JournalsrmiVol. 20, No. 3pp. 737–770

Maximal real Schottky groups

  • Rubén A. Hidalgo

    Universidad Técnica Federico Santa María, Valparaíso, Chile
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Abstract

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.

Cite this article

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

DOI 10.4171/RMI/406