Cohomology and the combinatorics of words for Magnus formations (Q6614200)

Let \(G\) be a free pro \(p\)-group on a basis \(X\). It is well known that the first cohomology group of \(G\) with coefficient \(\mathbb{Z}/p\) describes the generators of it and the second cohomology group decribes the relations between generators. The author proves an isomorphism theorem for the second cohomology group in terms of Lyndon words and the shuffle algebra on the basis \(X\).\N\NThe author considers \(X\) as an alphabet with a fixed total order and define \(X^*\) as the set of words in \(X\). Lyndon words are the non-empty words \(w\) in \(X^*\) which are smaller in the alphabetical order than all their proper suffixes. For each such word \(w\), one can consider its Lie element \(\tau_w\) which is an iterated commutator in the free pro-\(p\) group \(G\). The author defines the closed subgroup \(G^\Phi\) of \(G\) which is generated by some specific \(p\)-powers of \(\tau_w\).\N\NThe shuffle \(\mathbb{Z}\)-algebra Sh\((X)\) over \(X\) is the \(\mathbb{Z}\)- module \(\bigoplus_{w\in X^*}\mathbb{Z}w\) with the shuffle product. Obviously one can consider the shuffle \(\mathbb{Z}_p\)-algebra \(\mathrm{Sh}(X)\otimes\mathbb{Z}_p \). Dividing it by the submodule generated by all shuffle products of nonempty words one gets the indecomposable quotient Sh\((X)_{\mathrm{indec}}\) of the shuffle algebra.\N\NThe main result of the paper is:\N\NFor sufficiently large \(p\), there is a canonical isomorphism of \(\mathbb{Z}/p\)-linear spaces \[(\bigoplus_{s\in I}\mathrm{Sh}(X)_{\mathrm{indec},s})\otimes \mathbb{Z}/p\cong H^2(G/G^{\Phi})\].\N\NThis isomorphism is proved in other papers for some special cases.\N\NIn [\textit{J. P. Labute}, Can. J. Math. 19, 106--132 (1967; Zbl 0153.04202)], the proof is given for the case, \(G^\Phi=G^p[G,G]\) and \(I=\{1,2\}\).\N\NIn [\textit{I. Efrat}, Doc. Math. 22, 973--997 (2017; Zbl 1437.20047); J. Pure Appl. Algebra 224, No. 6, Article ID 106260, 13 p. (2020; Zbl 1506.20086)] the author considers the lower \(p\)-central filtration and prove the main results for \(G^\Phi=G^{(n,p)}\) where \(n<p\) and \(I=\{1,2,....,n\}\).\N\NIn [\textit{I. Efrat}, J. Inst. Math. Jussieu 22, No. 2, 961--983 (2023; Zbl 1517.12005)], the main result is proved for \(G^\Phi=G_{(n,p)}\) where \(n<p\) and \(I=\{1,n\}\) and \(G_{(n,p)}\) is the term in \(p\)-Zassenhaus filtration of \(G\).\N\NThe main result of the current paper generalizes all these results. .