Pairs of symmetries of Riemann surfaces

Let F-g be a compact Riemann surface of genus g. A symmetry S of F-g is an anticonformal involution acting on F-g. The fixed-point set of a symmetry is a collection of disjoint simple closed curves, called the mirrors of the symmetry. The number of mirrors \S\ of a symmetry of a surface of genus g can be any integer k with 0 less than or equal to k less than or equal to g + 1. However, if a Riemann surface F-g admits a symmetry S-1 with k mirrors then work of Bujalance and Costa [1] and Natanzon [9] on symmetries with g + 1 mirrors suggest that there may possibly be restrictions on the number of mirrors of another symmetry S-2 of F-g. In the first three sections of this work we show that the number of such restrictions is few and only occur if one of the symmetries has g + 1 or 0 mirrors. The main result of Sections 1-3 is Theorem 1.1 below. In Section 4 we study a finer classification than the number of mirrors, namely the species of a symmetry. The k mirrors of a symmetry S may or may not separate the surface F-g into two non-empty components. If the mirrors do separate, then we say that S has species +k, and if the mirrors do not separate then we say that the species is -k. (See [5].) The species of S determines S up to topological conjugacy. In Section 4 we investigate which pairs of species can occur for two symmetries S-1, S-2 of F-g. There are many more restrictions than when we just ask for the number of mirrors.