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K-theoretic Farrell–Jones Conjecture for Group Rings

Establish that for every discrete group G and every (unital) ring R, the assembly map μ: H^G_*(E_{VCyc} G; K_R) → K_*(R[G]) from equivariant homology of the classifying space for the family of virtually cyclic subgroups to the algebraic K-theory of the group ring R[G] is an isomorphism.

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Background

The Farrell–Jones conjecture in algebraic K-theory asserts that the algebraic K-theory of a group ring R[G] can be computed via an assembly map from equivariant homology of the classifying space for virtually cyclic subgroups. It is a central open problem with far-reaching implications, as it entails several other famous conjectures such as the Bass conjectures and Kaplansky’s idempotent conjecture.

The conjecture has been verified for many classes of groups (e.g., hyperbolic groups, CAT(0)-groups, mapping class groups), but remains open in general. In this paper the authors use the conjecture as an assumption to develop K-theory computations for crossed products and wreath products.

References

The $K$-theoretic Farrell--Jones conjecture predicts that for any discrete group $G$ and any ring $R$, the assembly map \begin{equation}\label{eq:FJ} \mu\colon HG_*(\mathcal E_{\mathrm{VCyc}G,\mathbb K_R)\to K_*(R[G]) \end{equation} is an isomorphism.

eq:FJ:

μ ⁣:H∗G(EVCycG,KR)→K∗(R[G])\mu\colon H^G_*(\mathcal E_{\mathrm{VCyc}}G,\mathbb K_R)\to K_*(R[G])

— Bernoulli shifts on additive categories and algebraic $K$-theory of wreath products (2401.14806 - Kranz et al., 26 Jan 2024) in Section 1 (Introduction), Equation (1), label eq:FJ