Families preserving isomorphisms via techniques in anabelian geometry: with an application to a generalized Neukirch-Uchida theorem

Families preserving automorphisms of profinite groups are automorphisms that preserve each of the conjugacy classes of pro-cyclic subgroups. This notion appears in the context of verification of the property that every normal automorphism [i.e., an automorphism that preserves each of the normal closed subgroups] of a certain profinite group such as a nonabelian free profinite group or the absolute Galois group of a $p$-adic local field is an inner automorphism, which was proved by Jarden/Jarden-Ritter. In the present paper, we revisit the notion of a families preserving automorphism from the viewpoint of anabelian geometry. We introduce a natural generalized version of this notion for isomorphisms between general closed subgroups of a profinite group, which we shall refer to as families preserving isomorphisms in the profinite group. With regard to this generalized version, we prove that a large class of profinite groups satisfies a property that families preserving isomorphisms of certain closed subgroups of the profinite groups arise from inner automorphisms of them. For instance, the class includes the absolute Galois groups of Henselian discrete valuation fields of positive residue characteristic or Hilbertian fields. Moreover, as an application of this property, together with delicate considerations on pro-cyclic subgroups of the absolute Galois groups of infinite algebraic extension fields of $\mathbb{Q}$, we obtain a generalized version of the Neukirch-Uchida theorem for $l$-quasi-number fields, where an $l$-quasi-number field is defined to be an algebraic extension field $K$ of $\mathbb{Q}$ whose Galois closure $L$ over $\mathbb{Q}$ satisfies that $[L : \mathbb{Q}]$ is not divisible by $l^{\infty}$. Surprisingly, this result includes the Neukirch-Uchida-type result for the class of subfields of the maximal pro-prime-to-$l$ extension fields of arbitrary number fields.