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Fuchsian groups and algebraic number fields


Authors: P. L. Waterman and C. Maclachlan
Journal: Trans. Amer. Math. Soc. 287 (1985), 353-364
MSC: Primary 20H10; Secondary 11F06, 32G15
DOI: https://doi.org/10.1090/S0002-9947-1985-0766224-5
MathSciNet review: 766224
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Abstract: Given the signature of a finitely-generated Fuchsian group, we find the minimal extension of the rationals for which there is a Fuchsian group having the required signature, whose matrix entries lie in this field.


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  • [1] L. Bers, On boundaries of Teichmüller spaces and on Kleinian groups. I, Ann. of Math. (2) 91 (1970), 570-600. MR 0297992 (45:7044)
  • [2] A. L. Edmonds, J. H. Ewing and R. S. Kulkarni, Torsion free subgroups of Fuchsian groups and tessellations of surfaces, Invent. Math. 69 (1982), 331-346. MR 679761 (84d:20046)
  • [3] -, Regular tessellations of surfaces and $ (p,q,2)$-triangle groups, Ann. of Math. (2) 116 (1982), 113-132. MR 662119 (83h:57005)
  • [4] A. L. Edmonds, R. S. Kulkarni and R. E. Stong, Realizability of branched coverings of surfaces (preprint). MR 732119 (85k:57005)
  • [5] L. Greenberg, Homomorphisms of triangle groups into $ {\text{PSL}}(2,C)$, Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 167-182. MR 624813 (83b:10025)
  • [6] -, Homomorphisms of triangle groups into $ {\text{SO}}(3)$, Modular Functions in Analysis and Number Theory, (T. Metzer, ed.), Univ. of Pittsburgh, 1983.
  • [7] -, Homomorphisms and quadratic differentials of triangle groups (unpublished).
  • [8] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., Oxford, 1979. MR 0067125 (16:673c)
  • [9] A. W. Knapp, Doubly generated Fuchsian groups, Michigan Math. J. 15 (1968), 289-304. MR 0248231 (40:1483)
  • [10] R. S. Kulkarni, An extension of a theorem of Kurosh and applications to Fuchsian groups (preprint). MR 725780 (84k:10018)
  • [11] J. Lehner and M. Newman, Real $ 2$-dimensional representations of the modular group and related groups, Amer. J. Math. 87 (1965), 945-954. MR 0188298 (32:5737)
  • [12] -, Real $ 2$-dimensional representations of the free product of two finite cyclic groups, Proc. Cambridge Philos. Soc. 62 (1966), 135-141. MR 0193159 (33:1380)
  • [13] W. Magnus, Rational representations of Fuchsian groups and non-parabolic subgroups of the modular group, Nachr. Akad. Wiss. Göttingen II: Math. Phys. K1. 1973, 179-189. MR 0346069 (49:10795)
  • [14] J. Mennicke, A note on regular coverings of closed orientable surfaces, Proc. Glasgow Math. Assoc. 5 (1961), 49-66. MR 0161910 (28:5114)
  • [15] M. H. Millington, Subgroups of the classical modular group, J. London Math. Soc. (2) 1 (1969), 351-357. MR 0244160 (39:5477)
  • [16] K. Takeuchi, Fuchsian groups contained in $ {\text{SL}}(2,Q)$, J. Math. Soc. Japan 23 (1971), 82-94. MR 0271342 (42:6225)
  • [17] P. L. Waterman, Fuchsian groups and algebraic number fields, Thesis, Aberdeen, 1982.
  • [18] A. Weil, On discrete subgroups of Lie groups, Ann. of Math. (2) 72 (1960), 369-384. MR 0137792 (25:1241)

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DOI: https://doi.org/10.1090/S0002-9947-1985-0766224-5
Article copyright: © Copyright 1985 American Mathematical Society

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