The Farrell-Jones Conjecture is known for hyperbolic, CAT(0), arithmetic, and many other groups, and it implies a long list of conjectures about group rings and aspherical manifolds.
On the Farrell-Jones conjecture for localising invariants
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abstract
We show the Farrell-Jones conjecture with coefficients in left-exact $\infty$-categories for finitely $\mathcal{F}$-amenable groups and, more generally, Dress-Farrell-Hsiang-Jones groups. Our result subsumes and unifies arguments for the K-theory of additive categories and spherical group rings and extends it for example to categories of perfect modules over $\mathbb{E}_{1}$-ring spectra.
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Survey on the Farrell-Jones Conjecture
The Farrell-Jones Conjecture is known for hyperbolic, CAT(0), arithmetic, and many other groups, and it implies a long list of conjectures about group rings and aspherical manifolds.