Imprecise Kashiwara–Vergne Symmetry Relation

Definitions and mathematical statement

The supplied record uses $\mathrm{KV}$ to denote a symmetry group of solutions of the Kashiwara--Vergne problem and refers to a prounipotent Grothendieck--Teichmüller group. It does not specify the versions of these groups, their base field, the solution space, or the symmetry action. These descriptions therefore identify the intended objects only provisionally.

Source-level assertion. The symmetry group $\mathrm{KV}$ of Kashiwara--Vergne solutions is conjecturally closely related to the prounipotent Grothendieck--Teichmüller group.

Unresolved formulation. The phrase "closely related" specifies no mathematical relation, map, equality, isomorphism, or quantified property. According to the supplied source-verification note, the report describes an approach illuminating a relationship conjectured elsewhere, without stating a precise standalone conjecture. The record is marked statement_status: unrecoverable, and no clean statement is supplied. Consequently, the evidence does not determine a precise mathematical proposition; the assertion above is retained for provenance and requires recovery of the intended conjecture before formalization.

The supplied literature assessment also describes an extracted welded-foam claim, but no welded-foam objects or precise claim are given. Its statements that the Kashiwara--Vergne problem itself is solved and that its symmetry group is related to Grothendieck--Teichmüller groups are supplied background assessments, not independently verified conclusions of this draft.

Short English statement

The record suggests a conjectural relationship between Kashiwara--Vergne symmetries and a prounipotent Grothendieck--Teichmüller group, but does not specify what that relationship means. It cannot presently be turned into a precise standalone conjecture. The supplied open label concerns this imprecise extracted relationship, while the supplied assessment describes the Kashiwara--Vergne problem itself as solved.

Sources

• Local UnsolvedMath registry record, permanent ID $30004480$, source problem code catalogue entry, title "Imprecise Kashiwara--Vergne Symmetry Relation," in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2020$ and no named proposer.
• Supplied report citation: Topologie (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/21, pp. 1083--1137. https://doi.org/10.4171/owr/2020/21. The supplied background describes Oberwolfach Reports as documenting workshops and their open-problem discussions.
• Dancso--Halacheva--Robertson, A topological characterisation of the KV groups. Supplied literature source: https://arxiv.org/abs/2106.02373.
• Dancso--Hogan--Robertson, A knot-theoretic approach to KV and GT groups. Supplied literature source: https://arxiv.org/abs/2211.11370.
• OWR Topology report. Supplied literature source: https://ems.press/content/serial-article-files/46858.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the extracted problem open, describes the Kashiwara--Vergne problem itself as solved, and characterizes the extracted welded-foam relationship as conjectural, without a proved equality or complete formulation. The supplied background requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification note says that the official report does not formulate a precise standalone claim and that the original record is retained publicly for provenance. The conjecture said to occur elsewhere is not identified precisely in the supplied evidence.