The m-step solvable anabelian geometry of finitely generated fields

A celebrated theorem of Neukirch and Uchida states that two number fields are isomorphic if their absolute Galois groups are isomorphic. The Grothendieck birational conjecture predicts a similar result for all finitely generated fields. This has been proved by Pop, using in an essential way the Neukirch-Uchida theorem for global fields.
In a recent work, joint with Akio Tamagawa, we prove that two number fields are isomorphic if their 3-step solvable Galois groups are isomorphic. More generally we prove that two finitely generated fields of (kronecker) dimension d are isomorphic if their 3+[3d(d+1)/2]-step solvable Galois groups are isomorphic. These results bring (birational) anabelian geometry close to class field theory.
In my talk I will explain the proof of our results in the number field case as well as the ideas and key ingredients of proof in general.