The Cohomology of canonical quotients of free groups and Lyndon words

2016/04/26 by Ido Efrat, Efrat, Ido · 1 citation
Mathematics · #68R15 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Number Theory (math.NT) #Primary 12G05 #Secondary 20J06

paper · pdf · doi:10.48550/arxiv.1604.07566

openalex publication_date 2016/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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