Topology Seminar: C*-superrigidiity of nilpotent groups

It is a classical problem to recover a discrete group from various rings or algebras associated with it, such as the integral group ring (cf. the Whitehead group and the Whitehead torsion). By analogy, in an operator algebraic framework we want to recover torsion-free groups from certain topological completions of the complex group ring, such as the reduced group C*-algebra. Groups for which this is possible are called C*-superrigid. In recent joint work with Caleb Eckhardt, we could prove C*-superrigidity for arbitrary finitely generated, torsion-free, 2-step nilpotent groups by combining K-theoretic methods with certain bundle decompositions of C*-algebras.

In this talk, I will introduce the relevant notion of reduced group C*-algebras and put it in the context of the more familiar complex group algebra.  Further, I will provide a first motivation to study C*-superrigidity. Then I will discuss our result with Caleb Eckhardt, focusing on methods that have analogies in topology such as bundle decompositions and topological K-theory. The talk will finish with a persepective on relevance of C*-superrigidity in other areas of mathematics.