The topological K-theory of certain crystallographic groups

Abstract

Let $\Gamma$ be a semidirect product of the form $\mathbb{Z}^n \rtimes_{\rho} \mathbb{Z}/p$ 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 $B Gamma$ and $\underline{B} \Gamma$. The latter is the quotient of the induced $\mathbb{Z}/p$-action on the torus $T^n$.