Relating the Farrell Nil-groups to the Waldhausen Nil-groups

Every virtually cyclic group Γ that surjects onto the infinite dihedral group D∞ contains an index two subgroup Π of the form H ⋊α Z. We show that the Waldhausen Nil-group of Γ vanishes if and only if the Farrell Nil-group of Π vanishes. 1. Statement of results. The Bass Nil-groups, Farrell Nil-groups, and Waldhausen Nil-groups appear respectively as pieces in the computation of the algebraicK-theory of direct products, semi-direct products, and amalgamations. While the Bass Nil-groups have been extensively studied, much less is known for both the Farrell Nil-groups and the Waldhausen Nil-groups. For the purposes of computing the algebraic K-theory of infinite groups, the Nil-groups of virtually cyclic groups yield obstructions to certain assembly maps being isomorphisms. In particular, the vanishing/non-vanishing of Nil-groups is of crucial importance for computational aspects of algebraicK-theory. In this short note we prove the following result: Main Theorem. Let Γ be a virtually cyclic group that surjects onto the infinite dihedral group D∞, and Γ = G1 ∗HG2 be the corresponding splitting of groups (with H of index two in both G1 and G2). Let Π = H⋊αZ ≤ Γ be the canonical subgroup of Γ of index two, obtained by taking the pre-image of the canonical index two Z subgroup of D∞. Then for i = 0, 1, the following two statements are equivalent: (A) The Waldhausen Nil-group NKi(ZH ;Z[G1 −H ],Z[G2 −H ]) for the group Γ = G1 ∗H G2 vanishes. (B) The Farrell Nil-group NKi(ZH,α) for the group Π = H ⋊α Z vanishes. The proof of our Main Theorem will be completed in Section 2, with some concluding remarks in Section 3. Next, let us recall that the Farrell-Jones Isomorphism Conjecture for a finitely generated group Γ states that the assembly map: H n (EVC(Γ);KZ ) −→ H n (EALL(Γ);KZ ) = Kn(ZΓ) is an isomorphism. The term on the left is the generalized equivariant homology theory of the space EVC(Γ) with coefficients in the integral K-theory spectrum, where the space EVC(Γ) is a classifying space for Γ-actions with isotropy in the family VC of virtually cyclic subgroups. The term on the right gives the algebraic K-theory of the integral group ring of Γ. Explicit models for the classifying space EVC(Γ) are known for few classes of groups: virtually cyclic groups (take EVC(Γ) to be a point with trivial action), crystallographic groups (by work of Alves and Ontaneda [AO06]), hyperbolic groups