On the $K$-theory of crystallographic groups

Abstract:

For any crystallographic group $\Gamma$ we show that the groups ${K_i}(\Gamma )$ are isomorphic, via the forget control map, to the controlled $K$-groups ${K_i}{(\Gamma )_c}$, for all $i \leqslant 1$ and for an appropriate choice of the control map. By using this result and under a mild hypothesis on the crystallographic group $\Gamma$, it is proved that ${K_i}(\Gamma ) = 0$ for all $i \leqslant - 2$ and ${N^j}{K_i}(\Gamma ) = 0$ for all $i \leqslant - 1$ and $j > 0$.