On the automorphism groups of a compact bordered Riemann surface of genus five

Abstract

Let 5 be a compact bordered Riemann surface of genus g with k boundary components. If 2g-\-k—1^2, the automorphism group of S is a finite group. Then, we put N(g, k) the maximum order of automorphism groups of S where the maximum is taken over all S of genus g with k boundary components. It is well known that N(g, k) is equal to the maximum order of automorphism groups of compact Riemann surfaces of genus g deleted k points, and every automorphism group of S is isomorphic to that of a compact Riemann surface (Oikawa [7]). For every k^O, N(0, k), N(l, k), N{2, k), N(3, k) and 7V(4, k) are determined by Heins [2], Oikawa [7], Tsuji [8], Tsuji [9] and Kato [4], respectively. In the present paper, we shall determine N(5, k). Theorem. JV(5, k) is