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

Abstract:

Poincaré’s Theorem asserts that a group $\Gamma$ of isometries of hyperbolic space $\mathbb {H}$ is discrete if its generators act suitably on the boundary of some polyhedron in $\mathbb {H}$, and when this happens a presentation of $\Gamma$ can be derived from this action. We explain methods for deducing the precise hypotheses of the theorem from calculation in $\Gamma$ when $\Gamma$ is "algorithmically defined", and we describe a file of Fortran programs that use these methods for groups $\Gamma$ acting on the upper half space model of hyperbolic 3-space ${\mathbb {H}^3}$. We exhibit one modest example of the application of these programs, and we summarize computations of repesentations of groups ${\text {PSL}}(2,\mathcal {O})$ where $\mathcal {O}$ is an order in a complex quadratic number field.