JournalslemVol. 58, No. 1/2pp. 189–203

Rank of mapping tori and companion matrices

  • Gilbert Levitt

    Université de Caen Basse-Normandie, Caen, France
  • Vassilis Metaftsis

    University of the Aegean, Karlovassi, Greece
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Given an element φGL(d,Z)\varphi\in \mathrm {GL}(d,\mathbb Z), consider the mapping torus defined as the semidirect product G=ZdφZG=\mathbb Z^d\rtimes_\varphi\mathbb Z. We show that one can decide whether GG has rank 22 or not (i.e.\ whether GG may be generated by two elements). When GG is 2-generated, one may classify generating pairs up to Nielsen equivalence. If φ\varphi has infinite order, we show that the rank of ZdφnZ\mathbb Z^d\rtimes_{\varphi^n}\mathbb Z is at least 3 for all nn large enough; equivalently, φn\varphi^n is not conjugate to a companion matrix in GL(d,Z)(d,\mathbb Z) if nn is large.

Cite this article

Gilbert Levitt, Vassilis Metaftsis, Rank of mapping tori and companion matrices. Enseign. Math. 58 (2012), no. 1, pp. 189–203

DOI 10.4171/LEM/58-1-9