Revista Matemática Iberoamericana

Maximal real Schottky groups

Rubén A. Hidalgo

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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.

Article information

Rev. Mat. Iberoamericana, Volume 20, Number 3 (2004), 737-770.

First available in Project Euclid: 27 October 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F10: Compact Riemann surfaces and uniformization [See also 14H15, 32G15] 30F40: Kleinian groups [See also 20H10] 30F50: Klein surfaces

Schottky groups Riemann surfaces Riemann matrices


Hidalgo, Rubén A. Maximal real Schottky groups. Rev. Mat. Iberoamericana 20 (2004), no. 3, 737--770.

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