Applications of a computer implementation of Poincaré’s theorem on fundamental polyhedra

PoincarCs Theorem asserts that a group F of isometries of hyperbolic space H is discrete if its generators act suitably on the boundary of some polyhedron in H, and when this happens a presentation of F can be derived from this action. We explain methods for deducing the precise hypotheses of the theorem from calculation in F when F is "algorithmi- cally defined", and we describe a file of Fortran programs that use these methods for groups F acting on the upper half space model of hyperbolic 3-space H. We exhibit one modest example of the application of these programs, and we summarize computations of repesenta- tions of groups PSL(2, 0) where 6 is an order in a complex quadratic number field. In the early 1880's H. Poincare discovered a general theorem allowing one to deduce the discreteness of, and a presentation for, a group G of isometries of hyperbolic space from its action on a hyperbolic polyhedron under certain condi- tions. Theorems of this sort are part of the foundations of his theories of Fuchsian and Kleinian groups that have become very popular again, and H. Seifert has recently given us a modern proof of a fairly general version of Poincare's Theorem in (12); see also (7). This theorem has been little used over the past century, perhaps partly because its hypotheses have seemed very difficult to verify for a given group G except in very special circumstances. One reason for doubting that Poincare's Theorem is unreasonably difficult to apply to fairly general discrete groups is that no general alternative method for accomplishing its tasks has been proposed. The present paper is devoted to demonstrating that Poincare's Theorem can indeed be applied to given groups in apparently difficult cases and that much of the work can be done by a computer. Our experience suggests that the theorem is really very helpful in guiding the user to an understanding of the details of the action of G starting from a state of near ignorance. An outline of this paper is as follows. In Section 1 we begin by stating Seifert's version of Poincare's Theorem and explaining how we would apply it to an " algorithmically defined" group G of isometries of hyperbolic space H 'o. In Section 2 we specialize to the situation, 00 , for which we wrote our file, Poincare', of Fortran