Resumen: Square-tiled surfaces (aptly named) are surfaces obtained by gluing together a collection of unit squares along their sides (the square torus being the simplest example). These surfaces are special cases of translation surfaces, whose moduli space carries a natural action of the group SL(2,R). The famous works of Eskin–Mirzakhani and Eskin–Mirzakhani–Mohammadi were concerned with understanding the orbits of this action.
This SL(2,R)-action restricts to an action of SL(2,Z) on square-tiled surfaces, and the orbits under this restricted action can be transformed into finite regular graphs. It is a long-standing conjecture of McMullen that a specific family of these orbit graphs in genus two forms a family of expander graphs. I will discuss the history of this conjecture and recent progress towards it obtained in joint work with Carlos Matheus.
Es en el salón 25 de FCEA.
