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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Abstract

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. 1/2, pp. 189–203

DOI 10.4171/LEM/58-1-9