5. Quasiplatonic Surfaces, and Automorphisms

Quasiplatonic Riemann surfaces or algebraic curves, sometimes also called curves with many automorphisms or triangle curves, can be characterised in many equivalent ways, for example as those curves having a regular dessin, one with the greatest possible degree of symmetry. The sphere and the torus each support infinitely many regular dessins, easily described in both cases. For each genus g > 1 there are, up to isomorphism, only finitely many regular dessins; this chapter gives complete lists for genera 2, 3 and 4, and discusses methods for counting and classifying them. These methods often involve counting generating triples for finite groups, in some cases with the aid of character theory (which we briefly summarise) and Möbius inversion. We present several important infinite families of quasiplatonic curves, such as Hurwitz and Macbeath-Hurwitz curves, Lefschetz and Accola-Maclachlan curves. We prove that like their counterparts, the curves with trivial automorphism group, quasiplatonic curves can be defined over their field of moduli. Many of the automorphism groups appearing in this chapter are 2-dimensional linear or projective groups over finite fields, so we summarise their most relevant properties in the final section.