Automorphism groups of Riemann surfaces and dessins d'enfants

Viernes 11 de septiembre. Auditorio Prof. Manuel López Ramírez. de Matemática y Estadística UFRO. 16:00 Hrs.

Conferencista: Gareth A. Jones, University of Southampton


In 1960 Greenberg proved that every countable group A is isomorphic to the automorphism group of a Riemann surface S, and in 1973 he proved that if A is finite then S can be chosen to be compact. The proofs are long, complicated and not constructive. I will give a short and explicit algebraic proof of Greenberg's Theorems for finitely generated groups A, using results of Macbeath, Singerman and Takeuchi on triangle groups, and of Margulis on arithmeticity. In each case, S is a quotient of the hyperbolic plane by an explicit subgroup of a triangle group. When A is finite it follows from this and from Belyi's Theorem that S can be defined, as a complex algebraic curve, over an algebraic number field.