TOPOLOGICAL K-THEORY OF THE GROUP C-ALGEBRA OF A SEMI-DIRECT PRODUCT Z ⋊ Z/m FOR A FREE CONJUGATION ACTION

We compute the topological K-theory of the group C-algebra C r (Γ) for a group extension 1 → Z → Γ → Z/m → 1 provided that the conjugation action of Z/m on Z is free outside the origin. Introduction Throughout this paper let 1 → Z → Γ → Z/m → 1 be a group extension such that conjugation action of Z/m on Z is free outside the origin. Our main goal is to compute the topological K-theory of the group C-algebra C r (Γ). This generalizes results of Davis-Lück [7], where m was assumed to be a prime. Except ideas from that paper, the proof of a Conjecture due to Adem-Ge-Pan-Petrosyan in Langer-Lück [11] is a key ingredient. The calculation and its result are surprisingly complicated. It will play an important role in a forthcoming paper by Li-Lück [12]. There the computation of the topological K-theory of a C-algebra associated to the ring of integers in an algebraic number field will be carried out in general, thus generalizing the work of Cuntz and Li [6] who had to assume that +1 and −1 are the only roots of unity. 0.1. Main Result. Our main result is the following theorem. In the sequel C r (G) is the reduced group C-algebra of a group G. We denote by EG the classifying space of proper actions of a group G and by BG its quotient space G\EG. Let Ĥ(G;M) be the Tate cohomology of a group G with coefficients in a ZG-module M . Denote by ΛZ the i-th exterior power. Theorem 0.1 (Computation of the topological K-theory). Consider the extension of groups 1 → Z → Γ → Z/m→ 1 such that the conjugation action of Z/m on Z is free outside the origin 0 ∈ Z. Let M be the set of conjugacy classes of maximal finite subgroups of Γ. (i) We obtain an isomorphism ω1 : K1(C ∗ r (Γ)) ∼= −→ K1(BΓ). Restriction with the inclusion k : Z → Γ induces an isomorphism k : K1(C ∗ r (Γ)) ∼= −→ K1(C ∗ r (Z )). Induction with the inclusion k yields a homomorphism k∗ : Z⊗Z[Z/m] K1(C ∗ r (Z )) → K1(C ∗ r (Γ)). Date: September, 5th, 2011. 2010 Mathematics Subject Classification. 19L47,46L80.