Group homology and Connes’ periodicity operator

Given a commutative ring kk, a group GG and an element g∈Gg∈G of infinite order with centralizer C(g)C(g), we study the inverse system⋯⟶H2n(C(g)/〈g〉,k)⟶H2n-2(C(g)/〈g〉,k)⟶⋯⋯⟶H2n(C(g)/〈g〉,k)⟶H2n-2(C(g)/〈g〉,k)⟶⋯arising from Burghelea's computation [D. Burghelea, The cyclic homology of group rings, Comment. Math. Helv. 60 (1985) 354–365] of the cyclic homology of the group algebra kGkG and Connes’ periodicity operator S:HC2n(kG)⟶HC2n-2(kG)S:HC2n(kG)⟶HC2n-2(kG). A vanishing theorem for the limit of this inverse system is proved for groups in the class AA introduced in Emmanouil and Passi [A contribution to Bass’ conjecture, J. Group Theory 7 (2004) 409–420], thereby contributing to a conjecture by Burghelea [The cyclic homology of group rings, Comment. Math. Helv. 60 (1985) 354–365]. The homological condition defining the class AA is closely examined; in particular, it is shown that this class properly contains the class studied in Emmanouil [On a class of groups satisfying Bass’ conjecture, Invent. Math. 132 (1998) 307–330].