# The topological K-theory of certain crystallographic groups

### James F. Davis

Indiana University, Bloomington, USA### Wolfgang Lück

University of Bonn, Germany

## Abstract

Let $Γ$ be a semidirect product of the form $Z_{n}⋊_{ρ}Z/p$ where $p$ is prime and the $Z/p$-action $ρ$ on $Z_{n}$ is free away from the origin. We will compute the topological K-theory of the real and complex group C*-algebra of $Γ$ and show that $Γ$ 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 $B Γ$. The latter is the quotient of the induced $Z/p$-action on the torus $T_{n}$.

## Cite this article

James F. Davis, Wolfgang Lück, The topological K-theory of certain crystallographic groups. J. Noncommut. Geom. 7 (2013), no. 2, pp. 373–431

DOI 10.4171/JNCG/121