K-theoretic Farrell–Jones isomorphism conjecture
Canonical statement
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For every discrete group \(G\), every associative unital ring \(R\), and every integer \(n\), the assembly map \[ H^{G}_{n}\!\left(E_{\mathrm{VCyc}}G;\mathbf K_R\right) \longrightarrow K_n(RG) \] is an isomorphism. Here \(E_{\mathrm{VCyc}}G\) is the terminal \(G\)-CW complex whose \(H\)-fixed points are contractible for virtually cyclic \(H\le G\) and empty otherwise, and \(H^G_*(-;\mathbf K_R)\) is the equivariant homology theory whose value at \(G/H\) is \(K_*(RH)\).Notes
The Farrell–Jones conjecture in algebraic -theory predicts that for every discrete group , every associative unital ring , and every integer , the assembly map is an isomorphism, where is the classifying space of for the family of virtually cyclic subgroups. Informally, the -theory of a group ring should be assembled from the -theory of group rings of the virtually cyclic subgroups. The conjecture was formulated by Farrell and Jones in 1993 in their work relating -theory to dynamics [FarrellJones1993KTheoryDynamics].
The conjecture has strong consequences, including vanishing results for Whitehead groups and, together with its -theoretic companion, the Borel rigidity conjecture. It has been verified for large classes of groups, among them hyperbolic groups, finite-dimensional groups, and virtually solvable groups [BartelsLueckReich2008FJApplications] [Lueck2025FJSurvey]. The present formulation fixes ring coefficients; -theoretic and additive-category versions carry further coefficient and decoration data.
No counterexample is known and the class of verified groups continues to grow, but the assertion for an arbitrary discrete group remains open.
References (3)
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