Splitting formulas for certain Waldhausen Nil-groups

For a group G that splits as an amalgamation of A and B over a common subgroup C, there is an associated Waldhausen Nil-group, measuring the “failure” of Mayer-Vietoris for algebraic K-theory. Assume that (1) the amalgamation is acylindrical, and (2) the groups A,B,G satisfy the Farrell-Jones isomorphism conjecture. Then we show that the Waldhausen Nil-group splits as a direct sum of Nil-groups associated to certain (explicitly describable) infinite virtually cyclic subgroups of G. We note that a special case of an acylindrical amalgamation includes any amalgamation over a finite group C.

💡 Research Summary

The paper investigates the structure of Waldhausen Nil‑groups that arise when a group G is expressed as an amalgamated free product G = A *_{C} B under two principal hypotheses. First, the amalgamation is required to be acylindrical, meaning that the associated Bass‑Serre tree admits only finite edge stabilizers and that any non‑trivial element fixes a set of points of uniformly bounded diameter. This condition ensures that the action of G on the tree is “thin” enough to allow controlled topological techniques. Second, the groups A, B, and the whole amalgam G are assumed to satisfy the Farrell‑Jones conjecture (FJC) in algebraic K‑theory, which guarantees that the assembly map from the equivariant homology of the classifying space for the family of virtually cyclic subgroups to the K‑theory of the group ring is an isomorphism.

Under these assumptions the authors prove a decomposition theorem for the Waldhausen Nil‑group N_{W}(R