The projective class group of the fundamental group of a surface is trivial

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 and $T$ an infinite cyclic group. Let $D{ \times _\alpha }T$ be the semidirect product of $D$ and $T$ with respect to $\alpha$. We prove that the Whitehead group of $D{ \times _\alpha }T$ and the projective class group of the integral group ring $Z(D{ \times _\alpha }T)$ are trivial. The second result implies that the projective class group of the integral group ring over the fundamental group of a surface is trivial.