The external fundamental group of an algebraic number field

We associate to every algebraic number field K/Q a hyperbolic surface lamination and an external fundamental group degrees Gamma(K): a generalization of the fundamental germ construction of Gendron that necessarily contains external (not first order definable) elements. The external fundamental group degrees Gamma(Q) is an extension of the absolute Galois group (Gamma) over cap (Q), that conjecturally contains a subgroup whose abelianization is isomorphic to the idele class group.