Multiply Quasiplatonic Riemann Surfaces

Abstract

The aim of this article is the study of the circumstances under which a compact Riemann surface may contain two regular dessin d'enfants of different types. In terms of Fuchsian groups, an equivalent condition is the uniformizing group being normally contained in several different triangle groups. The question is answered in a graph-theoretical way, providing algorithms that decide if a surface that carries a regular dessin (a quasiplatonic surface) can also carry other regular dessins. The multiply quasiplatonic surfaces are then studied depending on their arithmetic character. Finally, the surfaces of lowest genus carrying a large number of nonarithmetic regular dessins are computed.