Duplicate Free-Group Word-Measure Conjecture

Definitions and mathematical statement

Let $n\in{\mathbb{N}}$ with $n\geq 2$, and let $F=F(x_1,\ldots,x_n)$ be a free group with the indicated free basis. For any finite group $G$ and any word $w\in F$, define the word map \[ w_G:G^n\longrightarrow G,\qquad (g_1,\ldots,g_n)\longmapsto w(g_1,\ldots,g_n), \] where evaluation substitutes $g_i$ for $x_i$. The associated word measure is the pushforward of the uniform probability measure on $G^n$: for every subset $S\subseteq G$, \[ P_{w,G}(S)=\frac{\lvert w_G^{-1}(S)\rvert}{\lvert G\rvert^n}. \] Here $\operatorname{Aut}(F)$ denotes the group of automorphisms of $F$.

Conjecture 4, as reconstructed from the supplied extraction. For every finite rank $n\geq 2$, every free group $F$ of rank $n$, and every pair $w,w'\in F$, \[ \left[\text{for every finite group }G\text{ and every }S\subseteq G,\quad P_{w,G}(S)=P_{w',G}(S)\right] \ \Longrightarrow\ \left[\text{there exists }\varphi\in\operatorname{Aut}(F)\text{ such that }\varphi(w)=w'\right]. \] Both word measures use the same chosen basis of $F$. No exclusion of the identity word or restriction to a particular class of finite groups is supplied.

Unresolved source qualifications. The displayed definition of the word measure is corrupted in the supplied extraction; the formula above explicitly adopts its usual interpretation as the distribution of word evaluation on independent uniform inputs. The source record is marked unrecoverable and titled "Duplicate Free-Group Word-Measure Conjecture." Its supplied verification note says that contextual material precedes the same Conjecture 4 preserved in the next record; no permanent identifier for that next record is supplied. Thus this is a statement-only reconstruction of the explicitly quoted conjecture, retaining permanent ID $30006197$, and requires human review. The preceding discussion of profinite rigidity, surface subgroups, and branched surfaces depends on an absent Theorem 2 and incompletely defined objects; it is not imposed as an additional hypothesis on this conjecture.

Short English statement

The extracted conjecture asks whether two words in the same non-abelian free group must be related by a free-group automorphism whenever their evaluations have identical probability distributions in every finite group. The supplied record is flagged as an unrecoverable duplicate, and its probability formula needs reconstruction. Its partially solved label and listed special cases are supplied dated curation, not independently verified conclusions.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30006197$, title "Duplicate Free-Group Word-Measure Conjecture," exact source code catalogue entry; Oberwolfach Reports Open Problems collection, proposed year 2025. No individual proposer is supplied.
• Geometric Structures in Group Theory (2025), Oberwolfach Reports, pp. 275--338, DOI 10.4171/owr/2025/6: https://doi.org/10.4171/owr/2025/6. This is the supplied report citation and the literature-source link labelled "OWR report." The extracted target is labelled Conjecture 4. The supplied verification field reports that the official report confirms duplication with the next record; that verification has not been independently performed here.
• Supplied literature source labelled "Hanany--Meiri--Puder": https://doi.org/10.1016/j.jalgebra.2020.03.013. No article title or theorem-level attribution for the listed special cases is supplied.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem partially solved and says that the conjecture remains open in general, with substantial cases proved for primitive words, powers, commutators, and partial-surface words. This is supplied dated curation derived from the Oberwolfach material, not an independently verified literature assessment. The source explicitly requests independent verification of time-sensitive status and cited claims.
• The extracted contextual passage attributes a surface-subgroup result to Wilton under reference label [6], but the bibliographic entry for [6] is absent from the supplied record. Its references to Theorem 2 likewise lack that theorem's statement.