Isomorphism Conjecture for homotopy K-theory and groups acting on trees

Abstract

We discuss an analogon to the Farrell–Jones Conjecture for homotopy algebraic K-theory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result can be used to get rational injectivity results for the assembly map in the Farrell–Jones Conjecture in algebraic K-theory.