k ≥ 0, as above These ‘jumps’ arise when we analyze the filtration for the group Ui Z/pj of unitriangular (i + 1) × (i + 1)-matrices over Z/pj. The correspondence in the Main Theorem demonstrates deep connections between the p-Zassenhaus filtration and its cohomology and the n-fold Massey product H1(G)n → H2(G) It was shown in [7] that when S is a free profinite group, S(n,p)/S(n+1,p) is dual to the subgroup of H2 S/S(n,p) generated by all such products. [28] and the references therein)

Hall sets
Lie algebras
The p-Zassenhaus filtration
The fundamental matrix
Unitriangular matrices
The Lyndon bases
Shuffle relations
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call