Talk

A linear algebraic group G over a field k is called a Cayley group if
it admits a Cayley map, i.e., a G-equivariant birational isomorphism
over k between the group variety G and the Lie algebra Lie(G). A
Cayley map can be thought of as a partial algebraic analogue of the
exponential map. A prototypical example is the classical "Cayley
transform" for the special orthogonal group SO_n defined by Arthur
Cayley in 1846. A k-group G is called stably Cayley if the product of
G with a split r-dimensional k-torus is Cayley for some r=0,1,2,....

• Prove the best version(s) of the homotopy hypothesis

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.