Riemann surfaces with maximal real symmetry

Abstract

Let S be a compact Riemann surface of genus g>1, and let τ:S→S be any anti-conformal automorphism of S, of order 2. Such an anti-conformal involution is known as a symmetry of S, and the species of all conjugacy classes of all symmetries of S constitute what is known as the symmetry type of S. The surface S is said to have maximal real symmetry if it admits a symmetry τ:S→S such that the compact Klein surface S/τ has maximal symmetry (which means that S/τ has the largest possible number of automorphisms with respect to its genus). If τ has fixed points, which is the only case we consider here, then the maximum number of automorphisms of S/τ is 12(g−1). In the first part of this paper, we develop a computational procedure to compute the symmetry type of every Riemann surface of genus g with maximal real symmetry, for given small values of g>1. We have used this to find all of them for 1<g≤101, and give details for 1<g≤25 (in an appendix). In the second part, we determine the symmetry types of four infinite families of Riemann surfaces with maximal real symmetry. We also determine the full automorphism group of the Klein surface S/τ associated with each symmetry τ:S→S.