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On neoclassical Schottky groups


Authors: Rubén Hidalgo and Bernard Maskit
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
Published electronically: October 31, 2005
MathSciNet review: 2231871
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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
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

DOI: https://doi.org/10.1090/S0002-9947-05-03792-X
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
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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