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# L’Enseignement Mathématique

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**Volume 58, Issue 1/2, 2012, pp. 189–203**

**DOI: 10.4171/LEM/58-1-9**

Published online: 2012-06-30

Rank of mapping tori and companion matrices

Gilbert Levitt^{[1]}and Vassilis Metaftsis

^{[2]}(1) Université de Caen Basse-Normandie, Caen, France

(2) University of the Aegean, Karlovassi, Greece

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.

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Levitt Gilbert, Metaftsis Vassilis: Rank of mapping tori and companion matrices. *Enseign. Math.* 58 (2012), 189-203. doi: 10.4171/LEM/58-1-9