The symbol length for elementary type pro-p groups and Massey products

Abstract

For a prime number p and an integer m≥2, we prove that the symbol length of all elements of m-fold Massey products in H2(G,Fp), for pro-p groups G of elementary type, is bounded by (m2/4)+m. Assuming the Elementary Type Conjecture, this applies to all finitely generated maximal pro-p Galois groups G=GF(p) of fields F which contain a root of unity of order p. More generally, we provide such a uniform bound for the symbol length of all pullbacks ρ⁎(ω¯) of a given cohomology element ω¯∈Hn(G¯,Fp), where G¯ is a finite p-group, n≥2, and ρ:G→G¯ is a pro-p group homomorphism.