The Cohomology of Canonical Quotients of Free Groups and Lyndon Words

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 $H^{2}(S^{[n,p]},{\
Z}/p)$, 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.

2010 Mathematics Subject Classification: Primary 12G05, Secondary 20J06, 68R15

Keywords and Phrases: Profinite cohomology, lower p-central filtration, Lyndon words, Shuffle relations, Massey products

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