A Note on Strongly Quasidiagonal Groups

In this note we address a question of Don Hadwin: "Which groups have strongly quasidiagonal C*-algebras?" In recent work we showed that all finitely generated virtually nilpotent groups have strongly quasidiagonal C*-algebras, while together with Carri\'on and Dadarlat we showed that most wreath products fail to have strongly quasidiagonal C*-algebras. These two results raised the question of whether or not strong quasidiagonality could characterize virtual nilpotence among finitely generated groups. The purpose of this note is to provide examples of finitely generated groups (in fact of the form $\Z^3\rtimes \Z^2$) that are not virtually nilpotent yet have strongly quasidiagonal C*-algebras. Moreover we show these examples are the "simplest" possible by proving that a group of the form $\Z^d\rtimes \Z$ is virtually nilpotent if and only if its group C*-algebra is strongly quasidiagonal.