Given a ring \(R\) and a discrete group \(G\), the Farrell-Jones Conjecture tells how the algebraic \(K\)- and \(L\)-theory of the group ring \(R[G]\) may be expressed in terms of the algebraic \(K\)-theory of \(R\) and the group theory of \(G\). The Baum-Connes conjecture gives a relation between the topological \(K\)-theory of the reduced \(C^*\)-algebra of a group and the \(K\)-homology of the classifying space of proper actions of that group.
The authors of the paper under review consider the isomorphism conjectures (Farrell-Jones and Baum-Connes conjectures). They prove the fact that isomorphism conjectures hold for any torsion-free acyclic group implies that the assembly maps are injective for any torsion-free group. As an interesting consequence they obtain that the isomorphism conjectures hold for any torsion-free group if and only if the assembly maps are surjective for any torsion-free group.