Finite Index of Arithmetic Dehn-Twist Subgroups

Definitions and mathematical statement

Let $p$ be a prime and let $K$ be a $p$-adic field, understood in the usual sense of a finite extension of ${\mathbb{Q}}_p$. Fix a separable closure $K^{\mathrm{sep}}$, and put \[ G_K=\operatorname{Gal}(K^{\mathrm{sep}}/K),\qquad G_K(p)=G_K\Big/\bigcap_{\substack{N\trianglelefteq G_K\text{ open}\\[G_K:N]\text{ a power of }p}}N. \] Thus $G_K(p)$ is the maximal pro-$p$ quotient of the absolute Galois group. Write \[ \operatorname{Out}(G_K(p))= \operatorname{Aut}_{\mathrm{cont}}(G_K(p))/\operatorname{Inn}(G_K(p)). \] For the subgroup-generation question, use the usual topology on continuous automorphisms and the induced quotient topology on the outer automorphism group.

Let $\mathcal T_K\subseteq\operatorname{Out}(G_K(p))$ denote the family referred to in the supplied corrected statement as all arithmetic Dehn twists, and interpret the profinite subgroup they generate as \[ D_K=\overline{\langle\mathcal T_K\rangle} \subseteq\operatorname{Out}(G_K(p)). \] The supplied record does not define arithmetic Dehn twists or specify which constructions and choices determine $\mathcal T_K$. Consequently, this notation is a placeholder for the intended source-defined family, and the displayed definition of $D_K$ is conditional on identifying that family. Taking the closure in the usual outer automorphism topology is an explicit interpretation of the phrase "profinite subgroup generated"; the record does not specify the topology itself.

Question. For every $p$-adic field $K$, is \[ [\operatorname{Out}(G_K(p)):D_K]<\infty? \] No restrictions on $p$ or additional hypotheses on $K$ are stated in the supplied formulation. The original excerpt says "all Dehn twists," whereas the corrected statement specifies "all arithmetic Dehn twists." Their intended identification and any hypotheses required by the source construction remain to be checked.

Short English statement

For a $p$-adic field $K$, consider the maximal pro-$p$ quotient of its absolute Galois group. The question asks whether the closed subgroup generated by all arithmetic Dehn twists has finite index in its continuous outer automorphism group. The supplied record leaves the twist family undefined, so this formulation requires clarification from the cited source. The problem's open status is supplied curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006075$, title "Finite Index of Arithmetic Dehn-Twist Subgroups," exact source problem code catalogue entry. No individual proposer is supplied; the proposed year is recorded as 2024.
• Anabelian Geometry and Representations of Fundamental Groups (2025), Oberwolfach Reports, pp. 2621--2670, DOI 10.4171/owr/2024/45: https://doi.org/10.4171/owr/2024/45. The original extracted question carries the page marker 2641. The supplied literature list labels this reference "Oberwolfach Report 45/2024." The citation year 2025 and report designation 2024 are retained as supplied.
• Gropper, Surfaces and p-adic fields I: https://arxiv.org/abs/2303.04309. This is the second reference in the supplied literature list; no more detailed bibliographic information is supplied.
• Supplied literature assessment dated exactly 2026-08-22: the record labels the problem open and reports that arithmetic Dehn twists have infinite order in the relevant outer automorphism groups, while the finite-index question remains unanswered. This is supplied dated curation, not independently verified here; the record itself requests independent verification of time-sensitive status and cited claims. Its corrected_verified statement label and assertion that the official report was checked are also supplied provenance, not verification performed for this draft.