On the vanishing of certain K-theory Nil groups

Let Γ i , i = 0, 1, be two groups containing C p , the cyclic group of prime order p, as a subgroup of index 2. Let Γ = Γ0 * Cp Γ1. We show that the Nil-groups appearing in Waldhausen’s splitting theorem for computing K j (ℤΓ) (j≤ 1) vanish. Thus, in low degrees, the K-theory of ℤΓ can be computed by a Mayer-Vietoris type exact sequence involving the K-theory of the integral group rings of the groups Γ0, Γ1 and C p .