Center-freeness of finite-step solvable groups arising from anabelian geometry
Naganori Yamaguchi
TL;DR
The paper investigates center-freeness for maximal $m$-step solvable quotients of étale fundamental groups in the setting of anabelian geometry, focusing on hyperbolic curves over characteristic $0$. It develops a pro-$\Sigma$ Fox calculus framework and the Blanchfield–Lyndon sequence to compute centralizers in free $m$-step solvable groups, establishing explicit results such as $\mathrm{C}_{\mathcal{F}^{m}}(x^{n})=\langle x\rangle$ and slimness when $r\neq 1$. These algebraic Insights enable the authors to prove center-freeness for $\Delta_{X}^{m}$ and $\Pi_{X}^{(m)}$ in the geometric setting, and they derive the injectivity part of the $m$-step Grothendieck conjecture for hyperbolic curves by relating geometric isomorphisms to $G_{k}$-isomorphisms of solvable quotients. Together, the results extend anabelian rigidity to solvable quotients and bolster the $m$-step Grothendieck program, linking centralizer structure to reconstruction of curves from solvable Galois data.$
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.