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

Anabelian geometry suggests that, for suitably geometric objects, their étale fundamental groups determine the geometric objects up to isomorphism. From a group-theoretic viewpoint, this philosophy requires rigidity properties, which often follow from their center-freeness of the associated étale fundamental groups. In fact, some profinite groups arising from anabelian geometry are center-free. For any integer $m\geq 2$, we investigate how such center-freeness behaves under passage to the maximal $m$-step solvable quotients. In particular, we show that the maximal $m$-step solvable quotients of the étale and tame fundamental groups of a hyperbolic curve over a separably closed field are torsion-free and center-free. Furthermore, we show that this implies the rigidity property of the $m$-step solvable Grothendieck conjecture.