The finite-by-cyclic assembly conjecture for algebraic K-theory

Let $G$ be a discrete group and $R$ a ring. Let $\mathcal{\mathrm{fbc}}$ denote the family of finite-by-cyclic subgroups, $\mathcal{\mathrm{all}}$ the family of all subgroups, and $\mathbf{K}_R$ the algebraic K-theory functor. Finite-by-cyclic assembly conjecture. There is an isomorphism

$$H_n^G(E_{\mathcal{\mathrm{fbc}}}G;\mathbf{K}_R)\longrightarrow H_n^G(E_{\mathcal{\mathrm{all}}}G;\mathbf{K}_R)=K_n(R[G]).$$

Moreover, for every epimorphism $\varphi:\Gamma\to G$, there is an isomorphism

$$H^\Gamma_n(E_{\varphi^*\mathcal{\mathrm{fbc}}(G)}\Gamma;\mathbf{K}_R)\longrightarrow H^\Gamma_n(E_{\mathcal{\mathrm{all}}}\Gamma;\mathbf{K}_R)=K_n(R[\Gamma]).$$

The theorem immediately preceding this statement proves the comparison from the finite-by-cyclic family to the virtually cyclic family, so this conjecture is presented as an equivalent finite-by-cyclic formulation of the Farrell–Jones conjecture.