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

TL;DR

This paper proves that groups acting on trees satisfy an analog of the Farrell-Jones Conjecture for homotopy K-theory, extending the conjecture's validity to a broader class of groups and enabling rational injectivity results.

Contribution

It establishes that if all isotropy groups of a group acting on a tree satisfy the conjecture, then the entire group does, providing a new method to verify the conjecture for complex groups.

Findings

Groups acting on trees satisfy the homotopy K-theory analog of the Farrell-Jones Conjecture.

The result enables rational injectivity of the assembly map in algebraic K-theory.

The approach reduces the problem to verifying the conjecture for isotropy groups.

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.