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44 A. Bartels, S. Echterhoff, and W. Lück1.5 Colimits of hyperbolic groups. In [23, Section 7] Higson, Lafforgue and Skandalisconstruct counterexamples to the Baum–Connes conjecture with coefficients, actuallywith a commutative C -algebra as coefficients. They formulate precise propertiesfor a group G which imply that it does not satisfy the Baum–Connes conjecture withcoefficients. Gromov [20] describes the construction of such a group G as a colimitover a directed system of groups fG i j i 2 I g, where each G i is hyperbolic.This construction did raise the hope that these groups G may also be counterexamplesto the Baum–Connes conjecture with trivial coefficients. But – to the authors’knowledge – this has not been proved and no counterexample to the Baum–Connesconjecture with trivial coefficients is known.Of course one may wonder whether such counterexamples to the Baum–Connesconjecture with coefficients or with trivial coefficients respectively may also be counterexamplesto the Farrell–Jones conjecture or the Bost conjecture with coefficients orwith trivial coefficients respectively. However, this can be excluded by the followingresult.Theorem 1.9. Let G be the colimit of the directed system fG i j i 2 I g of groups (withnot necessarily injective structure maps). Suppose that each G i is hyperbolic. LetK G be a subgroup. Then:(i) The group K satisfies for every ring R on which K acts by ring automorphismsthe Farrell–Jones conjecture for algebraic K-theory with coefficients in R, i.e.,the assembly map (1.1) is bijective for all n 2 Z.(ii) The group K satisfies for every ring R on which K acts by ring automorphismsthe Farrell–Jones conjecture for homotopy K-theory with coefficients in R, i.e.,the assembly map (1.2) is bijective for all n 2 Z.(iii) The group K satisfies for every C -algebra A on which K acts by C -automorphismsthe Bost conjecture with coefficients in A, i.e., the assembly map (1.4)is bijective for all n 2 Z.Proof. If G is the colimit of the directed system fG i j i 2 I g, then the subgroup K Gis the colimit of the directed system fi 1 .K/ j i 2 I g, where i W G i ! G is thestructure map. Hence it suffices to prove Theorem 1.9 in the case G D K. This casefollows from Theorem 1.8 (i) as soon as one can show that the Farrell–Jones conjecturefor algebraic K-theory, the Farrell–Jones conjecture for homotopy K-theory, or theBost conjecture respectively holds for every subgroup H of a hyperbolic group G witharbitrary coefficients R and A respectively.Firstly we prove this for the Bost conjecture. Mineyev and Yu [30, Theorem 17]show that every hyperbolic group G admits a G-invariant metric d O which is weaklygeodesic and strongly bolic. Since every subgroup H of G clearly acts properly on Gwith respect to any discrete metric, it follows that H belongs to the class C 0 as describedby Lafforgue in [27, page 5] (see also the remarks at the top of page 6 in [27]). Nowthe claim is a direct consequence of [27, Theorem 0.0.2].