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, pp. 189–203DOI 10.4171/LEM/58-1-9