
Professor Gareth A. Jones at University of Southampton gave the talk “Automorphism groups of Riemann surfaces and dessins d’enfants" at our seminar. Sep 11, 2020 
On this opportunity, Professor Dr. Jones gave a short and explicit algebraic proof of Greenberg’s Theorems for finitely generated groups, using results of Macbeath, Singerman and Takeuchi on triangle groups, and of Margulis on arithmeticity. In each case, a Riemann surface S is a quotient of the hyperbolic plane by an explicit subgroup of a triangle group. When the group is finite it follows from this and from Belyi’s Theorem that the surface can be defined, as a complex algebraic curve, over an algebraic number field.

