Center-freeness of finite-step solvable groups arising from anabelian geometry

Abstract: Anabelian geometry suggests that, for suitably geometric objects, their étale fundamental group determines the object up to isomorphism. From a group-theoretic viewpoint, this philosophy requires rigidity properties of the associated étale fundamental groups, which often follow from their center-freeness. In fact, some profinite groups arising from anabelian geometry are center-free. In the present paper, we investigate how such center-freeness behaves when passing to maximal $m$-step solvable quotients for any integer $m\geq 2$. In particular, we show that the maximal $m$-step solvable quotient of the geometric étale fundamental group of a hyperbolic curve over a field of characteristic $0$ is center-free. Furthermore, we show that this implies the injectivity statement, i.e., the rigidity property, of the $m$-step solvable Grothendieck conjecture.