Maximal real Schottky groups
type: Item
instance of: scholarly article
title: Maximal real Schottky groups
zbMATH Open document ID: 1065.30040
DOI: 10.4171/RMI/406
author: Rubén A. Hidalgo
published in: Revista Matemática Iberoamericana
publication date: 2005-04-15T00:00:00Z
full work available at URL: https://eudml.org/doc/39731
review text: It is well-known that each closed Riemann surface of genus \(g\geq 2\) can be uniformized by Schottky groups and also by algebraic curves, but in general there is no explicit relation between the two points of view. The paper under review concerns an aspect of this relation. The results are as follows. Let \(S\) be a closed Riemann surface equipped with an anticonformal involution with fixed points \(\tau: S\to S\). A well-known fact due to \textit{C. L. May} [``A bound for the number of automorphisms of a compact Klein surface with boundary'', Proc. Am. Math. Soc. 63, 273--280 (1977; Zbl 0422.30038)] states that the group of all automorphisms of \(S\) which commutes with \(\tau\) has order at most \(24(g-1)\). The surface \(S\) is said to be maximally symmetric if this bound is attained. In this paper, the author constructs real Schottky uniformizations for all maximally symmetric Riemann surfaces of genus \(g\leq 5\). Here, a real Schottky group is a Schottky group which keeps invariant a circle \(L\) in the Riemann sphere; equivalently, there is an involution of the Riemann sphere with a circle \(L\) as its set of fixed points which commutes with every element of the group. The method used to construct the real Schottky uniformizations is due to Burnside (1892), and it consists in building a basis for the holomorphic one forms in terms if the real Schottky group and, in particular, to compute Riemann period matrices for them. The paper also contains a few results for the case of genus 2 and 3.
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 30F50
zbMATH DE Number: 2156857
zbMATH Keywords: Schottky groups
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: uniformization
zbMATH Keywords: Riemann matrices
MaRDI profile type: Publication
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OpenAlex ID: W2082569223
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description: scientific article; zbMATH DE number 2156857