Problems in group theory

The thesis is concerned with groups acting on three dimensional spaces. The groups are assumed to have a compact simply-connected fundamental region. The action of the group is given partially by its action on the boundary of the fundamental region. This boundary is naturally split up by its intersection with transforms of the fundamental region. We assume that each such intersection is a single proper face of the boundary unless the transforming element is of order two, in which case there can be two faces. We also assume that any point of the boundary has only a finite number of transforms under the group which lie on the boundary. This enables one to give generators and defining relations for the group. The generators correspond to faces of the boundary inequivalent under the group, and defining relations to inequivalent lines. In these circumstances two questions arise: Is the three-dimensional space a manifold? Is the group finite? If the space is not a manifold, then the group cannot be finite. So an answer to the first question gives some information about the second. Another theorem which is a corollary of the methods used in proving the first theorem is:- Given a group acting on a three-dimensional space with a fundamental region satisfying the conditions above then the group has a soluble word problem.