Nuclear Dimension and Rigidity Results for Virtually Abelian Groups

Let $G$ be a finitely generated virtually abelian group. We show that the Hirsch length, $h(G)$, is equal to the nuclear dimension of its group $C^{\ast }$-algebra, ${\dim }_{nuc}(C^{\ast }(G))$. We then specialize our attention to a generalization of crystallographic groups dubbed $\mathit{\text{crystal-like}}$. We demonstrate that in this scenario a $\mathit{\text{point}}\mathit{\text{group}}$ is well defined and the order of this point group is preserved by $C^{\ast }$-isomorphism. We close by using these tools to demonstrate that crystallographic (as a group property) is preserved by $C^{\ast }$-isomorphism. These three tools combine to prove that $2D$ crystallographic groups are $C^{\ast }$-superrigid.