Applications of a computer implementation of Poincaré's theorem on fundamental polyhedra
Author:
Robert Riley
Journal:
Math. Comp. 40 (1983), 607632
MSC:
Primary 20H10; Secondary 11F06, 2004, 22E40, 51M20, 57N10
MathSciNet review:
689477
Fulltext PDF Free Access
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Abstract: Poincaré's Theorem asserts that a group of isometries of hyperbolic space is discrete if its generators act suitably on the boundary of some polyhedron in , and when this happens a presentation of can be derived from this action. We explain methods for deducing the precise hypotheses of the theorem from calculation in when is "algorithmically defined", and we describe a file of Fortran programs that use these methods for groups acting on the upper half space model of hyperbolic 3space . We exhibit one modest example of the application of these programs, and we summarize computations of repesentations of groups where is an order in a complex quadratic number field.
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 H. Cohn, A Second Course in Number Theory, Wiley, New York, 1962. MR 0133281 (24:A3115)
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 T. Jørgensen, "On discrete groups of Möbius transformations," Amer. J. Math., v. 98, 1976, pp. 739749. MR 0427627 (55:658)
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 B. Maskit, "On Poincaré's theorem for fundamental polygons," Adv. in Math., v. 1971, pp. 219230. MR 0297997 (45:7049)
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 H. Seifert, "Komplexe mit Seitenzuordnung," Göttingen Nachrichten (1975), 4980. MR 0383219 (52:4100)
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 J. Sommer, Introduction à la Théorie des Nombres Algébriques, Paris, 1911.
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 R. Swan, "Generators and relators for certain special linear groups," Adv. in Math., v. 6, 1971, pp. 177. MR 0284516 (44:1741)
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 W. Thurston, The geometry and topology of 3manifolds. (To appear.)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198306894772
PII:
S 00255718(1983)06894772
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
