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 H and |ε| = 1.
(Received January 25 1990)