On the vanishing of the lower K-theory of the holomorph of a free group on two generators (Q2905098)

Let \(F_2\) be the free product on two generators. The holomorph group, \(\mathrm{Hol}(F_2)\), of \(F_2\) is the universal split extension of \(F_2\) by \(\mathrm{Aut}(F_2)\), hence it fits in an extension \(1\to F_2\to \mathrm{Hol}(F_2) \to \mathrm{Aut}(F_2)\to 1\). The authors prove in this paper that the fibered isomorphism conjecture for the pseudoisotopy functor is valid for the group \(\mathrm{Hol}(F_2)\). They also classify up to isomorphism the virtually cyclic subgroups that appear as subgroups of \(\mathrm{Hol}(F_2)\). The list of such groups is very limited, for example, the only infinite virtually cyclic subgroup of \(\mathrm{Hol}(F_2)\) is isomorphic to \(\mathbb{Z}/2\times \mathbb{Z}\). The main theorem is the following:NEWLINENEWLINE Main Theorem. The group \(\mathrm{Hol}(F_2)\) satisfies the Farrell-Jones fibered isomorphism conjecture for the pseudoisotopy functor. Furthermore, for any subgroup \(\Gamma<\mathrm{Hol}(F_2)\) it follows that \( Wh(\Gamma)=\widetilde{K}_0(\mathbb{Z}[\Gamma])=K_i(\mathbb{Z}[\Gamma])=0\) for \(i\leq -1\).