The topological K-theory of certain crystallographic groups

TL;DR

This paper computes the topological K-theory of certain crystallographic groups' group C*-algebras and verifies the Gromov-Lawson-Rosenberg Conjecture for these groups, involving analysis of classifying spaces and homology.

Contribution

It provides explicit calculations of K-theory for a class of crystallographic groups and confirms the conjecture in this context, extending understanding of their topological properties.

Findings

Computed topological K-theory of real and complex group C*-algebras for Gamma.

Verified the unstable Gromov-Lawson-Rosenberg Conjecture for these groups.

Analyzed the (co-)homology and K-theory of classifying spaces BGamma and ar{B}Gamma.

Abstract

Let Gamma be a semidirect product of the form Z^n rtimes Z/p where p is prime and the Z/p-action 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 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 underbar{B}Gamma. The latter is the quotient of the induced Z/p-action on the torus T^n.