process

58 A. Bartels, S. Echterhoff, and W. LückExplicitly K i Df.k; g i / 2 K G i j .k/ D i .g i /g. Let x i;j W K i ! K j be themap induced by i;j W G i ! G j ,id K and id G and the pullback property. One easilychecks by inspecting the standard model for the colimit over a directed set that weobtain a directed system x i;j W K i ! K j of groups indexed by the directed set I andŠthe system of maps x i W K i ! K yields an isomorphism colim i2I K i ! K. Thefollowing diagram commutes:colim i2I H K inS i E F .G/ t K n . E F .G//ŠHn K. E F .G//colim i2I H K in .fg/tn K.fg/Š HKn .fg/,where the vertical arrows are induced by the obvious projections onto fg and thehorizontal maps are the isomorphisms from Lemma 3.4. Notice that S i E F .G/ is amodel for E Si F .K i/ D E ii F .K i /. Hence each map H K inS i E F .G/ !H K in .fg/ is bijective since .G i ;ı i / satisfies the fibered isomorphism conjecture fori F and hence .K i; ı i ı i / satisfies the isomorphism conjecture for i i F .This implies that the left vertical arrow is bijective. Hence the right vertical arrow isan isomorphism. Since E F .G/ is a model for E F .K/, this means that .K; ı/ satisfies the isomorphism conjecture for F . Since W K ! G is any grouphomomorphism, .G; / satisfies the fibered isomorphism conjecture for F .The proof of the following results are analogous to the one in [3, Lemma 1.6] and [4,Lemma 1.2], where only the case Df1g is treated.Lemma 5.3. Let .G; / be a group over and let F G be families of subgroupsof G. Suppose that .G; / satisfies the fibered isomorphism conjecture for the family F .Then .G; / satisfies the fibered isomorphism conjecture for the family G .Lemma 5.4. Let .G; / be a group over . Let W K ! G be a group homomorphismand let F be a family of subgroups of G. If .G; / satisfies the fibered isomorphismconjecture for the family F , then .K; ı/satisfies the fibered isomorphism conjecturefor the family F .For the remainder of this section fix a class of groups C closed under isomorphisms,taking subgroups and taking quotients, e.g., the families F in or VCyc.Lemma 5.5. Let .G; / be a group over . Suppose that the fibered isomorphismconjecture holds for .G; / and C.G/. Let H G be a subgroup.Then the fibered isomorphism conjecture holds for .H; j H / and C.H /.Proof. This follows from Lemma 5.4 applied to the inclusion H ! G since C.H / DC.G/j H .Theorem 5.6. Let .G; / be a group over .