We associate to every algebraic number field a geometrization by hyperbolic surface laminations as well as a corresponding "external fundamental group". The external fundamental group is a generalization of the fundamental germ: the latter being an "irrational" version of the ordinary fundamental group that is appropriate for laminations. Whereas the fundamental germ consists of internal (i.e. first order definable) diophantine approximations of "irrational" loops, the external fundamental group necessarily includes external (not first order definable) approximations. The external fundamental group is a split extension of the absolute Galois group; we conjecture that its abelianization (or that of a subgroup) is isomorphic to the idele class group.