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 $A\in G{L}_2\mathbb{Z}$ and $G_A={\mathbb{Z}}^2{⋊}_A\mathbb{Z}$ is generated by two elements, there is a single Nielsen equivalence class of $2$-element generating sets for $G_A$ unless $A$ is conjugate to $\pm \left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$, in which case there are two.

arXiv comment6 pages, 3 figures, v2 includes corrections suggested by the referee