From the classification of Cuntz-Li-$C^*$-algebras to the group cohomology of crystallographic groups

The talk consists of three parts.
1.) Cuntz-Li associated to the ring of algebraic integers of a number
field a $C^*$-algebra.
We will discuss a preprint by Li and Lueck, where  a full classification
of these $C^*$-algebras is presented.
2.) There is a Conjecture due to Adem-Ge-Pan-Petroysan about the
cohomology of certain crystallographic groups
with cyclic holonomy. In papers by Langer and Lueck the conjecture has
been disproved in general, but proved
in the case where the action of the holonomy group is free.
3.) We answer the question what the relation between these two on the
first glance unlinked problems is.
Actually, we needed the positive answer to 2.) in order to handle the
classification appearing in 1.)
This is done by an interesting tour from group cohomology, divided power
algebras, characteristic classes
of Charlap-Vasquez, classifying spaces for proper group actions,
equivariant topological $K$-theory of group $C^*$-algebras,
the Atiyah-Segal Completion Theorem for infinite discrete groups, the
Baum-Connes Conjecture, and the classification theorem of Kirchberg.