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The residual finiteness of a class of 1-relator groups

Published online by Cambridge University Press:  18 May 2009

R. B. J. T. Allenby  and
C. Y. Tang
R. B. J. T. Allenby
Affiliation:
School of Mathematics, The University, Leeds LS2 9JT, England
C. Y. Tang
Affiliation:
Department of Pure Mathematics, The University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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In [3] Gilbert Baumslag asserted that, for non-zero integers α, β, γ, δ such that α + γ ≠ 0 ≠ β + δ, the group G = <a, b:aα, bβaγbδ> is residually finite (RF). This result has been quoted in the literature: for example, in [2]. At the “Groups '85” meeting at St. Andrews, the second author learned, indirectly, that Professor Baumslag could not recall all the details of the rather complicated (unpublished) proof he had constructed and that he referred those asking for a proof to the present authors. It thus seems worthwhile formally to record the following fairly short proof of the above claim.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 29 , Issue 2 , July 1987 , pp. 267 - 269
Copyright
Copyright © Glasgow Mathematical Journal Trust 1987

Footnotes

†

The work of the second author was partly supported by Grant No. A-4064 from the NSERC of Canada.

References

1. Allenby, R. B. J. T., Moser, L. E. and Tang, C. Y., The residual finiteness of certain 1-relator groups, Proc. Amer. Math. Soc. 78 (1980), 8–10.CrossRefGoogle Scholar
2. Allenby, R. B. J. T. and Tang, C. Y., Residual finiteness of certain 1-relator groups; extensions of results of Gilbert Baumslag, Math. Proc. Cambridge Philos. Soc. 97 (1985), 225–230.CrossRefGoogle Scholar
3. Baumslag, Gilbert, Some problems on 1-relator groups, Proceedings of the Second International Conference on the Theory of Groups (Lecture Notes in Mathematics 372, Springer, 1974), 75–81.CrossRefGoogle Scholar