Rank of mapping tori and companion matrices

  • Gilbert Levitt

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

    University of the Aegean, Karlovassi, Greece


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