A remark on the Isomorphism Conjectures

TL;DR

This paper demonstrates that for certain classes of groups, the validity of the Isomorphism Conjecture for acyclic groups implies the injectivity or bijectivity of the assembly map for those classes.

Contribution

It establishes a link between the Isomorphism Conjecture's truth for acyclic groups and the assembly map properties across various natural group classes.

Findings

Injectivity or bijectivity of the assembly map follows from the Isomorphism Conjecture for acyclic groups.

The results apply to various natural classes of groups and K- and L-theoretic functors.

Provides a new approach to verify the Isomorphism Conjecture via acyclic groups.

Abstract

We show that for various natural classes of groups and appropriately defined K- and L-theoretic functors, injectivity or bijectivity of the assembly map follows from the Isomorphism Conjecture being true for acyclic groups lying within that class.