Application Of Hoare's Theorem To Symmetries Of Riemann Surfaces

. Let X be a compact Riemann surface of genus g ? 1 . A symmetry S of X is an anticonformal involution. We write jSj for the number of connected components of the fixed points set of S . Suppose that X admits two distinct symmetries S 1 and S 2 ; then we find a bound for jS 1 j + jS 2 j in terms of the genus of X and the order of S 1 S 2 . We discuss circumstances in which the bound is attained, showing that this occurs only for hyperelliptic surfaces. In this way we generalize a theorem of S.M. Natanzon. 1. Introduction Let X be a compact Riemann surface of genus g ? 1 . A symmetry S of X is an anticonformal involution S: X ! X and a Riemann surface that admits a symmetry is called symmetric. By Harnack's theorem [1, 5, 9], the fixed-point set of S is either empty or consists of k g + 1 disjoint simple closed curves or, as we shall call them, mirrors. We write jSj for the number of mirrors of S , so that jSj is the number of components of the fixed-point set of S . Suppose that X..