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The \(K\)-theoretic Farrell-Jones conjecture for hyperbolic groups. (English) Zbl 1143.19003

Let \(G\) be a group and \(R\) a ring with unit. The Farrell-Jones conjecture for \(K\)-theory and for the group ring \(R[G]\) predicts that the algebraic \(K\)-groups \(K_{i}(R[G])\) may be computed from the corresponding \(K\)-groups of group rings of the form \(R[V]\), where \(V\) runs over the family of virtually cyclic subgroups of \(G\) and homological data. The paper under review proves this conjecture in a general setup when the group is hyperbolic.
The main theorem is the following: Let \(G\) be a hyperbolic group. Then \(G\) satisfies the \(K\)-theoretic Farrell-Jones conjecture with coefficients, i.e., if \(\mathcal{A}\) is an additive category with right \(G\)-action, then for every \(n\in\mathbb{Z}\) the assembly map
\[ H^{G}_{n}(E_{\mathcal{VC}}G;\mathbf{K}_{\mathcal{A}})\to H^{G}_{n}(pt;\mathbf{K}_{\mathcal{A}})\cong K_{n}( \mathcal{A}*_{G}pt) \]
is an isomorphism.
The authors show many applications, by considering different coefficient rings, such as Bass’ conjectures concerning the Hattori-Stallings rank, the Kaplansky conjecture on existence of nontrivial idempotents in group rings, generalizations of Moody’s induction theorems and behaviour of nil groups.

MSC:

19D55 \(K\)-theory and homology; cyclic homology and cohomology
19A31 \(K_0\) of group rings and orders
19B28 \(K_1\) of group rings and orders
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