On the automorphism groups of a compact bordered Riemann surface of genus five (Q5903385)

Let g and k be nonnegative integers satisfying \(2g+k-3\geq 0\). Let N(g,k) be the maximum of the orders of conformal automorphism groups of Riemann surfaces, where Riemann surfaces run over all the compact bordered Riemann surfaces of genus g and with k boundary components. It is a well known result that N(g,0)\(\leq 84(g-1)\) (Hurwitz). For \(k\geq 0\), Oikawa gave a general estimate such that \(N(g,k)\leq 12(g-1)+6k\). Several persons including the reviewer determined the number N(g,k) exactly, if \(g=0,...,4\) and k is arbitrary, or if \(k=1,2,3\) and g is arbitrary. In this paper the author determines N(5,k) for arbitrary k's. There are 18 possibilities of values of N(5,k).