Nielsen equivalence in mapping tori over the torus

TL;DR

This paper provides a new proof using Farey graph geometry that characterizes when certain mapping tori over the torus have a unique Nielsen equivalence class of generating sets, with specific exceptions.

Contribution

It offers an alternative geometric proof for the classification of Nielsen equivalence classes in these groups, highlighting the role of the Farey graph.

Findings

Unique Nielsen class for most matrices A

Two Nielsen classes when A is conjugate to ±[[2,1],[1,1]]

Geometric approach via Farey graph

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 $G_A=\mathbb Z^2 \rtimes_A \mathbb Z$ 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 $\pm \left(\begin {smallmatrix} 2 & 1 \\ 1 & 1 \end {smallmatrix}\right )$, in which case there are two.