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.
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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.