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The symmetry type of a Riemann surface. (English) Zbl 0545.30032
Let X be a compact Riemann surface of genus g. A symmetry T of X is an anticonformal involution T:$$X\to X$$. The first point set F(T) of T consists of $$0\leq k\leq g+1$$ disjoint Jordan curves and X-F(T) has one or two components. We define the species of T to be $$+k$$ (resp. -k) if F(T) has two (resp. one) components. A fixed point free symmetry has species 0. The symmetry type of X is defined to be the unordered list of the species of the symmetries of X one being listed for each conjugacy class of symmetries in the automorphism group of X. The symmetry type of the Riemann sphere is easily seen to be $$\{0,+1\}$$ and the symmetry types of complex tori were found in N. L. Alling and N. Greenleaf’s Foundations of the theory of Klein surfaces (1971; Zbl 0225.30001). In this paper we use some recent subgroup theorems on NEC groups to obtain all possible symmetry types for genus 2. A symmetric Riemann surface represents a real algebraic curve and the paper ends with some applications to real forms of symmetric Riemann surfaces of genus 2, an example being that every such surface has a real form which is not purely imaginary.

##### MSC:
 30F10 Compact Riemann surfaces and uniformization 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) 14Pxx Real algebraic and real-analytic geometry
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