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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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On neoclassical Schottky groups
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by Rubén Hidalgo and Bernard Maskit PDF
Trans. Amer. Math. Soc. 358 (2006), 4765-4792 Request permission

Abstract:

The goal of this paper is to describe a theoretical construction of an infinite collection of non-classical Schottky groups. We first show that there are infinitely many non-classical noded Schottky groups on the boundary of Schottky space, and we show that infinitely many of these are “sufficiently complicated”. We then show that every Schottky group in an appropriately defined relative conical neighborhood of any sufficiently complicated noded Schottky group is necessarily non-classical. Finally, we construct two examples; the first is a noded Riemann surface of genus $3$ that cannot be uniformized by any neoclassical Schottky group (i.e., classical noded Schottky group); the second is an explicit example of a sufficiently complicated noded Schottky group in genus $3$.
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Additional Information
  • Rubén Hidalgo
  • Affiliation: Departamento de Matemática, Universidad Tecnica Federico Santa Maria, Valpa- raíso, Chile
  • MR Author ID: 272770
  • ORCID: 0000-0003-4070-2819
  • Email: ruben.hidalgo@usm.cl
  • Bernard Maskit
  • Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
  • Email: bernie@math.sunysb.edu
  • Received by editor(s): March 25, 2002
  • Received by editor(s) in revised form: July 21, 2004
  • Published electronically: October 31, 2005
  • Additional Notes: This work was partially supported by Projects Fondecyt 1030252, 1030373, 7000715 and UTFSM 12.03.21
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 4765-4792
  • MSC (2000): Primary 30F10, 30F40
  • DOI: https://doi.org/10.1090/S0002-9947-05-03792-X
  • MathSciNet review: 2231871