Rank of mapping tori and companion matrices

Abstract

Given an element $\phi \in \mathrm{GL}(d,\mathbb{Z})$, consider the mapping torus defined as the semidirect product $G={\mathbb{Z}}^d{⋊}_{\phi }\mathbb{Z}$. We show that one can decide whether $G$ has rank $2$ or not (i.e.\ whether $G$ may be generated by two elements). When $G$ is 2-generated, one may classify generating pairs up to Nielsen equivalence. If $\phi$ has infinite order, we show that the rank of ${\mathbb{Z}}^d{⋊}_{{\phi }^n}\mathbb{Z}$ is at least 3 for all $n$ large enough; equivalently, ${\phi }^n$ is not conjugate to a companion matrix in GL$(d,\mathbb{Z})$ if $n$ is large.