JournalsrmiVol. 20, No. 3pp. 737–770

Maximal real Schottky groups

  • Rubén A. Hidalgo

    Universidad Técnica Federico Santa María, Valparaíso, Chile
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Let SS be a real closed Riemann surfaces together a reflection \mbox{τ:SS\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,τ)K(S,\tau), consisting on all automorphisms (conformal and anticonformal) of SS which commutes with τ\tau, has order at most 24(g1)24(g-1). The surface SS is called maximally symmetric Riemann surface if K(S,τ)=24(g1)|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 g5g \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 SS. 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 SS in terms of the parameters defining the real Schottky groups.

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Rubén A. Hidalgo, Maximal real Schottky groups. Rev. Mat. Iberoam. 20 (2004), no. 3, pp. 737–770

DOI 10.4171/RMI/406