Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-19T15:47:32.931Z Has data issue: false hasContentIssue false
  • English
  • Français

On one relator groups and HNN extensions

Published online by Cambridge University Press:  09 April 2009

James McCool  and
Paul E. Schupp
James McCool
Affiliation:
Department of Mathematics University of TorontoToronto 181, Canada
Paul E. Schupp
Affiliation:
Department of Mathematics University of IllinoisUrbana, Illinois 61801, U. S. A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In his work [5] on subgroups of one relator groups, Moldavanski observed that if G is a one relator group whose defining relator R is cyclically reduced and has exponent sum zero on some generator occurring in it, then G is an HNN extension of a one relator group H whose defining relator is shorter than R. This observation, together with Britton's Lemma, can be used to give rather easy proofs of the basic results on one relator groups. To exposit this point of view, we give here a proof of the Freiheitssatz, the solvability of the word problem for one relator groups, and the theorem classifying elements of finite order in one relator groups. In particular, the solution obtained for the word problem is often easy to apply. We also give a proof of the “Spelling Theorem” of Newman [6].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 2 , September 1973 , pp. 249 - 256
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Britton, J. L., ‘The word problem’, Annals of Math., 77 (1963), 16–32.CrossRefGoogle Scholar
[2]Graham, Higman, Neumann, B. H., and Hanna, Neumann, ‘Embedding theorems for groups’, J. London Math. Soc. 241 (1949), 247–254.Google Scholar
[3]Magnus, W., Karrass, A., and Solitar, D., Combinatorial group theory, (Interscience, New York, 1966.)Google Scholar
[4]Miller, C. F. III, ‘On Britton's theorem A’, Proc. Amer. Math. Soc. 19 (1968), 1151–1154.CrossRefGoogle Scholar
[5]Moldavanskii, D. I., ‘Certain subgroups of groups with one defining relator’, Siberian Math. J. 8 (1967), 1370–1384.CrossRefGoogle Scholar
[6]Newman, B. B., ‘Some results on one-relator groups’, Bull. Amer. Math. Soc. 74 (1968), 568–571.CrossRefGoogle Scholar
[7]Newman, B. B., Some aspects of one-relator groups, (Ph.D. Thesis, University of Queensland, 1968).CrossRefGoogle Scholar