A compact Riemann surface X of genus g is called an (M−1)-surface if it admits an anticonformal involution that fixes g simple closed curves, the second maximum number by Harnack’s theorem. In this paper we investigate the automorphism groups of (M−1)- surfaces with the M-property. | Turk J Math 27 (2003) , 349 – 367. ¨ ITAK ˙ c TUB Automorphism Groups of (M−1)-surfaces with the M-property Adnan Meleko˘glu Abstract A compact Riemann surface X of genus g is called an (M−1)-surface if it admits an anticonformal involution that fixes g simple closed curves, the second maximum number by Harnack’s theorem. If X also admits an automorphism of order g which cyclically permutes these g curves, then we shall call X an (M−1)-surface with the M-property. In this paper we investigate the automorphism groups of (M−1)surfaces with the M-property. Key Words: Riemann surface, (M−1)-surface. 1. Introduction Let X be a compact Riemann surface of genus g > 1. X is said to be symmetric if it admits an anticonformal involution T : X → X which we call a symmetry of X. The fixed point set of T is either empty or consists of k simple closed curves, each of which is called a mirror of T . Here k is a positive integer and by Harnack’s theorem 1 ≤ k ≤ g + 1. If T has g + 1 mirrors, then it is called an M-symmetry and X is called an M-surface. In [10] and [11], we defined an M-surface to have the M-property if it admits an automorphism of order g + 1 which cyclically permutes the mirrors of an M-symmetry, and we worked out the automorphism groups of M-surfaces with the M-property. Now let X be a Riemann surface of genus g > 2 and T a symmetry of X. If T has g mirrors, then it is called an (M−1)-symmetry and X is called an (M−1)-surface. Similarly, if X admits 2000 AMS Mathematics Subject Classification: 30F10 349 ˘ MELEKOGLU an automorphism of order g which cyclically permutes the mirrors of T , then we shall call it an (M−1)-surface with the M-property. In this paper we study the automorphism groups of (M−1)-surfaces with the M-property. The design of this paper is as follows. Sections 2 and 3 are devoted to the background material. In Sections 4 and 5, we work out the full automorphism groups of nonhyperelliptic and hyperelliptic (M−1)-surfaces with the .
