Whitehead groups of certain semidirect products of free groups

Author:
Koo Guan Choo

Journal:
Proc. Amer. Math. Soc. **43** (1974), 26-30

MSC:
Primary 18F25

DOI:
https://doi.org/10.1090/S0002-9939-1974-0338124-4

MathSciNet review:
0338124

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Abstract | References | Similar Articles | Additional Information

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**].

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*The projective class group of the fundamental group of a surface is trivial*, Proc. Amer. Math. Soc.**40**(1973), 42–46. MR**323869**, DOI https://doi.org/10.1090/S0002-9939-1973-0323869-1 - F. T. Farrell and W.-C. Hsiang,
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Keywords:
Whitehead group,
projective class group,
semidirect product of free groups,
twisted group ring

Article copyright:
© Copyright 1974
American Mathematical Society