Some remarks concerning the rank of mapping tori and ascending HNN-extensions of abelian groups

Abstract

Let $A$ be a matrix in GL${}_d(Z)$ of infinite order. In a recent paper, G. Levitt and V. Metaftsis prove that for any sufficiently large integer $n$ the matrix $A^n$ is not conjugated to a companion matrix. We first prove a local version of this theorem. Then, we give an effective statement, using diophantine methods. We also discuss several related problems.