Geodesic
The Cohomology of Canonical Quotients of Free Groups and Lyndon Words
Documenta mathematica, Tome 22 (2017), pp. 973-997.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

For a prime number $p$ and a free profinite group $S$, let $S^{(n,p)}$ be the $n$th term of its lower $p$-central filtration, and $S^{[n,p]}$ the corresponding quotient. Using tools from the combinatorics of words, we construct a canonical basis of the cohomology group $H^2(S^{[n,p]}, \Bbb Z/p)$, which we call the Lyndon basis, and use it to obtain structural results on this group. We show a duality between the Lyndon basis and canonical generators of $S^{(n,p)}/S^{(n+1,p)}$. We prove that the cohomology group satisfies shuffle relations, which for small values of $n$ fully describe it.
Classification : 20J06, 12G05, 20E18, 68R15
Keywords: profinite cohomology, lower $p$-central filtration, Lyndon words, Shuffle relations, Massey products 
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     author = {Efrat, Ido},
     title = {The {Cohomology} of {Canonical} {Quotients} of {Free} {Groups} and {Lyndon} {Words}},
     journal = {Documenta mathematica},
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     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2017__22__a25/}
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Efrat, Ido. The Cohomology of Canonical Quotients of Free Groups and Lyndon Words. Documenta mathematica, Tome 22 (2017), pp. 973-997. http://geodesic.mathdoc.fr/item/DOCMA_2017__22__a25/