Counterexamples to torsion-freeness in the $K$-theory of crystallographic group $C^*$-algebras

Abstract

We give a negative answer to a question of Sánchez Saldaña--Velásquez and Lück asking whether the topological K-theory of the reduced group C*-algebra of a split crystallographic group with finite cyclic holonomy is always torsion-free. For two explicit split crystallographic groups with translation subgroup of rank four and holonomy of order four, we compute the equivariant K-theory of the classifying space for proper actions and find nontrivial two-torsion. The equivariant universal coefficient theorem transfers this torsion to proper equivariant K-homology, and the Baum--Connes conjecture identifies the resulting groups with the topological K-theory of the corresponding reduced group C*-algebras.