Anabelian characterization conjecture for the geometric fundamental group

Let $X$ be the curve over the field $k$ considered in the paper, let $\bar{k}$ be an algebraic closure of $k$, and write $\Pi_X$ for its arithmetic fundamental group and $\Delta_X$ for its geometric fundamental group. Suppose that $\Pi_X$ is not topologically finitely generated and that $G_k\simeq\mathbb{Z}_p$. For a closed normal subgroup $N$ of $\Pi_X$ satisfying $\Pi_X/N\simeq\mathbb{Z}_p$, define, for $m\in\mathbb{Z}_{>0}$,

$$N_m:=\operatorname{Ker}\bigl(\Pi_X\twoheadrightarrow (\Pi_X/N)/p^m(\Pi_X/N)\bigr).$$

For an open subgroup $U$ of $\Pi_X$ and a prime $l\neq p$, define

$$R_m(U,l):=\operatorname{rank}_{\mathbb{Z}_l}\bigl((U\cap N_m)^l\bigr)^{\mathrm{ab/tor}}.$$

Anabelian characterization conjecture. The subgroup $\Delta_X$ is the unique closed normal subgroup $N$ of $\Pi_X$ satisfying: (i) $\Pi_X/N\simeq\mathbb{Z}_p$; and (ii) for every open subgroup $U$ of $\Pi_X$, there is a positive integer $R(U)$, depending only on $U$, such that for every prime $l\neq p$,

$$R(U)=\lim_{m\to\infty}R_m(U,l).$$

The conjecture proposes a group-theoretic recovery of the geometric fundamental group from the arithmetic fundamental group in the case where the latter is not topologically finitely generated. The preceding lemma supplies the relevant rank-growth characterization in the prime-to-$p$ case, while the uniqueness assertion for the subgroup with quotient $\mathbb{Z}_p$ remains conjectural.