Profinite Rigidity with Free Edge Groups

Definitions and mathematical statement

For a group $G$, its profinite completion is the topological group \[ \widehat G=\varprojlim_{\substack{N\trianglelefteq G\\[G:N]<\infty}}G/N. \] Here an isomorphism between profinite completions is understood to be an isomorphism of topological groups.

Let $H$ be an arbitrary finitely generated free group. Consider either of the following constructions:

• Free groups $F_1,F_2$ and injective homomorphisms $\iota_j:H\hookrightarrow F_j$, giving the amalgamated product \[ \Lambda=F_1*_{\iota_1(H)=\iota_2(H)}F_2. \]
• A free group $F_1$ and injective homomorphisms $\alpha,\beta:H\hookrightarrow F_1$, giving the HNN extension \[ \Lambda=\left\langle F_1,t\ \middle|\ t^{-1}\alpha(h)t=\beta(h)\text{ for every }h\in H\right\rangle. \]

Let $\Gamma$ be a free group or a surface group. The supplied question asks whether, for every such choice, \[ \widehat\Lambda\cong\widehat\Gamma \quad\Longrightarrow\quad \Lambda\cong\Gamma. \] The conclusion concerns an isomorphism of abstract groups.

Unresolved domain qualifications. The supplied reconstructed statement explicitly requires $H$ to be finitely generated, whereas the retained original excerpt says only that $H$ is an arbitrary free group. The allowed ranks of the vertex groups and of a free target $\Gamma$ are not specified. A surface group means a fundamental group $\pi_1(S)$, but the intended class of connected surfaces $S$, including orientability, compactness, boundary, and genus restrictions, is not specified. Consequently, the displayed question preserves the supplied formulation but does not determine a unique fully specified domain of quantification. No residual-finiteness assumption on $\Lambda$ is supplied.

The supplied dated assessment reports that the cyclic-edge case has been proved and that the arbitrary-free-edge case remains open. The precise hypotheses and scope of that reported cyclic result are not supplied here; these are source qualifications, not independently verified mathematical conclusions.

Short English statement

Must an amalgamated product or HNN extension of free groups over a finitely generated free group be isomorphic to a free or surface group whenever their profinite completions are isomorphic? The intended rank restrictions and class of surfaces require clarification, and the original excerpt does not explicitly impose finite generation on the edge group.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30006196$, title Profinite Rigidity with Free Edge Groups, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2025; no individual proposer is supplied.
• Geometric Structures in Group Theory (2025), Oberwolfach Reports, pp. 275--338, DOI 10.4171/owr/2025/6. Supplied report reference: https://doi.org/10.4171/owr/2025/6.
• Supplied literature source, labelled "OWR 2025/6 geometric group theory report": https://oa.tib.eu/renate/server/api/core/bitstreams/6f61d540-8e50-41f3-97a8-6e8214672460/content.
• Supplied literature source, labelled "Jaikin-Zapirain and Morales, free and surface groups": https://arxiv.org/abs/2211.12390.
• Literature assessment dated 2026-08-22: supplied dated curation, not independently verified. It labels the problem open, reports the cyclic case as proved, and states that nearby results for free and surface groups do not settle the general free-edge case. The background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. This records supplied provenance; it does not constitute independent verification in this draft. The retained original excerpt states only: "The case when H is an arbitrary free group remains open."