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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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


Author: Koo Guan Choo
Journal: Proc. Amer. Math. Soc. 40 (1973), 42-46
MSC: Primary 18G99; Secondary 55A05
MathSciNet review: 0323869
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0323869-1
Keywords: Projective class group, Whitehead group, fundamental group, direct product of free groups
Article copyright: © Copyright 1973 American Mathematical Society