The geometrically m-step solvable Grothendieck conjecture

Let $k$ be a field of characteristic $p\geq 0$, finitely generated over the prime field, with absolute Galois group $G_k$. Let $U$ be an anabelian variety over $k$ with a geometric point $\ast$, let $k^{\mathrm{sep}}$ be a separable closure of $k$, and define its geometrically maximal $m$-step solvable quotient by

$$\pi^{\acute{e}\mathrm{t}}_1(U,\ast)^{(m)}=\pi^{\acute{e}\mathrm{t}}_1(U,\ast)/\pi^{\acute{e}\mathrm{t}}_1(U_{k^{\mathrm{sep}}},\ast)^{[m]},$$

where $\pi^{\acute{e}\mathrm{t}}_1(U_{k^{\mathrm{sep}}},\ast)^{[m]}$ is the $m$-th commutator subgroup. The quotient has its natural projection to $G_k$. The geometrically $m$-step solvable Grothendieck conjecture. The variety $U$ may be reconstructed group-theoretically from $\pi^{\acute{e}\mathrm{t}}_1(U,\ast)^{(m)}$ and its natural projection to $G_k$. This is stronger than reconstruction from the full étale fundamental group because the quotient retains less information. The conjecture has been studied for hyperbolic curves, while the paper investigates genus-zero hyperbolic curves over finitely generated fields.