A continuity theorem for Fuchsian groups
Author:
C. K. Wong
Journal:
Trans. Amer. Math. Soc. 166 (1972), 225239
MSC:
Primary 30A58
MathSciNet review:
0301192
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Abstract: On a given Riemann surface, fix a discrete (finite or infinite) sequence of points and associate to each an ``integer'' (which may be . This sequence of points and ``integers'' is called a ``signature'' on the Riemann surface. With only a few exceptions, a Riemann surface with signature can always be represented by a Fuchsian group. We investigate here the dependence of the group on the number . More precisely, keeping the points fixed, we vary the numbers in such a way that the signature tends to a limit signature. We shall prove that the corresponding representing Fuchsian group converges to the Fuchsian group which corresponds to the limit signature.
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 L. Bers, Riemann surfaces, Notes, New York University, New York, 1958.
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 , Uniformization by Beltrami equations, Comm. Pure Appl. Math. 14 (1961), 215228. MR 24 #A2022. MR 0132175 (24:A2022)
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 M. Heins, On Fuchsoid groups that contain parabolic transformations, Contribution to Function Theory, Internat. Colloq. Function Theory (Bombay, 1960), Tata Institute of Fundamental Research, Bombay, 1960, pp. 203210. MR 27 #1597. MR 0151613 (27:1597)
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 O. Kellogg, Foundations of potential theory, Ungar, New York, 1929.
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 P. Koebe, Ueber die Uniformisierung beliebiger analytischer Kurven, Gött. Nachr. (1907), 191210.
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 J. Lehner, Discontinuous groups and automorphic functions, Math. Surveys, no. 8, Amer. Math. Soc., Providence, R.I., 1964. MR 29 #1332. MR 0164033 (29:1332)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197203011922
PII:
S 00029947(1972)03011922
Keywords:
Riemann surfaces with signature,
the limit circle theorem of Koebe,
Fuchsian groups,
Poincaré metrics
Article copyright:
© Copyright 1972 American Mathematical Society
