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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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

Author: Robert Riley
Journal: Math. Comp. 40 (1983), 607-632
MSC: Primary 20H10; Secondary 11F06, 20-04, 22E40, 51M20, 57N10
MathSciNet review: 689477
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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.

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Keywords: Poincaré's Theorem on fundamental polyhedra, fundamental domain, discrete group, group presentation, Kleinian group, Bianchi group, hyperbolic space
Article copyright: © Copyright 1983 American Mathematical Society