In the present paper I show that one can recover much of the inertia structure of Zariski (quasi) divisors of a function field K|k over an algebraically closed base field k from the maximal pro-` abelian-by-central Galois theory of K. The results play a central role in the birational anabelian geometry and related questions.

We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A. Kishimoto’s result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and K-groups of our algebras.