Fundamental isomorphism conjecture via noncommutative motives

Given a group, we construct a “fundamental localizing invariant” on its orbit category. We prove that any isomorphism conjecture valid for this fundamental invariant implies the same isomorphism conjecture for all localizing invariants, like non-connectiveK-theory, Hochschild homology, cyclic homology, and so on. Then, we discuss how to reduce such a fundamental isomorphism conjecture to essentiallyK-theoretic ones. Finally, we develop the analogue additive results.