Burghelea conjecture and asymptotic dimension of groups

We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum–Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which together imply the Burghelea conjecture for such groups. We prove both conjectures for many classes of groups. It is known that the Burghelea conjecture does not hold for all groups, although no finitely presentable counterexample was known. We construct a finitely presentable (even type [Formula: see text]) counterexample based on Thompson’s group [Formula: see text]. We construct as well a finitely generated counterexample with finite decomposition complexity.