Deciding whether a mapping torus is of full rank

Abstract

The mapping torus induced by an automorphism $\varphi$ of the free abelian group ${\mathbb{Z}}^n$ is a semi-direct product $G={\mathbb{Z}}^n{⋊}_{\varphi }\mathbb{Z}$. We show that whether the rank of $G$ is equal to $n+1$ is decidable. As a corollary, the rank of ${\mathbb{Z}}^3{⋊}_{\varphi }\mathbb{Z}$ is decidable.