Juemin Lin

Deciding whether a mapping torus is of full rank

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