We use the geometry of the Farey graph to give an alternative proof of thefact that if $A \in GL_2\mathbb Z$ and $G_A=\mathbb Z^2 \rtimes_A \mathbb Z$ isgenerated by two elements, there is a single Nielsen equivalence class of$2$-element generating sets for $G_A$ unless $A$ is conjugate to $\pm\left(\begin {smallmatrix} 2 & 1 \\ 1 & 1 \end {smallmatrix}\right )$, in whichcase there are two.
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机译：我们使用Farey图的几何来提供事实的另一种证明，如果GL_2 \ mathbb Z $和$ G_A = \ mathbb Z ^ 2 \ rtimes_A \ mathbb Z $由两个元素生成，则只有一个除非$ A $与$ \ pm \ left（\ begin {smallmatrix} 2＆1 \\ 1＆1 \ end {smallmatrix} \ right）$共轭，否则$ G_A $的$ 2 $元生成集的Nielsen等价类。在这种情况下，有两个。
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