ON THE GROTHENDIECK CONJECTURE FOR AFFINE HYPERBOLIC CURVES OVER KUMMER-FAITHFUL FIELDS

Abstract

In the present paper, we discuss the Grothendieck conjecture for hyperbolic curves over Kummer-faithful fields. In particular, we prove that every point-theoretic and Galois-preserving outer isomorphism between the étale/tame fundamental groups of affine hyperbolic curves over Kummer-faithful fields arises from a uniquely determined isomorphism between the original hyperbolic curves. This result generalizes results of Tamagawa and Mochizuki, i.e., our main result in the case where the basefields are either finite fields or mixed-characteristic local fields.