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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

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.
References
  • Lars V. Ahlfors, Möbius transformations in several dimensions, Ordway Professorship Lectures in Mathematics, University of Minnesota, School of Mathematics, Minneapolis, Minn., 1981. MR 725161
  • A. F. Beardon, The geometry of discrete groups, Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975) Academic Press, London, 1977, pp. 47–72. MR 0474012
  • Alan F. Beardon and Bernard Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974), 1–12. MR 333164, DOI 10.1007/BF02392106
  • Luigi Bianchi, Opere. Vol. I, Parte prima, Edizioni Cremonese della Casa Editrice Perrella, Roma, 1952 (Italian). A cura dell’Unione Matematica Italiana e col contributo del Consiglio Nazionale delle Ricerche. MR 0051755
  • Harvey Cohn, A second course in number theory, John Wiley & Sons, Inc., New York-London, 1962. MR 0133281
  • Troels Jørgensen, On discrete groups of Möbius transformations, Amer. J. Math. 98 (1976), no. 3, 739–749. MR 427627, DOI 10.2307/2373814
  • Bernard Maskit, On Poincaré’s theorem for fundamental polygons, Advances in Math. 7 (1971), 219–230. MR 297997, DOI 10.1016/S0001-8708(71)80003-8
  • Robert Riley, A quadratic parabolic group, Math. Proc. Cambridge Philos. Soc. 77 (1975), 281–288. MR 412416, DOI 10.1017/S0305004100051094
  • Robert Riley, Discrete parabolic representations of link groups, Mathematika 22 (1975), no. 2, 141–150. MR 425946, DOI 10.1112/S0025579300005982
  • Robert Riley, An elliptical path from parabolic representations to hyperbolic structures, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977) Lecture Notes in Math., vol. 722, Springer, Berlin, 1979, pp. 99–133. MR 547459
  • R. Riley, Seven excellent knots, Low-dimensional topology (Bangor, 1979) London Math. Soc. Lecture Note Ser., vol. 48, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 81–151. MR 662430
  • Herbert Seifert, Komplexe mit Seitenzuordnung, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 6 (1975), 49–80 (German). MR 383219
  • J. Sommer, Introduction à la Théorie des Nombres Algébriques, Paris, 1911.
  • Richard G. Swan, Generators and relations for certain special linear groups, Advances in Math. 6 (1971), 1–77 (1971). MR 284516, DOI 10.1016/0001-8708(71)90027-2
  • W. Thurston, The geometry and topology of 3-manifolds. (To appear.)
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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: https://doi.org/10.1090/S0025-5718-1983-0689477-2
  • MathSciNet review: 689477