Farrell–Jones conjecture for totally disconnected groups

Establish that for every totally disconnected topological group G, the Farrell–Jones assembly map K(E_Cop) → K(G) is an equivalence of spectra, where E_Cop is the smooth G-space classifying compact open subgroups (the universal G-space for the family of compact open subgroups), and K(−) denotes the non-connective K-theory spectrum associated to the derived category of smooth representations.

Background

The paper builds a functor K from the category of transitive G-sets with open isotropy to spectra, and then left Kan extends it to smooth G-spaces, producing a canonical map K(E_Cop) → K(G). Here E_Cop is modeled by the extended Bruhat–Tits building with compact open isotropy, and the map encapsulates the idea of assembling K-theory of G from the K-theory of its compact open subgroups.

The authors recall the classical Farrell–Jones conjecture in this totally disconnected setting. While they prove related assembly results for rank-one reductive p-adic groups and discuss known proofs for reductive algebraic groups over uniformly regular rings, the conjecture in the stated generality (for all totally disconnected groups) is not resolved and remains the central open assertion.