Rank of mapping tori and companion matrices

Abstract

Given an element $\varphi\in \mathrm {GL}(d,\mathbb Z)$, consider the mapping torus defined as the semidirect product $G=\mathbb Z^d\rtimes_\varphi\mathbb Z$. 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 $\varphi$ has infinite order, we show that the rank of $\mathbb Z^d\rtimes_{\varphi^n}\mathbb Z$ is at least 3 for all $n$ large enough; equivalently, $\varphi^n$ is not conjugate to a companion matrix in GL$(d,\mathbb Z)$ if $n$ is large.