Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-29T18:17:18.138Z Has data issue: false hasContentIssue false
  • English
  • Français

Spherical diagrams and equations over groups

Published online by Cambridge University Press:  24 October 2008

James Howie
James Howie
Affiliation:
Department of Mathematics, University of Glasgow
Get access Rights & Permissions [Opens in a new window]

Extract

Spherical diagrams were introduced by Lyndon and Schupp[14] in order to study asphericity in group presentations and in 2-complexes. They have since been studied by several authors [2, 3, 5]. In particular, some technical loopholes in the original approach were closed in [5]. For many purposes the dual notion of pictures, introduced by Rourke[17], is more useful. These arise naturally through transversality. Pictures have also been studied and applied in [2, 5, 6, 12, 19].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 2 , September 1984 , pp. 255 - 268
Copyright
Copyright © Cambridge Philosophical Society 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Adams, J. F.. A new proof of a theorem of W. H. Cockcroft. J. London Math. Soc. (2) 49 (1955), 482–488.CrossRefGoogle Scholar
[2]Brown, R. and Huebschmann, J.. Identities among relations. In Low-Dimensional Topology, London Math. Soc. Lecture Note Ser. 48 (1982), 153–202.Google Scholar
[3]Chiswell, I. M., Collins, D. J. and Huebschmann, J.. Aspherical group presentations. Math. Z. 187 (1981), 1–36.Google Scholar
[4]Cohen, M. M., Metzler, W. and Zimmermann, A.. What does a basis of F(a, b) look like? Math. Ann. 257 (1981), 435–445.Google Scholar
[5]Collins, D. J. and Huebschmann, J.. Spherical diagrams and identities among relations. Math. Ann. 261 (1982), 155–183.CrossRefGoogle Scholar
[6]Fenn, R. A.. Techniques of Geometric Topology. London Math. Soc. Lecture Note Ser. 57 1983.Google Scholar
[7]Gersten, S. M. Conservative groups, indicability, and a conjecture of Howie. J. Pure Appl. Algebra 29 (1983), 59–74.CrossRefGoogle Scholar
[8]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.Google Scholar
[9]Howie, J.. On pairs of 2-complexes and systems of equations over groups. J. Reine Angew. Math. 324 (1981), 165–174.Google Scholar
[10]Howie, J.. The solution of length three equations over groups. Proc. Edinburgh Math. Soc. (2) 26 (1983), 89–96.CrossRefGoogle Scholar
[11]Howie, J. and Pride, S. J.. A spelling theorem for staggered generalized 2-complexes, with applications. Invent. Math. 76 (1984), 55–74.CrossRefGoogle Scholar
[12]Huebschmann, J.. Aspherical 2-complexes and an unsettled problem of J. H. C. Whitehead. Math. Ann. 258 (1981), 17–37.Google Scholar
[13]Levin, F.. Solutions of equations over groups. Bull. Amer. Math. Soc. 68 (1962), 603–604.Google Scholar
[14]Lyndon, R. C. and Schupp, P. E.. Combinatorial group theory. Ergebnisse der Math. vol. 89 (Springer-Verlag, 1977).Google Scholar
[15]Peiffer, R.. Über Identitäten zwischen Relationen. Math. Ann. 121 (1949), 67–99.Google Scholar
[16]Rothaus, O. S.. On the nontriviality of some group extensions given by generators and relations. Ann. of Math. (2) 106 (1977), 599–612.CrossRefGoogle Scholar
[17]Rourke, C. P.. Presentations and the trivial group. In Topology of Low-Dimensional Manifolds, Lecture Notes in Math. vol. 722 (Springer-Verlag, 1979), 134–143.Google Scholar
[18]Seifert, H. and Threlfall, W.. A Textbook of Topology (Academic Press, 1980).Google Scholar
[19]Short, H. B.. Topological methods in group theory: the adjunction problem. Ph.D. Thesis, Warwick, 1981.Google Scholar
[20]Thurston, W. P.. The geometry and topology of 3-manifolds, Lecture notes (Princeton, 1977).Google Scholar
[21]Stallings, J.. Surfaces in 3-manifolds and non-singular equations in groups. Math. Z. 184 (1983), 1–17.Google Scholar