The \(K\)-theoretic Farrell-Jones conjecture for CAT(0)-groups
The isomorphism conjecture stated by Farrell-Jones is the most important conjecture in geometric topology. It states that certain assembly maps are homotopy equivalences. In the paper under review, the isomorphism conjecture for \(K\)-theory is proved for CAT(0)-groups. Actually, a more general conjecture is proved, the isomorphism conjecture with coefficients in an additive category \(\mathcal{A}\). The methods used generalize the ones used for hyperbolic groups ([\textit{A. Bartels, W. Lück} and \textit{H. Reich}, ``The \(K\)-theoretic Farrell-Jones conjecture for hyperbolic groups'', Invent. Math. 172, No. 1, 29--70 (2008; Zbl 1143.19003)] and preprints by the first two authors).NEWLINENEWLINEThe key idea is to show that the homotopy fiber of the assembly map is contractible. In the previous papers that were mentioned, the conjecture was proved for hyperbolic groups and for the lower \(K\)-theory of Cat(0)-groups. The author uses similar methods, with the suitable modifications, to extend the proofs. The central idea is an action of the group on a compact metric space by isometries, imitating the metric properties of the action of a group of isometries on the compactification of a hyperbolic manifold. The conditions that are imposed allow for the proof that the higher homotopy of the homotopy fiber of the assembly map vanishes. The actions that are considered, in this case, are the actions of the group on a large ball in a CAT(0)-space. This action satisfies the axioms of the strong homotopy action. The second important piece in the proof is that CAT(0)-groups are strongly transfer reducible over the family of virtually cyclic subgroups. This property allows for the construction of transfer maps. NEWLINENEWLINENEWLINE NEWLINEA corollary of the methods used in the proof is that the class of groups that satisfy the Farrell-Jones isomorphism conjecture with coefficients is closed in taking finite direct products and finite free products.
scientific article; zbMATH DE number 6024086