The cohomology of canonical quotients of free groups and Lyndon words

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For a prime number p and a free profinite group S, let S(n,p) be the nth 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.

Original languageAmerican English
Pages (from-to)973-997
Number of pages25
JournalDocumenta Mathematica
Issue number2017
StatePublished - 1 Jan 2017


  • Lower p-central filtration
  • Lyndon words
  • Massey products
  • Profinite cohomology
  • Shuffle relations

All Science Journal Classification (ASJC) codes

  • General Mathematics


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