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

DOI:
https://doi.org/10.1090/S0025-5718-1983-0689477-2

MathSciNet review:
689477

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Abstract | References | Similar Articles | Additional Information

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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Additional Information

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