Co-compact discrete group actions and the assembly map

A discrete group $\Gamma$ can act freely and properly on $S^n \times  R^m$, for some
$n, m >0$ if and only if $\Gamma$ is a countable group with periodic Farrell cohomology:
Connolly-Prassidis (1989) assuming $vcd(\Gamma)$ finite, Adem-Smith (2001).
For free  co-compact actions there are additional restrictions, but no general  sufficient
conditions are known. The talk will survey this problem and its connection to the
Farrell-Jones assembly map in K-theory and L-theory.