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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Applications of a computer implementation of Poincaré’s theorem on fundamental polyhedra
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by Robert Riley PDF
Math. Comp. 40 (1983), 607-632 Request permission


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
  • © Copyright 1983 American Mathematical Society
  • Journal: Math. Comp. 40 (1983), 607-632
  • MSC: Primary 20H10; Secondary 11F06, 20-04, 22E40, 51M20, 57N10
  • DOI:
  • MathSciNet review: 689477