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Spherical diagrams and equations over groups

Published online by Cambridge University Press:  24 October 2008

James Howie
Affiliation:
Department of Mathematics, University of Glasgow

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
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1]Adams, J. F.. A new proof of a theorem of W. H. Cockcroft. J. London Math. Soc. (2) 49 (1955), 482488.CrossRefGoogle Scholar
[2]Brown, R. and Huebschmann, J.. Identities among relations. In Low-Dimensional Topology, London Math. Soc. Lecture Note Ser. 48 (1982), 153202.Google Scholar
[3]Chiswell, I. M., Collins, D. J. and Huebschmann, J.. Aspherical group presentations. Math. Z. 187 (1981), 136.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), 435445.Google Scholar
[5]Collins, D. J. and Huebschmann, J.. Spherical diagrams and identities among relations. Math. Ann. 261 (1982), 155183.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), 5974.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), 15311533.Google Scholar
[9]Howie, J.. On pairs of 2-complexes and systems of equations over groups. J. Reine Angew. Math. 324 (1981), 165174.Google Scholar
[10]Howie, J.. The solution of length three equations over groups. Proc. Edinburgh Math. Soc. (2) 26 (1983), 8996.CrossRefGoogle Scholar
[11]Howie, J. and Pride, S. J.. A spelling theorem for staggered generalized 2-complexes, with applications. Invent. Math. 76 (1984), 5574.CrossRefGoogle Scholar
[12]Huebschmann, J.. Aspherical 2-complexes and an unsettled problem of J. H. C. Whitehead. Math. Ann. 258 (1981), 1737.Google Scholar
[13]Levin, F.. Solutions of equations over groups. Bull. Amer. Math. Soc. 68 (1962), 603604.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), 6799.Google Scholar
[16]Rothaus, O. S.. On the nontriviality of some group extensions given by generators and relations. Ann. of Math. (2) 106 (1977), 599612.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), 134143.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), 117.Google Scholar