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18.43 (2014)

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a) Do there exist elements $u, v$ of a 2-generator free group $F(a, b)$ such that $u, v$ are not conjugate in $F(a, b)$ but for any matrices $A, B \in \text{GL}(3, \mathbb{C})$ we have $\text{trace}(u(A, B)) = \text{trace}(v(A, B))$?

b) The same question if we replace $\text{GL}(3, \mathbb{C})$ by $\text{SL}(3, \mathbb{C})$.

Contributor: I. Kapovich

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