Nonconjugate $\operatorname{SL}_3$-Character-Equivalent Free-Group Words

Definitions and mathematical statement

Let $F_2=\langle a,b\rangle$ be the free group on two generators. Define \[ \operatorname{SL}_3({\mathbb{C}})=\{A\in M_3({\mathbb{C}}):\det A=1\}. \] For a group homomorphism $\rho:F_2\to\operatorname{SL}_3({\mathbb{C}})$, its character is the function $\chi_\rho(w)=\operatorname{tr}(\rho(w))$, where $\operatorname{tr}$ is the sum of the diagonal entries. Elements $g,g'\in F_2$ are $\operatorname{SL}_3({\mathbb{C}})$-character equivalent if \[ \operatorname{tr}(\rho(g))=\operatorname{tr}(\rho(g')) \quad\text{for every group homomorphism }\rho:F_2\to\operatorname{SL}_3({\mathbb{C}}). \] They are conjugate in $F_2$ if $g'=hgh^{-1}$ for some $h\in F_2$.

The question is whether \[ \exists g,g'\in F_2\quad \left[ \begin{gathered} \forall\rho\in\operatorname{Hom}(F_2,\operatorname{SL}_3({\mathbb{C}})),\quad \operatorname{tr}(\rho(g))=\operatorname{tr}(\rho(g')),\\ \forall h\in F_2,\quad g'\ne hgh^{-1}. \end{gathered} \right] \] The quantifier ranges over all such representations, with no additional restriction. The supplied background mentions an analogous trace characterization for two generators without specifying that characterization; it is not used as an additional hypothesis here. The supplied open status and source-verification labels are curation claims, not independently verified conclusions of this draft.

Short English statement

Do there exist two nonconjugate elements of the free group on two generators whose traces agree under every representation into $\operatorname{SL}_3({\mathbb{C}})$?

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002136$, source problem code catalogue entry; supplied collection: Oberwolfach Reports Open Problems. The record gives proposed year 2012 and supplies no individual proposer.
• Supplied source citation: Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2013), Oberwolfach Reports, pp. 2139--2203, DOI 10.4171/owr/2012/35. https://doi.org/10.4171/owr/2012/35. The collection describes these reports as documenting mathematical workshops and their open-problem discussions. The citation year 2013 and the supplied proposed year 2012 are retained as given; their relationship has not been independently checked.
• Supplied literature reference: Lawton--Louder--McReynolds, traces and representations. https://arxiv.org/abs/1312.1261.
• Supplied literature reference: Published traces and representations paper. https://doi.org/10.4171/GGD/393.
• Supplied literature reference: Oberwolfach Report 35. https://doi.org/10.4171/owr/2012/35.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that the primary literature reports an analogous trace characterization for two generators while leaving the existence of an $\operatorname{SL}_3$-character-equivalent nonconjugate pair in the free group unresolved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its status and cited claims require independent checking.
• The supplied statement status is corrected_verified. Its verification note says that the mathematical content was checked against the official Oberwolfach report and that notation, context, and question form were normalized without changing the source-level claim. This draft records that supplied assertion without claiming independent source verification.