Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Aspherical relative presentations

W. A. Bogleya1 and S. J. Pridea2

a1 2055 Grant Street Eugene, OR 97405, U.S.A.

a2 Department of Mathematics University of Glasgow University Gardens Glasgow G12 8QW, Scotland

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 xs2208 H and |ε| = 1.

(Received January 25 1990)