15.84 (2002)

Yu. I. Merzlyakov (Sov. Math. Dokl., 19 (1978), 64–68) proved that if the complex numbers $\alpha, \beta, \gamma$ are each at least 3 in absolute value, then the matrices $\begin{pmatrix} 1 & \alpha \\ 0 & 1 \end{pmatrix}$, $\begin{pmatrix} 1 & 0 \\ \beta & 1 \end{pmatrix}$, and $\begin{pmatrix} 1-\gamma & -\gamma \\ \gamma & 1+\gamma \end{pmatrix}$ generate a free group of rank three. Are there rational numbers $\alpha, \beta, \gamma$, each less than 3 in absolute value, with the same property?