The topological K-theory of crystallographic groups with holonomy $\mathbb{Z}/2$

TL;DR

This paper computes the topological K-theory of the reduced C*-algebra for certain crystallographic groups with holonomy $\

Contribution

It provides a complete calculation of the topological K-theory for semidirect products of $\

Findings

Explicit K-theory computations for the groups considered.

Extension of previous results to more general conjugacy actions.

Application of Rosenberg, Davis, and Luck's results to new group classes.

Abstract

In this note we present a complete computation of the topological K-theory of the reduced C*-algebra of a semidirect product of the form $\Gamma=\mathbb{Z}^n\rtimes_\rho\mathbb{Z}/2$ with no further assumptions about of the conjugacy action $\rho$. For this, we use some results for $\mathbb{Z}/2$-equivariant K-theory proved by Rosenberg and previous results of Davis and Luck when the conjugacy action $\rho$ is free outside the origin.