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Transactions of the American Mathematical Society

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A continuity theorem for Fuchsian groups

Author: C. K. Wong
Journal: Trans. Amer. Math. Soc. 166 (1972), 225-239
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 $ \{ {P_k}\} ,k = 1,2,3, \ldots ,$ and associate to each $ {P_k}$ an ``integer'' $ {\nu _k}$ (which may be $ 1,2,3, \ldots ,{\text{or}}\;\infty )$. 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 $ {\nu _k}$. More precisely, keeping the points $ {P_k}$ fixed, we vary the numbers $ {\nu _k}$ 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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Keywords: Riemann surfaces with signature, the limit circle theorem of Koebe, Fuchsian groups, Poincaré metrics
Article copyright: © Copyright 1972 American Mathematical Society