Two-step nilpotent extensions are not anabelian

We prove the existence of two non-isomorphic number fields K and L such that the maximal two-step nilpotent quotients of their absolute Galois groups are isomorphic. In particular, one may take K and L to be any of the fields ℚ(√(-11)) , ℚ(√(-19)) , ℚ(√(-43)) , ℚ(√(-67)) or ℚ(√(-163)) . Furthermore, we give an explicit combinatorial description of these Galois groups in terms of a generalization of the Rado graph. A critical ingredient in our proofs is the back-and-forth method from model theory.