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^*-algebra, nuc(C^*(G)). We then specialize our attention to a generalization of crystallographic groups dubbed crystal-like. We demonstrate that in this scenario a point group is well defined and the order of this point group is preserved by C^*-isomorphism. We close by using these tools to demonstrate that crystallographic (as a group property) is preserved by C^*-isomorphism. These three tools combine to prove that $2D$ crystallographic groups are C^*-superrigid.