Kourovka Notebook Problem 18.43

Definitions and mathematical statement

Let $F(a,b)$ be the free group on the two generators $a,b$. Write \[ \operatorname{GL}_3({\mathbb{C}})=\{M\in M_3({\mathbb{C}}):\det M\ne 0\},\qquad \operatorname{SL}_3({\mathbb{C}})=\{M\in M_3({\mathbb{C}}):\det M=1\}. \] For $H\in\{\operatorname{GL}_3({\mathbb{C}}),\operatorname{SL}_3({\mathbb{C}})\}$ and $A,B\in H$, let $\rho_{A,B}:F(a,b)\to H$ be the unique group homomorphism satisfying $\rho_{A,B}(a)=A$ and $\rho_{A,B}(b)=B$. For $w\in F(a,b)$, set $w(A,B)=\rho_{A,B}(w)$; thus inverse letters are evaluated using matrix inverses. For $M=(m_{ij})\in M_3({\mathbb{C}})$, let $\operatorname{tr}(M)=\sum_{i=1}^{3}m_{ii}$. Two elements $u,v\in F(a,b)$ are conjugate if $v=gug^{-1}$ for some $g\in F(a,b)$.

1. (a) Do there exist $u,v\in F(a,b)$ such that \[ \left(\forall g\in F(a,b),\ v\ne gug^{-1}\right) \quad\text{and}\quad \left(\forall A,B\in\operatorname{GL}_3({\mathbb{C}}),\ \operatorname{tr}(u(A,B))=\operatorname{tr}(v(A,B))\right)? \]
2. (b) Do there exist $u,v\in F(a,b)$ such that \[ \left(\forall g\in F(a,b),\ v\ne gug^{-1}\right) \quad\text{and}\quad \left(\forall A,B\in\operatorname{SL}_3({\mathbb{C}}),\ \operatorname{tr}(u(A,B))=\operatorname{tr}(v(A,B))\right)? \]

The existence questions in (a) and (b) are posed separately; their witnesses are not required to coincide. The supplied record imposes no additional restrictions on the words or matrix pairs. Its designation of the problem as open is supplied source metadata, not an independently verified mathematical or literature assessment.

Short English statement

Can two nonconjugate elements of the free group on two generators have equal matrix traces under every substitution of invertible complex $3\times3$ matrices? Ask the same question when both substituted matrices must have determinant $1$.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701472$, source code catalogue entry, title Kourovka Notebook Problem 18.43. The record labels the problem open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 18.43, 18th issue (2014); the supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. This bibliographic attribution has not been independently checked.
• Proposed by I. Kapovich, with proposed year 2014 according to the supplied record.