DOCUMENTA MATHEMATICA, Vol. 22 (2017), 973-997

Ido Efrat

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 $H2(S[n,p],{\ Z}/p)$, 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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