Torsion-freeness of K_*(C^*_r(Z^n ⋊ Z/m)) for arbitrary cyclic orders and actions

Determine whether the topological K-theory groups K_*(C^*_r(Z^n ⋊ Z/m)) are torsion free for every finite cyclic group Z/m when the action of Z/m on Z^n is arbitrary, with no restrictions on the action.

Background

The paper proves that K_(C^_r(Z^n ⋊ Z/m)) is finitely generated and torsion free when m is square-free, without imposing any conditions on the Z/m-action on Z^n. This generalizes prior results of Davis–Lück and Langer–Lück that required the action to be free outside the origin.

Motivated by these results and those in Langer–Lück (2012), the authors ask whether torsion-freeness persists for all finite cyclic groups, not only those of square-free order, and without any restriction on the action. They note a relation to a conjecture of Adem–Ge–Pan–Petrosyan concerning a spectral sequence collapse for cohomology of Z^n ⋊ Z/m, which is known to be false, suggesting potential avenues for counterexamples.