A combinatorial approach to the symmetries of M and M–l Riemann surfaces
E. Bujalance, Antonio F. Costa · Cambridge University Press eBooks · 1992
Introduction Let X be a compact Riemann surface of genus g . A symmetry of X is an anticonformal involution T : X → X . The topological nature of a symmetry T is determined by properties of its fixed point set F(T) . F(T) consists of k disjoint Jordan curves, where 0 ≤ k ≤ g + 1 (Harnack's theorem). X – F(T) has either one or two components. It consists of one component if X /〈 T 〉 is non-orientable and two components if X /〈 T 〉 is orientable. Let T be a symmetry of X and suppose that in F(T) there are k disjoint Jordan curves, then we shall say (see [4]) that the species of T is + k if X – F(T) has two components and – k if X – F(T) has one component. We shall say that a Riemann surface X is an M (respectively M – 1) Riemann surface if it admits a symmetry with g + 1 fixed curves (respectively g fixed curves). S. M. Natanzon in [9] and [10] announced some properties about the topological nature of the symmetries of M and M – 1 Riemann surfaces and, in the hyperelliptic case, the classification of such symmetries up to conjugation in the automorphism group. Later, in [11] and [12] the above results were proved by topological techniques.