In joint work with Caleb Eckhardt and Paul McKenney, we show that the C*-algebras of discrete, finitely generated, virtually nilpotent groups G are strongly quasidiagonal and have finite decomposition rank. Thus, the only remaining step required to show that primitive quotients of such virtually nilpotent groups G are classified by their Elliott invariant is to check that these C*-algebras satisfy the UCT.

Our proof of finite decomposition rank relies on a careful analysis of the relationship between primitive ideals of C*(G) and those of C*(N), where N is a finite-index normal subgroup of G. In the case when N is also nilpotent, we obtain a decomposition of C*(G) as a continuous field of twisted crossed products, which enables us to prove finite decomposition rank of C*(G) by analyzing the decomposition rank of the fibers.