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 \({\mathbb {Q}}(\sqrt{-11})\) $\mathbb{Q}(\sqrt{-11})$ , \({\mathbb {Q}}(\sqrt{-19})\) $\mathbb{Q}(\sqrt{-19})$ , \({\mathbb {Q}}(\sqrt{-43})\) $\mathbb{Q}(\sqrt{-43})$ , \({\mathbb {Q}}(\sqrt{-67})\) $\mathbb{Q}(\sqrt{-67})$ or \({\mathbb {Q}}(\sqrt{-163})\) $\mathbb{Q}(\sqrt{-163})$ . 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.