Tessellating the Moduli Space of Strictly Convex Projective Structures on the Once-Punctured Torus

We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of marked strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Goncharov, the action of the mapping class group as well as algorithmic real algebraic geometry. We also show that the decorated moduli space of marked strictly convex real projective structures of finite volume on the thrice-punctured sphere has a natural cell decomposition.