The External Fundamental Group of an Algebraic Number Field

TL;DR

This paper introduces the external fundamental group of algebraic number fields, linking hyperbolic surface laminations with Galois groups and idele class groups, revealing new structural insights.

Contribution

It defines the external fundamental group for algebraic number fields and explores its relation to Galois groups and idele class groups, extending classical concepts.

Findings

External fundamental group contains external elements beyond first order definability.

The group of rationals' external fundamental group is a split extension of the absolute Galois group.

Conjecturally, it contains a subgroup with abelianization isomorphic to the idele class group.

Abstract

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.