Nuclear Dimension of Twisted $C^{\ast }$-Algebras of Virtually Abelian Groups

Let $G$ be a finitely generated virtually abelian group and $[\sigma ]\in H^2(G;\mathbb{T})$ such that $\sigma (x,y)$ is always a root of unity. We show that the nuclear dimension of the twisted group $C^{\ast }$-algebra $C^{\ast }(G,\sigma )$ is equal to the rank of a finite index abelian subgroup of $G$. We also show that $di{m}_{\text{nuc}}(C^{\ast }({\mathbb{Z}}^r,\sigma ))=r$ if and only if $\sigma$ is type I.