The notion of matrix stability of discrete groups with respect to the operator norm is naturally strengthened to weak stability and C*-stability via existing notions of semiprojectivity applied to the group C*-algebras.In ongoing joint work with Dadarlat and Shulman, we strive to characterize when these three properties hold for virtually Abelian groups. I will present the full solution of the Abelian case, as well as partial results for non-Abelian virtually Abelian groups which are suggestive of how such a characterization would pan out.
