Luke Jeffreys

Euler characteristics of SL(2,ℤ)-orbit graphs of square-tiled surfaces

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. Providing indirect evidence for this conjecture, I will describe joint work with Carlos Matheus in which we prove that the absolute values of the Euler characteristics of the graphs in this family go to infinity (a requirement for any expander family). To do this, we are required to count a variety of objects, including integer points on algebraic hypersurfaces, pseudo-Anosov homeomorphisms of fixed dilatation, and orbifold points on certain algebraic curves in moduli space.