Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-05T05:02:56.338Z Has data issue: false hasContentIssue false
  • English
  • Français

The solution of length three equations over groups

Published online by Cambridge University Press:  20 January 2009

James Howie
James Howie
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW
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.

Let G be a group, and let r = r(t) be an element of the free product G * 〈G〉 of G with the infinite cyclic group generated by t. We say that the equation r(t) = 1 has a solution in G if the identity map on G extends to a homomorphism from G * 〈G〉 to G with r in its kernel. We say that r(t) = 1 has a solution over G if G can be embedded in a group H such that r(t) = 1 has a solution in H. This property is equivalent to the canonical map from G to 〈G, t|r〉 (the quotient of G * 〈G〉 by the normal closure of r) being injective.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 26 , Issue 1 , February 1983 , pp. 89 - 96
Copyright
Copyright © Edinburgh Mathematical Society 1983

References

REFERENCES

1.Brodskii, S. D., Equations over groups and groups with one defining relator (Russian), Uspekhi Mat. Nauk 35–4 (1980), 183.Google Scholar
2.Gerstenhaber, M. and Rothaus, O. S., The solution of sets of equations in groups, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1531–1533.CrossRefGoogle ScholarPubMed
3.Howie, J., On pairs of 2-complexes and systems of equations over groups, J. reine angew. Math. 324 (1981), 165–174.Google Scholar
4.Levin, F., Solutions of equations over groups, Bull. American Math. Soc. 68 (1962), 603–604.CrossRefGoogle Scholar
5.Lyndon, R. C., On Dehn's algorithm, Math. Ann. 166 (1966), 208–228.CrossRefGoogle Scholar
6.Rothaus, O. S., On the nontriviality of some group extensions given by generators andrelators, Ann. Math. 106 (1977), 559–612.CrossRefGoogle Scholar
7.Rourke, C. P., Presentations and the trivial group, Topology of low-dimensional manifolds, Proc. 2nd Sussex Conf. (Lecture Notes in Mathematics 722, 1979), 134–143.CrossRefGoogle Scholar
8.Schupp, P. E., On Dehn's algorithm and the conjugacy problem, Math. Ann. 178 (1968), 119–130.CrossRefGoogle Scholar
9.Short, H. B., Topological methods in group theory: the adjunction problem (Ph.D. Thesis, University of Warwick, 1981).Google Scholar