Billiards and the Arithmetic of Non-Arithmetic Groups

What are the slopes of periodic billiard paths in a regular polygon?

We will connect this question and others to:

        - cusps of thin groups,

        - curves on Hilbert modular varieties,

        - heights from Jacobians with real multiplication, and 

        - a spectral gap for the Galois orbits of triangles.

Along the way we will encounter issues of chaos and decidability, first appearing in polygons with 7 and 12 sides.