On the $K$-theory of groups of the form $\mathbb{Z}^n\rtimes \mathbb{Z}/m$ with $m$ square-free

TL;DR

This paper explicitly computes the topological K-theory groups of certain semidirect product groups involving free abelian groups and square-free order cyclic groups, generalizing previous results without restrictions on the group action.

Contribution

It provides a new explicit computation of K-theory for groups of the form Z^n ⋊ Z/m with m square-free, without imposing conditions on the action, extending prior work.

Findings

Explicit formulas for K-theory groups of Z^n ⋊ Z/m

Generalization to arbitrary actions without restrictions

Extension of previous computations by Lück-Davis and Langer-Lück

Abstract

We provide an explicit computation of the topological $K$-theory groups $K_*(C_r^*(\mathbb{Z}^n\rtimes \mathbb{Z}/m))$ of semidirect products of the form $\mathbb{Z}^n\rtimes \mathbb{Z}Z/m$ with $m$ square-free. We want to highlight the fact that we are not impossing any conditions on the $\Z/m$-action on $\mathbb{Z}^n$. This generalizes previous computations of L\"uck-Davis and Langer-L\"uck.