Return to Revisions

These two matrices generate a free group: $$ \left( \begin{matrix}{ccc} \frac{1}{3} & \frac{2 \sqrt{2}}{3} & 0 \\ -\frac{2 \sqrt{2}}{3} & \frac{1}{3} & 0 \\ 0 & 0 & 1 \end{matrix} \right), \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \frac{1}{3} & \frac{2 \sqrt{2}}{3} \\ 0 & -\frac{\sqrt{2}}{3} & \frac{1}{3} \end{array} \right). $$ To see that they do, consider a corresponding pair of matrices over the field $\mathbb{F}_3(i)$.