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

TL;DR

This paper studies the center-freeness of finite-step solvable quotients of fundamental groups in anabelian geometry, demonstrating torsion-freeness and center-freeness for certain quotients, which supports rigidity conjectures.

Contribution

It proves that maximal m-step solvable quotients of fundamental groups of hyperbolic curves are torsion-free and center-free, advancing understanding of their rigidity properties.

Findings

Maximal m-step solvable quotients are torsion-free.

These quotients are center-free.

Results support the m-step solvable Grothendieck conjecture.

Abstract

Anabelian geometry suggests that, for suitably geometric objects, their \'etale 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 \'etale 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 \'etale 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.