# Maximal real Schottky groups

### Rubén A. Hidalgo

Universidad Técnica Federico Santa María, Valparaíso, Chile

## Abstract

Let $S$ be a real closed Riemann surfaces together a reflection \mbox{$τ:S→S$}, that is, an anticonformal involution with fixed points. A well known fact due to C. L. May \cite{May 1977} asserts that the group $K(S,τ)$, consisting on all automorphisms (conformal and anticonformal) of $S$ which commutes with $τ$, has order at most $24(g−1)$. The surface $S$ is called maximally symmetric Riemann surface if $∣K(S,τ)∣=24(g−1)$ \cite{Greenleaf-May 1982}. In this note we proceed to construct real Schottky uniformizations of all maximally symmetric Riemann surfaces of genus $g≤5$. 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.

## 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