Symmetries of Accola-Maclachlan and Kulkarni surfaces

Abstract

For all g ≥ 2 g \ge 2 there is a Riemann surface of genus g g whose automorphism group has order 8 g + 8 8g+8 , establishing a lower bound for the possible orders of automorphism groups of Riemann surfaces. Accola and Maclachlan established the existence of such surfaces; we shall call them Accola-Maclachlan surfaces. Later Kulkarni proved that for sufficiently large g g the Accola-Maclachlan surface was unique for g = 0 , 1 , 2 mod 4 g= 0,1,2\mod 4 and produced exactly one additional surface (the Kulkarni surface) for g = 3 mod 4 g= 3\mod 4 . In this paper we determine the symmetries of these special surfaces, computing the number of ovals and the separability of the symmetries. The results are then applied to classify the real forms of these complex algebraic curves. Explicit equations of these real forms of Accola-Maclachlan surfaces are given in all but one case.