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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.

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

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