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On a Class of One-Relator Groups

Published online by Cambridge University Press:  20 November 2018

A. M. Brunner
A. M. Brunner*
Affiliation:
University of Wisconsin—Parkside, Kenosha, Wisconsin
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In this paper, we consider the class of groups G(l, m; k) which are defined by the presentation

where k, l, m are integers, and |l| > m > 0, k > 0. Groups in this class possess many properties which seem unusual, especially for one-relator groups. The basis for the results obtained below is the determination of endomorphisms.

For certain of the groups, we are able to calculate their automorphism groups. One consequence of this is to produce examples of one-relator groups with infinitely generated automorphism groups. This answers a question raised by G. Baumslag (in a colloquium lecture at the University of Waterloo). Our examples are, perhaps, the simplest possible; J. Lewin [10] has found an example of a finitely presented group with an infinitely generated automorphism group.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 414 - 420
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Anshel, M., The endomorphism s of certain one-relator groups and the generalized Hopfian problem, Bull. Amer. Math. Soc. 77 (1971), 348–350.Google Scholar
2. Anshel, M., Non-Hopfian groups with fully invariant kernels I, Trans. Amer. Math. Soc. 170 (1972), 231–237.Google Scholar
3. Anshel, M., Non-Hopfian groups with fully invariant kernels II, J. Algebr. 24 (1973), 473–485.Google Scholar
4. Baumslag, G. and Solitar, D., Some two-generator one-relator non-Hopfian'groups, Bull. Amer. Math. Soc. 68 (1962), 199–201.Google Scholar
5. Baumslag, G., A non-cyclic one-relator group all of whose quotients are cyclic, J. Austral. Math. Soc. 10 (1969), 497–498.Google Scholar
6. Brunner, A. M., Transitivity-systems of certain one-relator groups, Proc. Second Internat. Conf. Theory of Groups, Canberra (1973), Lecture Notes in Mathematics, No. 372, 141–149 (Springer-Verlag, Berlin).Google Scholar
7. Brunner, A. M., A group with an infinite number of Nielsen inequivalent one-relator presentations, J. Algebra 42 (1976), 81–84.Google Scholar
8. Collins, D. J., Some one-relator Hopfian groups, Trans. Amer. Math. Soc. (to appear).Google Scholar
9. Collins, D. J., The automorphism towers of some one-relator groups, Proc. London Math. Soc. (3). 36 (1978), 480–493.Google Scholar
10. Lewin, J., A finitely presented group whose group of automorphisms is infinitely generated, J. Lond. Math. Soc. 42 (1967), 610–613.Google Scholar
11. Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory, Pure and App. Math. Vol. 13 (Interscience, New York, London, Sydney, 1966).Google Scholar
12. Meskin, S., N on-residually finite one-relator groups, Trans. Amer. Math. Soc. 164 (1972), 105–114.Google Scholar
13. Miller III, C. F., On group-theoretic decision problems and their classification, Ann. Math. Studies 68 (Princeton University Press, Princeton, 1971).Google Scholar
14. Miller III, C. F. and Schupp, P. E., The geometry of Higman-Neumann-Neumann extensions, Comm. Pure Appl. Math. 26 (1973), 787–802.Google Scholar
15. Pietrowski, A., The isomorphism problem for one-relator groups with non-trivial centre, Math. Z. 136 (1974), 95–106.Google Scholar