The genus of compact riemann surfaces with maximal automorphism group

Abstract

Any compact Riemann surface with genus g > 1 has at most 84( g–1) conformal automorphisms. In this paper it is shown that there are just 32 integers g in the range 1 < g < 11905 for which there exist compact Riemann surfaces of genus g with this maximum possible number of automorphisms. The automorphism group of each such surface is isomorphic to a quotient Δ N of the (2, 3, 7) triangle group δ =⩽ x, y| x 2 = y 3 =( xy) 7 = 1⩾where N is one of 92 proper normal subgroups of Δ with index less than 10 6. These normal subgroups are found using elementary group-theoretic techniques.