The External Fundamental Group of an Algebraic Number Field

We associate to every algebraic number field a hyperbolic surface lamination and an external fundamental group: the latter a generalization of the fundamental germ that necessarily contains external (not first order definable) elements. The external fundamental group of the rationals is a split extension of the absolute Galois group, that conjecturally contains a subgroup whose abelianization is isomorphic to the idele class group.