Nielsen equivalence in mapping tori over the torus

Abstract:

We use the geometry of the Farey graph to give an alternative proof of the fact that if $A \in GL_2\mathbb {Z}$ and if $G_A=\mathbb {Z}^2 \rtimes _A \mathbb {Z}$ is generated by two elements, then 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 which case there are two.