The topological K-theory of certain crystallographic groups

Abstract

Let $\Gamma$ be a semidirect product of the form ${\mathbb{Z}}^n{⋊}_{\rho }\mathbb{Z}/p$ where $p$ is prime and the $\mathbb{Z}/p$-action $\rho$ on ${\mathbb{Z}}^n$ is free away from the origin. We will compute the topological K-theory of the real and complex group C*-algebra of $\Gamma$ and show that $\Gamma$ satisfies the unstable Gromov–Lawson–Rosenberg Conjecture. On the way we will analyze the (co-)homology and the topological K-theory of the classifying spaces $BGamma$ and $\underline{B}\Gamma$. The latter is the quotient of the induced $\mathbb{Z}/p$-action on the torus $T^n$.