Chasing Maximal Pro-p Galois Groups via 1-Cyclotomicity

Abstract

Let p be a prime. We prove that certain amalgamated free pro-p products of Demushkin groups with pro-p-cyclic amalgam cannot give rise to a 1-cyclotomic oriented pro-p group, and thus do not occur as maximal pro-p Galois groups of fields containing a root of 1 of order p. We show that other cohomological obstructions which are used to detect pro-p groups that are not maximal pro-p Galois groups—the quadraticity of Z/pZ-cohomology and the vanishing of Massey products—fail with the above pro-p groups. Finally, we prove that the Minač–Tân pro-p group cannot give rise to a 1-cyclotomic oriented pro-p group, and we conjecture that every 1-cyclotomic oriented pro-p group satisfy the strong n-Massey vanishing property for n=3,4.