Whitehead groups of certain semidirect products of free groups

Abstract:

Let $D = {F_1} \times {F_2} \times \cdots \times {F_n}$ be a direct product of $n$ free groups ${F_1},{F_2}, \cdots ,{F_n},\alpha$ an automorphism of $D$ which leaves all but one of the noncyclic factors in $D$ pointwise fixed, $T$ an infinite cyclic group and $F$ another free group. Let $D{ \times _\alpha }T$ be the semidirect product of $D$ and $T$ with respect to $\alpha$ and $(D{ \times _\alpha }T){ \times _{\alpha \times \text {id}T}}F$ the semidirect product of $D{ \times _\alpha }T$ and $F$ with respect to the automorphism $\alpha \times idT$ of $D{ \times _\alpha }T$ induced by $\alpha$. We prove that the Whitehead group of $(D{ \times _\alpha }T){ \times _{\alpha \times idT}}F$ and the projective class group of the integral group ring $Z((D{ \times _\alpha }T){ \times _{\alpha \times idT}}F)$ are trivial. These results extend that of [3].