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October, 2004 Maximal real Schottky groups
Rubén A. Hidalgo
Rev. Mat. Iberoamericana 20(3): 737-770 (October, 2004).

Abstract

Let $S$ be a real closed Riemann surfaces together a reflection \mbox{$\tau:S \to S$}, that is, an anticonformal involution with fixed points. A well known fact due to C. L. May \cite{May 1977} asserts that the group $K(S,\tau)$, consisting on all automorphisms (conformal and anticonformal) of $S$ which commutes with $\tau$, has order at most $24(g-1)$. The surface $S$ is called maximally symmetric Riemann surface if $|K(S,\tau)|=24(g-1)$ \cite{Greenleaf-May 1982}. In this note we proceed to construct real Schottky uniformizations of all maximally symmetric Riemann surfaces of genus $g \leq 5$. A method due to Burnside \cite{Burnside 1892} permits us the computation of a basis of holomorphic one forms in terms of these real Schottky groups and, in particular, to compute a Riemann period matrix for them. We also use this in genus 2 and 3 to compute an algebraic curve representing the uniformized surface $S$. The arguments used in this note can be programed into a computer program in order to obtain numerical approximation of Riemann period matrices and algebraic curves for the uniformized surface $S$ in terms of the parameters defining the real Schottky groups.

Citation

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Rubén A. Hidalgo. "Maximal real Schottky groups." Rev. Mat. Iberoamericana 20 (3) 737 - 770, October, 2004.

Information

Published: October, 2004
First available in Project Euclid: 27 October 2004

zbMATH: 1065.30040
MathSciNet: MR2124488

Subjects:
Primary: 30F10 , 30F40 , 30F50

Keywords: Riemann matrices , Riemann surfaces , Schottky groups

Rights: Copyright © 2004 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.20 • No. 3 • October, 2004
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