Josh Southerland

Affine homeomorphisms of translation surfaces

The study of the action of SL(2,Z) on the torus has a rich history. In this talk, we will look at a generalization: the action of the group of affine homeomorphisms on a translation surface. Translation surfaces are flat surfaces away from finitely many cone points, and affine homeomorphisms are akin to a linear action of a matrix on the flat structure. Using two different methods, one involving the moduli space of translation surfaces, we will study the quantitative density of the orbits of this action. There is a curious tension between the results, and many natural questions follow. This work is joint with Chris Judge.