The symmetry type of a Riemann surface.

*(English)*Zbl 0545.30032Let 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 |