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The Residual Finiteness of Polygonal Products—Two Counterexamples

Published online by Cambridge University Press:  20 November 2018

R. B. J. T. Allenby
R. B. J. T. Allenby*
Affiliation:
School of Mathematics, University of Leeds Leeds, LS2 9JT England
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Abstract

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We show that, even under very favourable hypotheses, a polygonal product of finitely generated torsion free nilpotent groups amalgamating infinite cyclic subgroups is, in general, not residually finite, thus answering negatively a question of C. Y. Tang. A second example shows similar kinds of limitations apply even when the factors of the product are free abelian groups.

Keywords

Primary: 20E0620E2620F18secondary: 20F05generalised free productpolygonal productnilpotent groupabelian groupresidual finiteness
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 4 , 01 December 1994 , pp. 433 - 436
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Allenby, R. B. J. T., The potency of cyclically pinched one-relator groups, Arch. Math. 36(1981), 204–210.Google Scholar
2. Allenby, R. B. J. T. and Tang, C. Y., On the residual finiteness of certain polygonal products, Canad. Math. Bull. 32(1989), 11–17.Google Scholar
3. Brunner, A. M., Frame, A. L., Lee, Y. W. and Wielenberg, N. J., Classifying the torsion-free subgroups of the Picard group, Trans. Amer. Math. Soc. 282(1984), 205–235.Google Scholar
4. Higman, G., A finitely generated infinite simple group, J. London Math. Soc. 26(1951), 61–64.Google Scholar
5. Karrass, A., Pietrowski, A. and Solitar, D., The subgroups of a polygonal product of groups, unpublished manuscript.Google Scholar
6. Kim, Goansu, On polygonal products of finitely generated abelian groups, Bull. Austral. Math. Soc. 45 (1992), 453–462.Google Scholar
7. Kim, G. and Tang, C. Y., On the residual finiteness of polygonal products of nilpotent groups, Canad. Math. Bull. 35(1992), 390–399.Google Scholar
8. Pride, Stephen J., Groups with presentations in which each defining relator involves exactly two generators, J. London Math. Soc. (2) 36(1987), 245–256.Google Scholar