Fuchsian groups and algebraic number fields
Authors:
P. L. Waterman and C. Maclachlan
Journal:
Trans. Amer. Math. Soc. 287 (1985), 353364
MSC:
Primary 20H10; Secondary 11F06, 32G15
MathSciNet review:
766224
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Abstract: Given the signature of a finitelygenerated 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), 570600. 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), 331346. MR 679761 (84d:20046)
 [3]
 , Regular tessellations of surfaces and triangle groups, Ann. of Math. (2) 116 (1982), 113132. 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 , Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 167182. MR 624813 (83b:10025)
 [6]
 , Homomorphisms of triangle groups into , 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 , Homomorphisms and quadratic differentials of triangle groups (unpublished).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), 289304. 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 dimensional representations of the modular group and related groups, Amer. J. Math. 87 (1965), 945954. MR 0188298 (32:5737)
 [12]
 , Real dimensional representations of the free product of two finite cyclic groups, Proc. Cambridge Philos. Soc. 62 (1966), 135141. MR 0193159 (33:1380)
 [13]
 W. Magnus, Rational representations of Fuchsian groups and nonparabolic subgroups of the modular group, Nachr. Akad. Wiss. Göttingen II: Math. Phys. K1. 1973, 179189. MR 0346069 (49:10795)
 [14]
 J. Mennicke, A note on regular coverings of closed orientable surfaces, Proc. Glasgow Math. Assoc. 5 (1961), 4966. MR 0161910 (28:5114)
 [15]
 M. H. Millington, Subgroups of the classical modular group, J. London Math. Soc. (2) 1 (1969), 351357. MR 0244160 (39:5477)
 [16]
 K. Takeuchi, Fuchsian groups contained in , J. Math. Soc. Japan 23 (1971), 8294. 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), 369384. MR 0137792 (25:1241)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507662245
PII:
S 00029947(1985)07662245
Article copyright:
© Copyright 1985
American Mathematical Society
