The metabelian Grothendieck conjecture for genus zero curves over finitely generated fields

In this paper, we prove the metabelian Grothendieck conjecture for genus-zero curves over finitely generated fields.

More precisely, we show that two hyperbolic genus-zero curves are isomorphic over the base field, up to Frobenius twist in positive characteristic, if and only if their geometrically maximal metabelian tame fundamental groups are isomorphic over the absolute Galois group of the base field.