Maximal real Schottky groups
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.
scientific article; zbMATH DE number 2156857
scientific article