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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A continuity theorem for Fuchsian groups
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by C. K. Wong PDF
Trans. Amer. Math. Soc. 166 (1972), 225-239 Request permission

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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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 166 (1972), 225-239
  • MSC: Primary 30A58
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0301192-2
  • MathSciNet review: 0301192