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Aspherical relative presentations

Published online by Cambridge University Press:  20 January 2009

W. A. Bogley  and
S. J. Pride
W. A. Bogley
Affiliation:
2055 Grant StreetEugene, OR 97405, U.S.A.
S. J. Pride
Affiliation:
Department of MathematicsUniversity of GlasgowUniversity GardensGlasgow G12 8QW, Scotland
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A geometric hypothesis is presented under which the cohomology of a group G given by generators and defining relators can be computed in terms of a group H defined by a subpresentation. In the presence of this hypothesis, which is framed in terms of spherical pictures, one has that H is naturally embedded in G, and that the finite subgroups of G are determined by those of H. Practical criteria for the hypothesis to hold are given. The theory is applied to give simple proofs of results of Collins-Perraud and of Kanevskiĭ. In addition, we consider in detail the situation where G is obtained from H by adjoining a single new generator x and a single defining relator of the form xaxbxεc, where a, b, c ∈ H and |ε| = 1.

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Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 35 , Issue 1 , February 1992 , pp. 1 - 39
Copyright
Copyright © Edinburgh Mathematical Society 1992

References

REFERENCES

1.Bogley, W. A., Local collapses for diagrammatic reducibility, in: Topology and Combinatorial Group Theory (Lecture Notes in Mathematics 1440 (1990), edited by Latiolais, P., Springer-Verlag).CrossRefGoogle Scholar
2.Brick, S. G., Normal-convexity and equations over groups, Invent. Math. 94 (1988), 81–104.CrossRefGoogle Scholar
3.Brown, R. and Huebschmann, J., Identities among relations, in: Low-dimensional topology (London Mathematical Society Lecture Note Series, 48 (1982), edited by R. Brown and T. L. Thickston).CrossRefGoogle Scholar
4.Collins, D. J. and Huebschmann, J., Spherical diagrams and identities among relations, Math. Ann. 261 (1982), 155–183.CrossRefGoogle Scholar
5.Collins, D. J. and Perraud, J., Cohomology and finite subgroups of small cancellation quotients of free products, Math. Proc. Cambridge Philos. Soc. 97 (1985), 243–259.CrossRefGoogle Scholar
6.Fennessey, E. J. and Pride, S. J., Equivalences of two-complexes, with applications to NEC-groups, Math. Proc. Cambridge Philos. Soc. 106 (1989), 215–228.CrossRefGoogle Scholar
7.Gersten, S. M., Reducible diagrams and equations over groups, in: Essays in group theory (MSRI Publications 8 (1987), edited by Gersten, S. M., Springer-Verlag).CrossRefGoogle Scholar
8.Gersten, S. M., On certain equations over torsion-free groups (University of Utah), preprint.Google Scholar
9.Hilton, P. J. and Stammbach, U., A course in homological algebra (Graduate Texts in Mathematics 4 (1971), Springer-Verlag).CrossRefGoogle Scholar
10.Howie, J., On the asphericity of ribbon disc complements, Trans. Amer. Math. Soc. 289 (1985), 281–302.CrossRefGoogle Scholar
11.Howie, J., How to generalize one-relator group theory, in: Combinatorial group theory and topology (Annals of Mathematics Studies, 111 (1987), edited by Gersten, S. M. and Stallings, J. R., Princeton University Press).Google Scholar
12.Howie, J., The quotient of a free product of groups by a single high-powered relator. I. Pictures. Fifth and higher powers, Proc. London Math. Soc. (3) 59 (1989), 507–540.CrossRefGoogle Scholar
13.Howie, J. and Pride, S. J., A spelling theorem for staggered generalized 2-complexes, with applications, Invent. Math. 76 (1984), 55–74.CrossRefGoogle Scholar
14.Huebschmann, J., Cohomology theory of aspherical groups and of small cancellation groups, J. Pure Appl. Algebra 14 (1979), 137–143.CrossRefGoogle Scholar
15.Huebschmann, J., Aspherical 2-complexes and an unsettled problem of J. H. C. Whitehead, Math. Ann. 258 (1981), 17–37.CrossRefGoogle Scholar
16.Kanevskiĭ, D. S., The structure of groups connected with automorphisms of cubic surfaces, Math. USSR-Sb. 32 (1977), 252–264.CrossRefGoogle Scholar
17.Kanevskiĭ, D. S., On cubic planes and groups connected with cubic varieties, J. Algebra 80 (1983), 559–565.CrossRefGoogle Scholar
18.Levin, F., Solutions of equations over groups, Bull. Amer. Math. Soc. 68 (1962), 603–604.CrossRefGoogle Scholar
19.Lyndon, R. C. and Schupp, P. E., Combinatorial group theory (Springer-Verlag, 1977).Google Scholar
20.Pride, S. J., Star-complexes and the dependence problems for hyperbolic complexes, Glasgow Math. J. 30 (1988), 155–170.CrossRefGoogle Scholar
21.Pride, S. J., Involutary presentations, with applications to Coxeter groups, NEC-groups and groups of Kanevskiĭ, J. Algebra 120 (1989), 200–223.CrossRefGoogle Scholar
22.Pride, S. J., The (co)homology of groups given by presentations in which each defining relator involves at most two types of generators, J. Austral. Math. Soc., to appear.Google Scholar
23.Pride, S. J. and Stöhr, R., Relation modules of groups with presentations in which each relator involves exactly two types of generators, J. London Math. Soc. (2) 38 (1988), 99–111.CrossRefGoogle Scholar
24.Serre, J.-P., Trees (Springer-Verlag, 1980).CrossRefGoogle Scholar
25.Sieradski, A., A coloring test for asphericity, Quart. J. Math. Oxford (2) 34 (1983), 97–106.CrossRefGoogle Scholar