Connectivity of group C*-algebras

Many topological invariants, like K-theory and K-homology, can be generalised to functors on non-commutative C*-algebras. It turns out that these invariants are not only natural with respect to *-homomorphisms, but also with respect to the much less rigid asymptotic morphisms.
The E-theory group of two C*-algebras is constructed from asymptotic morphisms between the suspensions of the two algebras. The use of suspensions in the definition is necessary to obtain a group structure. However, asymptotic morphisms between the unsuspended algebras contain a priori more geometric information. This motivates the question: When is it possible to "unsuspend" in E-theory?
For nuclear C*-algebras the suspension is obsolete precisely if the source algebra satisfies a homotopy invariant property called connectivity. Surprisingly, it has some interesting implications for group algebras: If the kernel of the trivial representation in C*(G) is connective then - as a direct consequence of connectivity - G satisfies the Kadison-Kaplansky conjecture and C*(G) is quasidiagonal.

It is known that torsion free nilpotent groups and crystallographic groups with cyclic holonomy fit into this picture. There are also examples of groups, which are not connective. The goal of the project is to identify more examples and counterexample among the group C*-algebras of discrete torsion free amenable groups.