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Transactions of the American Mathematical Society

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

 
 

 

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
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
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.