Profinite Closure Obstructions in Free Groups

Definitions and mathematical statement

Let $F$ be a finitely generated free group. For $g\in F$, define its conjugacy class by \[ g^F=\{xgx^{-1}:x\in F\}. \] For a positive integer $k$ and elements $g_1,\ldots,g_k\in F$, let \[ N=\langle\!\langle g_1,\ldots,g_k\rangle\!\rangle_F \] be the smallest normal subgroup of $F$ containing all the $g_i$. Define the product of their conjugacy classes by \[ g_1^F\cdots g_k^F =\{x_1g_1x_1^{-1}\cdots x_kg_kx_k^{-1}:x_1,\ldots,x_k\in F\}. \] The profinite topology on $F$ has the finite-index normal subgroups as a neighborhood basis at the identity. The closure of $N$ in this topology is \[ \overline N^{\mathrm{pf}} =\bigcap_{\substack{K\trianglelefteq F\\[F:K]<\infty}}NK. \] The supplied question asks whether there exist $k\geq 1$ and $g_1,\ldots,g_k\in F$ such that \[ g_1^F\cdots g_k^F\not\subseteq\overline N^{\mathrm{pf}}. \] Unresolved formulation defect. Under the stated definition of $N$, normality gives $g_i^F\subseteq N$ for every $i$, and hence \[ g_1^F\cdots g_k^F\subseteq N\subseteq\overline N^{\mathrm{pf}}. \] Consequently, the requested noncontainment is incompatible with the supplied normal-closure wording. This identifies a defect in the recorded formulation; it does not recover or resolve the intended conjecture. The record does not specify the rank of $F$, whether the question is intended for every finitely generated free group or for some such group, or the precise equivalence with a nonsoficity question mentioned in its background.

Short English statement

The record asks whether a finite product of conjugacy classes in a finitely generated free group can escape the profinite closure of the normal subgroup generated by their representatives. The stated normal subgroup already contains that product, so the intended open question requires clarification from the cited source.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30001793$, source problem code catalogue entry, title Profinite Closure Obstructions in Free Groups. The supplied record labels the problem open and its statement corrected_verified; these are supplied registry classifications, not independent mathematical verification.
• Finite-dimensional Approximations of Discrete Groups (2011), Oberwolfach Reports, pp. 1429--1467, DOI 10.4171/owr/2011/26. Supplied source URL: https://doi.org/10.4171/owr/2011/26. The record attributes the collection to Oberwolfach workshop reports and gives the proposed year as 2011, but names no proposer and supplies no precise location within the report for this question.
• Supplied literature reference, labeled "original conjecture": https://doi.org/10.4171/owr/2011/26.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, says the extracted sentence describes an equivalence with a profinite closure conjecture without giving a resolution, and states that the associated nonsoficity question remains open. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The supplied background expressly calls for independent verification of time-sensitive status and cited claims.