Splitting formulas for certain Waldhausen Nil-groups

TL;DR

This paper proves that for acylindrical group amalgamations satisfying the Farrell-Jones conjecture, Waldhausen Nil-groups decompose into sums over specific virtually cyclic subgroups, simplifying their structure.

Contribution

It establishes a splitting formula for Waldhausen Nil-groups in acylindrical amalgamations under the Farrell-Jones conjecture, providing explicit descriptions of the summands.

Findings

Waldhausen Nil-groups split as a direct sum over certain virtually cyclic subgroups.

The splitting applies to acylindrical amalgamations satisfying the Farrell-Jones conjecture.

Special case includes amalgamations over finite groups.

Abstract

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.